Mackay - Multimedia Environmental Models (2 ed)

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The Fugacity Approach
Second Edition
Multimedia
Environmental
Models
 
LEWIS PUBLISHERS
Boca Raton London New York Washington, D.C.
Donald Mackay
The Fugacity Approach
Second Edition
Multimedia
Environmental
Models
 
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Preface
 
This book is about the behavior of organic chemicals in our multimedia envi-
ronment or biosphere of air, water, soil, and sediments, and the diversity of biota
that reside in these media. It is a response to the concern that we have unwisely
contaminated our environment with a large number of chemicals in the mistaken
belief that the environment’s enormous capacity to dilute and degrade will reduce
concentrations to negligible levels. We now know that the environment has only a
finite capacity to dilute and degrade. Certain chemicals have persisted and accumu-
lated to levels that have caused adverse effects on wildlife and even humans. Some
chemicals have the potential to migrate from medium to medium, reaching unex-
pected destinations in unexpectedly high concentrations. We need to understand
these processes, not only qualitatively in the form of assertions that DDT evaporates
and bioaccumulates, but quantitatively as statements that DDT in a particular region
evaporates at a rate of 100 kg per year and bioaccumulates from water at a concen-
tration of 1 ng/L to fish at levels of 1 
 
µ
 
g/g.
 
We have learned that chemical behavior in the complex assembly of environ-
mental media is not a random process like leaves blowing in the wind. The chemicals
behave in accordance with the laws of nature, which dictate chemical partitioning
and rates of transport and transformation. Most fundamentally, the chemicals are
subject to the law of conservation of mass, i.e., a mass balance exists for the chemical
that is a powerful constraint on quantities, concentrations, and fluxes. By coupling
the mass balance principle with expressions based on our understanding of the laws
of nature, we can formulate a quantitative accounting of chemical inputs and outputs.
This book is concerned with developing and applying these expressions in the form
of mathematical statements or “models” of chemical fate. These accounts or models
are invaluable summaries of chemical behavior. They can form the basis of remedial
and proactive strategies. 
Such models can confirm (or deny) that we really understand chemical fate in
the environment. Since many environmental calculations are complex and repetitive,
they are particularly suitable for implementation on computers. Accordingly, for
many of the calculations described in this book, computer programs are described
and made available on the Internet with which a variety of chemicals can be readily
assessed in a multitude of environmental situations.
The models are formulated using the concept of fugacity, which was introduced
by G.N. Lewis in 1901 as a criterion of equilibrium and has proved to be a very
convenient and elegant method of calculating multimedia equilibrium partitioning.
It has been widely and successfully used in chemical processing calculations. In this
book, we exploit it as a convenient and elegant method of explaining and deducing
the environmental fate of chemicals. Since publication of the first edition of this
book ten years ago, there has been increased acceptance of the benefits of using
fugacity to formulate models and interpret environmental fate. Multimedia fugacity
models are now routinely used for evaluating chemicals before and after production.
Much of the experience gained in these ten years is incorporated in this second
edition. Mathematical simulations of chemical fate are now more accurate, compre-
 
Front Page v Monday, January 15, 2001 1:46 PM
 
hensive, and reliable, and they have gained greater credibility as decision-support
tools. No doubt this trend will continue, especially as young environmental scientists
and engineers take over the reins of environmental science and continue to develop
new fugacity models.
This book has been written as a result of the author teaching graduate-level
courses at the University of Toronto and Trent University. It is hoped that it will be
suitable for other graduate courses and for practitioners of the environmental science
of chemical fate in government, industry, and the private consulting sector. The
simpler concepts are entirely appropriate for undergraduate courses, especially as a
means of promoting sensitivity to the concept that chemicals, which provide modern
society with so many benefits, must also be more carefully managed from their
cradle, in the chemical synthesis plant, to their grave of ultimate destruction.
At the end of most chapters is a “Concluding Example” in which a student may
be asked to apply the principles discussed in that chapter to one or more chemicals
of their choice. Necessary data are given in Table 3.5 in Chapter 3. I have found
this useful as a method of assigning different problems to a large number of students,
while allowing them to explore the properties and fate of substances of particular
interest to them.
We no longer regard the environment as a convenient, low-cost dumping ground
for unwanted chemicals. When we discharge chemicals into the environment, it must
be with a full appreciation of their ultimate fate and possible effects. We must ensure
that mistakes of the past with PCBs, mercury, and DDT are not repeated. This is
best guaranteed by building up a quantitative understanding of chemical fate in our
total multimedia environment, how chemicals will be transported and transformed,
and where, and to what extent they may accumulate. It is hoped that this book is
one step toward this goal and will be of interest and use to all those who value the
environment and seek its more enlightened stewardship.
 
Donald Mackay
 
Front Page vi Monday, January 15, 2001 1:46 PM
 
Acknowledgments
 
It is a pleasure to acknowledge the contribution of many colleagues. Much of
the credit for the approaches devised in this book is due to the pioneering work by
George Baughman, who saw most clearly the evolution of multimedia environmental
modeling as a coherent and structured branch of environmental science amid the
often frightening complexity of the environment and the formidable number of
chemicals with which it is contaminated. BrockNeely, Russ Christman, and Don
Crosby were instrumental in encouraging me to apply the fugacity concept to
environmental calculations.
I am indebted to my former colleagues at the University of Toronto, especially
Wan Ying Shiu and Sally Paterson, whose collaboration has been crucial in devel-
oping the fugacity approach. I am grateful to my more recent colleagues at Trent
University, and our industrial and government partners who have made the Canadian
Environmental Modelling Centre a successful focus for the development, validation,
and dissemination of mass balance models.
This second edition was written in part when on research leave at the Department
of Environmental Toxicology at U.C. Davis, where Marion Miller, Don Crosby, and
their colleagues were characteristically generous and supportive. At Trent, I was
greatly assisted by David Woodfine, Rajesh Seth, Merike Perem, Lynne Milford,
Angela McLeod, Adrienne Holstead, Todd Gouin, Alison Fraser, Ian Cousins, Tom
Cahill, Jenn Brimecombe, and Andreas Beyer. I am particularly grateful to Steve
Sharpe for the figures, to Matt MacLeod and Christopher Warren for their critical
review and comments, and to Eva Webster for her outstanding scientific and editorial
contributions.
Without the support and diligent typing of my wife, Ness, this book would not
have been possible. Thank you.
I dedicate this book to Ness, Neil, Ian, Julia, and Gwen, and especially to Beth,
who was born as this edition neared completion. I hope it will help to ensure that
her life is spent in a cleaner, more healthful environment.
 
Front Page vii Monday, January 15, 2001 1:46 PM
 
Front Page viii Monday, January 15, 2001 1:46 PM
 
Contents
 
Chapter 1
 
Introduction................................................................................................................1
 
Chapter 2
 
Some Basic Concepts.................................................................................................5
2.1 Introduction ...............................................................................................5
2.2 Units ..........................................................................................................5
2.3 The Environment as Compartments..........................................................9
2.4 Mass Balances.........................................................................................11
2.5 Eulerian and Lagrangian Coordinate Systems ........................................20
2.6 Steady State and Equilibrium..................................................................22
2.7 Diffusive and Nondiffusive Environmental Transport Processes...........25
2.8 Residence Times and Persistence............................................................26
2.9 Real and Evaluative Environments .........................................................27
2.10 Summary .................................................................................................28
 
Chapter 3
 
Environmental Chemicals and Their Properties ......................................................29
3.1 Introduction and Data Sources ................................................................29
3.2 Identifying Priority Chemicals................................................................31
3.3 Key Chemical Properties and Classes.....................................................40
3.4 Concluding Example...............................................................................54
 
Chapter 4
 
The Nature of Environmental Media .......................................................................55
4.1 Introduction .............................................................................................55
4.2 The Atmosphere ......................................................................................55
4.3 The Hydrosphere or Water......................................................................58
4.4 Bottom Sediments ...................................................................................60
4.5 Soils.........................................................................................................63
4.6 Summary .................................................................................................65
4.7 Concluding Example...............................................................................65
 
Chapter 5
 
Phase Equilibrium....................................................................................................69
5.1 Introduction .............................................................................................69
5.2 Properties of Pure Substances .................................................................75
5.3 Properties of Solutes in Solution.............................................................78
5.4 Partition Coefficients ..............................................................................83
5.5 Environmental Partition Coefficients and Z Values ...............................94
5.6 Multimedia Partitioning Calculations .....................................................99
5.7 Level I Calculations ..............................................................................111
5.8 Concluding Examples ...........................................................................113
 
Chapter 6
 
Advection and Reactions .......................................................................................117
6.1 Introduction ...........................................................................................117
6.2 Advection ..............................................................................................119
 
Front Page ix Monday, January 15, 2001 1:46 PM
 
6.3 Degrading Reactions .............................................................................125
6.4 Combined Advection and Reaction ......................................................131
6.5 Unsteady-State Calculations .................................................................133
6.6 The Nature of Environmental Reactions...............................................135
6.7 Level II Computer Calculations ............................................................141
6.8 Summary ...............................................................................................142
6.9 Concluding Example.............................................................................142
 
Chapter 7
 
Intermedia Transport..............................................................................................145
7.1 Introduction ...........................................................................................145
7.2 Diffusive and Nondiffusive Processes ..................................................145
7.3 Molecular Diffusion within a Phase......................................................148
7.4 Turbulent or Eddy Diffusion within a Phase ........................................153
7.5 Unsteady-State Diffusion ......................................................................156
7.6 Diffusion in Porous Media ....................................................................159
7.7 Diffusion between Phases: The Two-Resistance Concept....................161
7.8 Measuring Transport D Values .............................................................167
7.9 Combining Series and Parallel D Values ..............................................173
7.10 Level III Calculations............................................................................175
7.11 Level IV Calculations ...........................................................................181
7.12 Concluding Examples ...........................................................................183
 
Chapter 8
 
Applications of Fugacity Models...........................................................................185
8.1 Introduction, Scope, and Strategies.......................................................185
8.2 LevelI, II, and III Models.....................................................................189
8.3 An Air-Water Exchange Model ............................................................191
8.4 A Surface Soil Model............................................................................194
8.5 A Sediment-Water Exchange Model ....................................................199
8.6 QWASI Model of Chemical Fate in a Lake..........................................201
8.7 QWASI Model of Chemical Fate in Rivers ..........................................208
8.8 QWASI Multi-segment Models ............................................................210
8.9 A Fish Bioaccumulation Model ............................................................213
8.10 Sewage Treatment Plants ......................................................................220
8.11 Indoor Air Models.................................................................................221
8.12 Uptake by Plants ...................................................................................223
8.13 Pharmacokinetic Models.......................................................................224
8.14 Human Exposure to Chemicals.............................................................226
8.15 The PBT–LRT Attributes......................................................................229
8.16 Global Models.......................................................................................230
8.17 Closure ..................................................................................................232
 
Appendix
 
Fugacity Forms ......................................................................................................235
References and Bibliography ................................................................................239
Index ......................................................................................................................257
 
Front Page x Monday, January 15, 2001 1:46 PM
 
1
 
C
 
HAPTER
 
 1
Introduction
 
Since the Second World War, and especially since the publication of Rachel Carson’s
 
Silent Spring
 
 in 1962, there has been growing concern about contamination of the
environment by “man-made” chemicals. These chemicals may be present in indus-
trial and municipal effluents, in consumer or commercial products, in mine tailings,
in petroleum products, and in gaseous emissions. Some chemicals such as pesticides
may be specifically designed to kill biota present in natural or agricultural ecosys-
tems. They may be organic, inorganic, metallic, or radioactive in nature. Many are
present naturally, but usually at much lower concentrations than have been estab-
lished by human activity. Most of these chemicals cause toxic effects in organisms,
including humans, if applied in sufficiently large doses or exposures. They may
therefore be designated as “toxic substances.”
There is a common public perception and concern that when these substances
are present in air, water, or food, there is a risk of adverse effects to human health.
Assessment of this risk is difficult (a) because the exposure is usually (fortunately)
well below levels at which lethal toxic effects and even sub-lethal effects can be
measured with statistical significance against the “noise” of natural population
variation, and (b) because of the simultaneous multiple toxic influences of other
substances, some taken voluntarily and others involuntarily. There is a growing
belief that it is prudent to ensure that the functioning of natural ecosystems is
unimpaired by these chemicals, not only because ecosystems have inherent value,
but because they can act as sensing sites or early indicators of possible impact on
human well-being.
Accordingly, there has developed a branch of environmental science concerned
with describing, first qualitatively and then quantitatively, the behavior of chemicals
in the environment. This science is founded on earlier scientific studies of the
condition of the natural environment—meteorology, oceanography, limnology,
hydrology, and geomorphology and their physical, energetic, biological, and chem-
ical sub-sciences. This newer branch of environmental science has been variously
termed 
 
environmental chemistry, environmental toxicology,
 
 or 
 
chemodynamics
 
.
 
CH01 Page 1 Monday, January 15, 2001 1:48 PM
 
2 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
It is now evident that our task is to design a society in which the benefits of
chemicals are enjoyed while the risk of adverse effects from them is virtually
eliminated. To do this, we must exert effective and cost-effective controls over the
use of such chemicals, and we must have available methods of calculating their
environmental behavior in terms of concentration, persistence, reactivity, and parti-
tioning tendencies between air, water, soils, sediments, and biota. Such calculations
are useful when assessing or implementing remedial measures to treat already-
contaminated environments. They become essential as the only available method for
predicting the likely behavior of chemicals that (a) may be newly introduced into
commerce or that (b) may be subject to production increases or introduction into
new environments.
In response to this societal need, this book develops, describes, and illustrates a
framework and procedures for calculating the behavior of chemicals in the environ-
ment. It employs both conventional procedures that are based on manipulations of
concentrations and procedures that use the concepts of activity and fugacity to
characterize the equilibrium that exists between environmental phases such as air,
water, and soil. Most of the emphasis is placed on organic chemicals, which are
fortunately more susceptible to generalization than metals and other inorganic chem-
icals when assessing environmental behavior. 
The concept of fugacity, which was introduced by G.N. Lewis in 1901 as a more
convenient thermodynamic equilibrium criterion than chemical potential, has been
widely used in chemical process calculations. Its convenience in environmental
chemical equilibrium or partitioning calculations has become apparent only in the
last two decades. It transpires that fugacity is also a convenient quantity for describ-
ing mathematically the rates at which chemicals diffuse, or are transported, between
phases; for example, volatilization of pesticides from soil to air. The transfer rate
can be expressed as being driven by, or proportional to, the fugacity difference that
exists between the source and destination phases. It is also relatively easy to trans-
form chemical reaction, advective flow, and nondiffusive transport rate equations
into fugacity expressions and build up sets of fugacity equations describing the quite
complex behavior of chemicals in multiphase, nonequilibrium environments. These
equations adopt a relatively simple form, which facilitates their formulation, solution,
and interpretation to determine the dominant environmental phenomena.
We develop these mathematical procedures from a foundation of thermodynam-
ics, transport phenomena, and reaction kinetics. Examples are presented of chemical
fate assessments in both real and evaluative multimedia environments at various
levels of complexity and in more localized situations such as at the surface of a lake. 
These calculations of environmental fate can be tedious and repetitive, thus there
is an incentive to use the computer as a calculating aid. Accordingly, computer
programs are made available for many of the calculations described later in the text.
It is important that the computer be viewed and used as merely a rather fast and
smart adding machine and not as a substitute for understanding. The reader is
encouraged to write his or her own programs and modify those provided.
The author was “brought up” to write computer programs in languages such as
FORTRAN, BASIC, and C. The first edition of thisbook was regarded as very
advanced by including a diskette of programs in BASIC. Such programs have the
 
CH01 Page 2 Monday, January 15, 2001 1:48 PM
 
INTRODUCTION 3
 
immense benefit that the sequence and details of calculations are totally transparent.
Executable versions can be run on any computer. Unfortunately, it is not always
easy to change input parameters or equations, and the output is usually printed tables.
The modern trend is to use spreadsheets, such as Microsoft EXCEL
 
®
 
, which have
improved input and output features, including the ability to draw graphs and charts.
Spreadsheets have the disadvantages that calculations are less transparent, there may
be problems when changing versions of the spreadsheet program, and not everyone
has the same spreadsheet.
Sufficient information is given on each mass balance model that readers can
write their own programs using the system of their choice. Microsoft Windows
 
®
 
software for performing model calculations is available from the Internet site
www.trentu.ca/envmodel. Older DOS-based models are also available. They are
updated regularly and are subject to revision. In all cases, the equations correspond
closely to those in this book (unless otherwise stated), and they are totally transpar-
ent. Some are used in a regulatory context, thus the user is prevented from changing
the coding, although all code can be viewed. 
Preparing a second edition of this book has enabled me to update, expand, and
reorganize much of the material presented in the first (1991) edition. I have benefited
greatly from the efforts of those who have sought to understand environmental
phenomena and who have applied the fugacity approach when deducing the fate of
chemicals in the environment. There is no doubt that, as we enter the new millen-
nium, environmental science is becoming more quantitative. It is my hope that this
book will contribute to that trend. 
 
CH01 Page 3 Monday, January 15, 2001 1:48 PM
 
CH01 Page 4 Monday, January 15, 2001 1:48 PM
 
5
 
CHAPTER
 
 2
Some Basic Concepts
 
2.1 INTRODUCTION
 
Much of the scientific fascination with the environment lies in its incredible
complexity. It consists of a large number of phases such as air, soil, and water, which
vary in properties and composition from place to place (spatially) and with time
(temporally). It is very difficult to assemble a complete, detailed description of the
condition (temperature, pressure, and composition) of even a small environmental
system or microcosm consisting, for example, of a pond with sediment below and
air above. It is thus necessary to make numerous simplifying assumptions or state-
ments about the condition of the environment. For example, we may assume that a
phase is homogeneous, or it may be in equilibrium with another phase, or it may
be unchanging with time. The art of successful environmental modeling lies in the
selection of the best, or “least-worst,” set of assumptions that yields a model that is
not so complex as to be excessively difficult to understand yet is sufficiently detailed
to be useful and faithful to reality. The excessively simple model may be misleading.
The excessively detailed model is unlikely to be useful, trusted, or even understand-
able. The aim is to suppress the less necessary detail in favor of the important
processes that control chemical fate.
In this chapter, several concepts are introduced that are used when we seek to
compile quantitative descriptions of chemical behavior in the environment. But first,
it is essential to define the system of units and dimensions that forms the foundation
of all calculations.
 
2.2 UNITS
 
The introduction of the “SI” or “Système International d’Unités” or International
System of Units in 1960 has greatly simplified scientific calculations and communi-
cation. With few exceptions, we adopt the SI system. The system is particularly
convenient, because it is “coherent” in that the basic units combine one-to-one to
 
CH02 Page 5 Monday, January 15, 2001 1:47 PM
 
6 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
give the derived units directly with no conversion factors. For example, energy
(joules) is variously the product of force (newtons) and distance (metres), or pressure
(pascals) and volume (cubic metres), or power (watts) and time (seconds). Thus, the
foot-pound, the litre-atmosphere, and the kilowatt-hour become obsolete in favor of
the single joule. Some key aspects of the SI system are discussed below. Conversion
tables from obsolete or obsolescent unit systems are available in scientific handbooks.
 
Length
 
 (metre, m)
 
This base unit is defined as the specified number of wavelengths of a krypton
light emission.
 
Area
 
Square metre (m
 
2
 
). Occasionally, the hectare (ha) (an area 100 
 
×
 
 100 m or 10
 
4
 
 m
 
2
 
)
or the square kilometre (km
 
2
 
) is used. For example, pesticide dosages to soils are
often given in kg/ha.
 
Volume
 
 (cubic metre, m
 
3
 
)
 
The litre (L) (0.001 m
 
3
 
) is also used because of its convenience in analysis, but
it should be avoided in environmental calculations. In the United States, the spellings
“meter” and “liter” are often used.
 
Mass
 
 (kilogram, kg)
 
Kilogram (kg). The base unit is the kilogram (kg), but it is often more convenient
to use the gram (g), especially for concentrations. For large masses, the megagram
(Mg) or the equivalent metric tonne (t) may be used.
 
Amount
 
 (mole abbreviated to mol)
 
This unit, which is of fundamental importance in environmental chemistry, is
really a number of constituent entities or particles such as atoms, ions, or molecules.
It is the actual number of particles divided by Avogadro’s number (6.0 
 
×
 
 10
 
23
 
),
which is defined as the number of atoms in 12 g of the carbon-12 isotope. When
reactions occur, the amounts of substances reacting and forming are best expressed
in moles rather than mass, since atoms or molecules combine in simple stoichio-
metric ratios. The need to involve atomic or molecular masses is thus avoided.
 
Molar Mass or Molecular Mass (or Weight)
 
 (g/mol)
 
This is the mass of 1 mole of matter and is sometimes (wrongly) referred to as
molecular weight or molecular mass. Strictly, the correct unit is kg/mol, but it is
often more convenient to use g/mol, which is obtained by adding the atomic masses
(weights). Benzene (C
 
6
 
H
 
6
 
) is thus approximately 78 g/mol or 0.078 kg/mol.
 
CH02 Page 6 Monday, January 15, 2001 1:47 PM
 
SOME BASIC CONCEPTS 7
 
Time 
 
(second or hour, s or h)
 
The standard unit of a second (s) is inconveniently short when considering
environmental processes such as flows in large lakes when residence times may be
many years. The use of hours, days, and years is thus acceptable. We generally use
hours as a compromise.
 
Concentration
 
The preferred unit is the mole per cubic metre (mol/m
 
3
 
) or the gram per cubic
metre (g/m
 
3
 
). Most analytical data are reported in amount or mass per litre (L),
because a litre is a convenient volume for the analytical chemist to handle and
measure. Complications arise if the litre is used in environmental calculations,
because it is not coherent with area or length. The common mg/L, which is often
ambiguously termed the “part per million,” is equivalent to g/m
 
3
 
. In some circum-
stances, the use of mass fraction, volume fraction, or mole fraction as concentrations
is desirable.
It is acceptable, and common, to report concentrations in units such as mol/L or
mg/L but, prior to any calculation, they should be converted to a coherent unit of
amount of substance per cubic metre.
Concentrations such as parts per thousand (ppt), parts per million (ppm), parts
per billion (ppb), and parts per trillion (also ppt) should 
 
not
 
 be used. There can be
confusion between parts per thousand and per trillion. The billionis 10
 
9
 
 in North
America and 10
 
12
 
 in Europe. The air ppm is usually on a volume/volume basis,
whereas the water ppm is usually on a mass/volume basis. The mixing ratio used
for air is the ratio of numbers of molecules or volumes and is often given in ppm.
Concentrations must be presented with no possible ambiguity.
 
Density
 
 (kg/m
 
3
 
)
 
This has identical units to mass concentrations, but the use of kg/m
 
3
 
 is preferred,
water having a density of 1000 kg/m
 
3
 
 and air a density of approximately 1.2 kg/m
 
3
 
.
 
Force
 
 (newton, N)
 
The newton is the force that causes a mass of 1 kg to accelerate at 1 m/s
 
2
 
. It is
10
 
5
 
 dynes and is approximately the gravitational force operating on a mass of 102 g
at the Earth’s surface.
 
Pressure
 
 (pascal, Pa)
 
The pascal or newton per square metre (N/m
 
2
 
) is inconveniently small, since it
corresponds to only 102 grams force over one square metre, but it is the standard
unit, and it is used here. The atmosphere (atm) is 101325 Pa or 101.325 kPa. The
torr or mm of mercury (mmHg) is 133 Pa and, although still widely used, should
be regarded as obsolescent.
 
CH02 Page 7 Monday, January 15, 2001 1:47 PM
 
8 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Energy
 
 (joule, J)
 
The joule, which is one N-m or Pa-m
 
3
 
, is also a small quantity. It replaces the
obsolete units of calorie (which is 4.184 J) and Btu (1055 J).
 
Temperature
 
 (K)
 
The kelvin is preferred, although environmental temperatures may be expressed
in degrees Celsius, °C, and not centigrade, where 0°C is 273.15 K. There is no
degree symbol prior to K.
 
Frequency
 
 (hertz, Hz)
 
The hertz is one event per second (s
 
–1
 
). It is used in descriptions of acoustic and
electromagnetic waves, stirring, and in nuclear decay processes where the quantity
of a radioactive material may be described in becquerels (Bq), where 1 Bq corre-
sponds to the amount that has a disintegration rate of 1 Hz. The curie (Ci), which
corresponds to 3.7 
 
×
 
 10
 
10
 
 disintegrations per second (and thus 3.7 
 
×
 
 10
 
10
 
 Bq), was
formerly used.
 
Gas Constant
 
 (R)
 
This constant, which derives from the gas law, is 8.314 J/mol K or Pa-m
 
3
 
/mol K.
An advantage of the SI system is that R values in diverse units such as cal/mol K
and cm
 
3
 
·atm/mol K become obsolete and a single universal value now applies.
 
2.2.1 Prefices
 
The following prefices are used:
Note that these prefices precede the unit. It is inadvisable to include more than one
prefix in a unit, e.g., ng/mg, although mg/kg may be acceptable, because the base
unit of mass is the kg. The equivalent µg/g is clearer. The use of expressions such
as an aerial pesticide spray rate of 900 g/km
 
2
 
 can be ambiguous, since a kilo(metre
 
2
 
)
is not equal to a square kilometre, i.e., a (km)
 
2
 
. The former style is not permissible.
 
Factor Prefix Factor Prefix
 
10
 
1
 
deka da 10
 
–1
 
deci d
10
 
2
 
hecto h 10
 
–2
 
centi c
10
 
3
 
kilo k 10
 
–3
 
milli m
10
 
6
 
mega M 10
 
–6
 
micro 
 
µ
 
10
 
9
 
giga G 10
 
–9
 
nano n
 10
 
12
 
tera T 10
 
–12
 
pico p
 10
 
15
 
peta P 10
 
–15
 
femto f
 10
 
18
 
exa E 10
 
–18
 
atto a
 
CH02 Page 8 Monday, January 15, 2001 1:47 PM
 
SOME BASIC CONCEPTS 9
 
Expressing the rate as 9 g/ha or 0.9 mg/m
 
2
 
 removes all ambiguity. The prefices deka,
hecto, deci, and centi are restricted to lengths, areas, and volumes. A common (and
disastrous) mistake is to confuse milli, micro, and nano.
We use the convention J/mol-K meaning J mol
 
–1
 
 K
 
–1
 
. Strictly, J/(mol-K) is correct
but, in the interests of brevity, the parentheses are omitted.
 
2.2.2 Dimensional Strategy and Consistency
 
When undertaking calculations of environmental fate, it is highly desirable to
adopt the practice of first converting all the supplied input data, in its diversity of
units, into the SI units described above and eliminate the prefices, e.g., 10 kPa should
become 10
 
4
 
 Pa. Calculations should be done using only these SI units. If necessary,
the final results can then be converted to other units for the convenience of the user.
When assembling quantities in expressions or equations, it is critically important
that the dimensions be correct and consistent. It is always advisable to write down
the units on each side of the equation, cancel where appropriate, and check that
terms that add or subtract have identical units. For example, a lake may have an
inflow or reaction rate of a chemical expressed as follows:
A flow rate: 
(water flow rate G m
 
3
 
/h) 
 
×
 
 (concentration C g/m
 
3
 
) = GC g/h
A reaction rate:
(volume V m
 
3
 
) 
 
×
 
 (rate constant k h
 
–1
 
) 
 
×
 
 (concentration C mol/m
 
3
 
) = VkC mol/h
Obviously, it is erroneous to express the above concentration in mol/L or the
volume in cm
 
3
 
. When checking units it may be necessary to allow for changes in
the prefices (e.g. kg to g), and for unit conversions (e.g., h to s).
 
2.2.3 Logarithms
 
The preferred logarithmic quantity is the natural logarithm to the base e or
2.7183, designated as ln. Base 10 logarithms are still used for certain quantities such
as the octanol-water partition coefficient and for plotting on log-log or log-linear
graph paper. The natural antilog or exponential of x is written either e
 
x
 
 or exp(x).
The base 10 log of a quantity is the natural log divided by 2.303 or ln10.
 
2.3 THE ENVIRONMENT AS COMPARTMENTS
 
It is useful to view the environment as consisting of a number of connected
 
phases
 
 or 
 
compartments
 
. Examples are the atmosphere, terrestrial soil, a lake, the
bottom sediment under the lake, suspended sediment in the lake, and biota in soil
or water. The phase may be continuous (e.g., water) or consist of a number of
particles that are not in contact, but all of which reside in one phase [e.g., atmospheric
 
CH02 Page 9 Monday, January 15, 2001 1:47 PM
 
10 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
particles (aerosols), or biota in water]. In some cases, the phases may be similar
chemically but different physically, e.g., the troposphere or lower atmosphere, and
the stratosphere or upper atmosphere. It may be convenient to lump all biota together
as one phase or consider them as two or more classes each with a separate phase.
Some compartments are in contact, thus a chemical may migrate between them (e.g.,
air and water), while others are not in contact, thus direct transfer is impossible (e.g.,
air and bottom sediment). Some phases are accessible in a short time to migrating
chemicals (e.g., surface waters), but others are only accessible slowly (e.g., deep
lake or ocean waters), or effectively not at all (e.g., deep soil or rock).
Some confusion is possible when expressing concentrations for mixed phases
such as water containing suspended solids (SS). An analysis may give a total or bulk
concentration expressed as amount of chemical per m
 
3
 
 of mixed water and particles.
Alternatively, the water may be filtered to give the concentration or amount of
chemical that is dissolved in water per m
 
3
 
 of water. The difference between these
is the amount of chemical in the SS phase per m
 
3
 
 of water. This is different from
the concentration in the SS phase expressed as amount of chemical per m
 
3
 
 of particle.
Concentrations in soils, sediments, and biota can be expressed on a dry or wet weight
basis. Occasionally, concentrations in biota are expressed on a lipid or fat content
basis. Concentrations must be expressed unambiguously. 
 
2.3.1 Homogeneity and Heterogeneity
 
A key modeling concept is that of phase homogeneity and heterogeneity. Well
mixed phases such as shallow pondwaters tend to be homogeneous, and gradients
in chemical concentration or temperature are negligible. Poorly mixed phases such
as soils and bottom sediments are usually heterogeneous, and concentrations vary
with depth. Situations in which chemical concentrations are heterogeneous are
difficult to describe mathematically, thus there is a compelling incentive to assume
homogeneity wherever possible. A sediment in which a chemical is present at a
concentration of 1 g/m
 
3
 
 at the surface, dropping linearly to zero at a depth of 10 cm,
can be described approximately as a well mixed phase with a concentration of 1 g/m
 
3
 
and 5 cm deep, or 0.5 g/m
 
3
 
 and 10 cm deep. In all three cases, the amount of chemical
present is the same, namely 0.05 g per square metre of sediment horizontal area.
Even if a phase is not homogeneous, it may be nearly homogeneous in one or
two of the three dimensions. For example, lakes may be well mixed horizontally
but not vertically, thus it is possible to describe concentrations as varying only in
one dimension (the vertical). A wide, shallow river may be well mixed vertically
but not horizontally in the cross-flow or down-flow directions.
 
2.3.2 Steady- and Unsteady-State Conditions
 
If conditions change relatively slowly with time, there is an incentive to assume
“steady-state” behavior, i.e., that properties are independent of time. A severe math-
ematical penalty is incurred when time dependence has to be characterized, and
“unsteady-state,” dynamic, or time-varying conditions apply. We discuss this issue
in more detail later.
 
CH02 Page 10 Monday, January 15, 2001 1:47 PM
 
SOME BASIC CONCEPTS 11
 
2.3.3 Summary
 
In summary, our simplest view of the environment is that of a small number of
phases, each of which is homogeneous or well mixed and unchanging with time.
When this is inadequate, the number of phases may be increased; heterogeneity may
be permitted in one, two, or three dimensions; and variation with time may be
included. The modeler’s philosophy should be to concede each increase in complex-
ity reluctantly, and only when necessary. Each concession results in more mathe-
matical complexity and the need for more data in the form of kinetic or equilibrium
parameters. The model becomes more difficult to understand and thus less likely to
be used, especially by others. This is not a new idea. William of Occam expressed
the same sentiment about 650 years ago, when he formulated his principle of
parsimony or “Occam’s Razor,” stating
 
Essentia non sunt multiplicanda praeter necessitatem
 
which can be translated as, “What can be done with fewer (assumptions) is done in
vain with more,” or more colloquially, “Don’t make models more complicated than
is necessary.”
 
2.4 MASS BALANCES
 
When describing a volume of the environment, it is obviously essential to define
its limits in space. This may simply be the boundaries of water in a pond or the air
over a city to a height of 1000 m. The volume is presumably defined exactly, as are
the areas in contact with adjoining phases. Having established this control “envelope”
or “volume” or “parcel,” we can write equations describing the processes by which
a mass of chemical enters and leaves this envelope.
The fundamental and now axiomatic law of conservation of mass, which was
first stated clearly by Antoine Lavoisier, provides the basis for all mass balance
equations. Rarely do we encounter situations in which nuclear processes violate this
law. Mass balance equations are so important as foundations of all environmental
calculations that it is essential to define them unambiguously. Three types can be
formulated and are illustrated below. We do not treat energy balances, but they are
set up similarly.
 
2.4.1 Closed System, Steady-State Equations
 
This is the simplest class of equation. It describes how a given mass of chemical
will partition between various phases of fixed volume. The basic equation simply
expresses the obvious statement that the total amount of chemical present equals the
sum of the amounts in each phase, each of these amounts usually being a product of
a concentration and a volume. The system is closed or “sealed” in that no entry or
exit of chemical is permitted. In environmental calculations, the concentrations are
usually so low that the presence of the chemical does not affect the phase volumes.
 
CH02 Page 11 Monday, January 15, 2001 1:47 PM
 
12 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Worked Example 2.1
 
A three-phase system consists of air (100 m
 
3
 
), water (60 m
 
3
 
), and sediment (3
m
 
3
 
). To this is added 2 mol of a hydrocarbon such as benzene. The phase volumes
are not affected by this addition, because the volume of hydrocarbon is small.
Subscripting air, water, and sediment symbols with A, W, and S, respectively, and
designating volume as V (m
 
3
 
) and concentration as C (mol/m3), we can write the
mass balance equation.
total amount = sum of amounts in each phase mol
2 = VACA + VWCW + VSCS = 100 CA + 60 CW + 3 CS mol
To proceed further, we must have information about the relationships between CA,
CW, and CS. This could take the form of phase equilibrium equations such as
CA/CW = 0.4 and CS/CW = 100
These ratios are usually referred to as partition coefficients or distribution coef-
ficients and are designated KAW and KSW, respectively. We discuss them in more
detail later.
We can now eliminate CA and CS by substitution to give
2 = 100 (0.4 CW) + 60 CW + 3(100CW) = 400 CW mol
Thus,
CW = 2/400 = 0.005 mol/m3
It follows that
CA = 0.4 CW = 0.002 mol/m3
CS = 100 CW = 0.5 mol/m3
The amounts in each phase (mi) mol are the VC products as follows:
This simple algebraic procedure has established the concentrations and amounts in
each phase using a closed system, steady-state, mass balance equation and equilib-
rium relationships. The essential concept is that the total amount of chemical present
mW = VWCW = 0.30 mol (15%)
mA = VACA = 0.20 mol (10%)
mS = VSCS = 1.50 mol (75%)
Total 2.00 mol
CH02 Page 12 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 13
must equal the sum of the individual amounts in each compartment. We later refer
to this as a Level I calculation. It is useful because it is not always obvious where
concentrations are high, as distinct from amounts.
Example 2.2
In this example, 0.04 mol of a pesticide of molar mass 200 g/mol is applied to
a closed system consisting of 20 m3 of water, 90 m3 of air, 1 m3 of sediment, and
2 L of biota (fish). If the concentration ratios are air/water 0.1, sediment/water 50,
and biota/water 500, what are the concentrations and amounts in each phase in both
gram and mole units?
Answer
The fish contains 0.1 g or 0.0005 mol at a concentration of 50 g/m3 or 0.25
mol/m3.
Example 2.3
A circular lake of diameter 2 km and depth 10 m contains suspended solids (SS)
with a volume fraction of 10–5, i.e., 1 m3 of SS per 105 m3 water, and biota (such as
fish) at a concentration of 1 mg/L. Assuming a density of biota of 1.0 g/cm3, a
SS/water partition coefficient of 104, and a biota/water partition coefficient of 105.
Calculate the disposition and concentrations of 1.5 kg of a PCB in this system.
Answer
In this case, 8.3% is present in each of SS and biota and 83% in water with a
concentration in water of 39.8 µg/m3.
2.4.2 Open System, Steady-State Equations
In this class of mass balance equation, we introduce the possibility of the
chemical flowing into and out of the system and possibly reacting or being formed.
The conditions within the system do not change with time, i.e., its condition looks
the same now as in the past and in the future. The basic mass balance assertion is
that the total rate of input equals the total rate of output, these rates being expressed
in moles or grams per unit time. Whereas the basic unit in the closed system balance
was mol or g, it is now mol/h or g/h.
Worked Example 2.4
A 104 m3 thoroughlymixed pond has a water inflow and outflow of 5 m3/h. The
inflow water contains 0.01 mol/m3 of chemical. Chemical is also discharged directly
into the pond at a rate of 0.1 mol/h. There is no reaction, volatilization, or other
losses of the chemical; it all leaves in the outflow water. 
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14 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
(i) What is the concentration (C) in the outflow water? We designate this as an
unknown C mol/m3.
total input rate = total output rate
5 m3/h × 0.01 mol/m3 + 0.1 mol/h = 0.15 mol/h = 5 m3/h × C mol/m3 = 5 C mol/h
Thus,
C = 0.03 mol/m3
The total inflow and outflow rates of chemical are 0.15 mol/h.
(ii) If the chemical also reacts in a first-order manner such that the rate is
VCk mol/h where V is the water volume, C is the chemical concentration in the well
mixed water of the pond, and k is a first-order rate constant of 10–3 h–1, what will
be the new concentration?
The output by reaction is VCk or 104 × 10–3 C or 10 C mol/h, thus we rewrite
the equation as:
0.05 + 0.1 = 5 C + 10 C = 15 C mol/h
Thus,
C = 0.01 mol/m3
The total input of 0.15 mol/h is thus equal to the total output of 0.15 mol/h, consisting
of 0.05 mol/h outflow and 0.10 mol/h reaction.
An inherent assumption is that the prevailing concentration in the pond is con-
stant and equal to the outflow concentration. This is the “well mixed” or “continu-
ously stirred tank” assumption. It may not always apply, but it greatly simplifies
calculations when it does.
The key step is to equate the sum of the input rates to the sum of the output
rates, ensuring that the units are equivalent in all the terms. This often requires some
unit-to-unit conversions.
Worked Example 2.5
A lake of area (A) 106 m2 and depth 10 m (volume V 107 m3) receives an input
of 400 mol/day of chemical in an effluent discharge. Chemical is also present in the
inflow water of 104 m3/day at a concentration of 0.01 mol/m3. The chemical reacts
with a first-order rate constant k of 10–3 h–1, and it volatilizes at a rate of (10–5 C)
mol/m2s, where C is the water concentration and m2 refers to the air-water area. The
outflow is 8000 m3/day, there being some loss of water by evaporation. Assuming
that the lake water is well mixed, calculate the concentration and all the inputs and
outputs in mol/day. Use a time unit of days in this case.
CH02 Page 14 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 15
Discharge = 400 mol/day
Inflow = 104 m3/day × 0.01 mol/m3 = 100 mol/day
Total input = 500 mol/day
Reaction rate = VCk = 107 m3 × C mol/m3 × 10–3 h–1 × 24 h/day = 24 × 104C
mol/day
Volatilization rate = 106 m2 × 10–5 C mol/m2 s × 3600 s/h × 24 h/day = 86.4 ×
104C mol/day
Outflow = 8000 m3/day × C mol/m3 = 0.8 × 104C mol/day
Thus, 
500 = 24 × 104C + 86.4 × 104C + 0.8 × 104C = 111.2 × 104C
C = 4.5 × 10–4 mol/m3
Reaction rate = 107.9 mol/day (i.e., 108 mol/day)
Volatilization rate = 388.5 mol/day (i.e., 390 mol/day)
Outflow = 3.6 mol/day
Total rate of loss = 500 mol/day = input rate
Until proficiency is gained in manipulating these multi-unit equations, it is best
to write out all quantities and units and check that the units are consistent. Judgement
should be exercised when selecting the number of significant figures to be carried
through the calculation. It is preferable to carry more than is needed, then go back
and truncate. Remember that environmental quantities are rarely known with better
than 5% accuracy. Avoid conveying an erroneous impression of accuracy by using
too many significant figures.
Example 2.6
A building, 20 m wide × 25 m long × 5 m high is ventilated at a rate of 200 m3/h.
The inflow air contains CO2 at a concentration of 0.6 g/m3. There is an internal
source of CO2 in the building of 500 g/h. What is the mass of CO2 in the building
and the exit CO2 concentration?
Answer
7.75 kg and 3.1 g/m3
Example 2.7
A pesticide is applied to a 10 ha field at an average rate of 1 kg/ha every 4
weeks. The soil is regarded as being 20 cm deep and well mixed. The pesticide
evaporates at a rate of 2% of the amount present per day, and it degrades micro-
bially with a rate constant of 0.05 days–1. What is the average standing mass of
pesticide present at steady state? What will be the steady-state average concentra-
tion of pesticide (g/m3), and in units of µg/g assuming a soil solids density of
2500 kg/m3? 
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16 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Answer
5.1 kg, 0.255 g/m3, 0.102 µg/g
In all these examples, chemical is flowing or reacting, but observed conditions
in the envelope are not changing with time, thus the steady-state condition applies.
In Example 2.7, the concentration will change in a “sawtooth” manner but, over the
long term, it is constant.
2.4.3 Unsteady-State Equations
Whereas the first two types of mass balances lead to simple algebraic equations,
unsteady-state conditions give differential equations in time. The simplest method
of setting up the equation is to write
d(contents)/dt = total input rate – total output rate 
The input and output rates should be in units of amount/time, e.g., mol/h or g/h.
The “contents” must be in consistent units, e.g., in mol or g, and dt, the time
increment, in units consistent with the time unit in the input and output terms, (e.g.,
h). The differential equation can then be solved along with an appropriate initial or
boundary condition to give an algebraic expression for concentration as a function
of time. The simplest example is the first-order decay equation.
Worked Example 2.8
A lake of 106 m3 with no inflow or outflow is treated with 10 mol of piscicide
(a chemical that kills fish), which has a first-order reaction (degradation or decay)
rate constant k of 10–2 h–1. What will the concentration be after 1 and 10 days,
assuming no further input, and when will half the chemical have been degraded? 
The contents are VC or 106C mol. The output is only by reaction at a rate of
VCk or 106 × 10–2C or 104C mol/h. There is zero input, thus,
d (106C)/dt = 106dC/dt = 0 – 104 C mol/h
Thus,
dC/dt = –10–2C mol/h
This differential equation is easily solved by separating the variables C and t to give
dC/C = –10–2 dt
Integrating gives
lnC = –10–2t + IC
CH02 Page 16 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 17
where IC is an integration constant that is usually evaluated from an initial condition,
i.e., C = Co when t = 0; thus, IC is lnCo and
ln(C/Co) = –10–2 t
or
C = Co exp (–10–2t)
Now, Co is (10 mol)/106m3 or 10–5 mol/m3 
Thus,
C = 10–5 exp (–10–2t) mol/m3
After 1 day (24 h), 
C will be 0.79 × 10–5 mol/m3, i.e., 79% remains
After 10 days (240 h),
C will be 0.091 × 10–5 mol/m3, i.e., 9.1% remains
Half the chemical will have degraded when
C/Co is 0.5; or 10–2 t is –ln 0.5 or 0.693; or t is 69.3 h
Note that the half-time t is 0.693/k.
This relationship, that the half-time is 0.693 divided by the rate constant, is very
important and is used extensively. It is also possible to have inflow and outflow as
well as reaction, as shown in the next example.
Worked Example 2.9
A well mixed lake of volume V 106 m3 containing no chemical starts to receive
an inflow of 10 m3/s containing chemical at a concentration of 0.2 mol/m3. The
chemical reacts with a first-order rate constant of 10–2 h–1, and it also leaves with
the outflow of 10 m3/s. By “first-order,” we specify that the rate is proportional to
C raised to the power one. What will be the concentration of chemical in the lake
one day after the start of the input of chemical?
Input rate = 10 × 0.2 = 2 mol/s (we choose a time unit of seconds here)
Output by reaction = (106 m3)(10–2 h–1)(1 h/3600s)C mol/m3 = 2.78 C mol/s
Output by flow = 10 C mol/s
Thus,
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18 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Input – Output = d(contents)/dt
2 – 2.78C – 10C = d(106 C)/dt
or
dC/(2 – 12.78C) = 10–6 dt
or
ln(2 – 12.78C)/(–12.78)= 10–6 t + IC
When t is zero, C is zero, thus,
IC = –ln(2)/12.78
and
ln[(2 – 12.78C)/2] = –12.78 × 10–6 t
or
(2 – 12.78 C) = 2 exp(–12.78 × 10–6 t)
or
C = (2/12.78)[1 – exp(–12.78 × 10–6 t)]
Note that when t is zero, exp(0) is unity and C is zero, as dictated by the initial
condition. When t is very large, the exponential group becomes zero, and C
approaches (2/12.78) or 0.157 mol/m3. At such times, the input of 2 mol/s is equal
to the total of the output by flow of 10 × 0.157 or 1.57 mol/s plus the output by
reaction of 2.78 × 0.157 or 0.44 mol/s. This is the steady-state solution, which the
lake eventually approaches after a long period of time.
When t is 1 day or 86400s, C will be 0.105 mol/m3 or 67% of the way to its
final value. C will be halfway to its final value when 12.78 × 10–6 t is 0.693 or t is
54200 s or 15 h. This time is largely controlled by the residence time of the water
in the lake, which is
(106 m3)/(10 m3/s) or 105 s or 27.8 h
Worked Example 2.10
A well mixed lake of 105 m3 is initially contaminated with chemical at a con-
centration of 1 mol/m3. The chemical leaves by the outflow of 0.5 m3/s, and it reacts
with a rate constant of 10–2 h–1. What will be the chemical concentration after 1 and
10 days, and when will 90% of the chemical have left the lake?
CH02 Page 18 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 19
Input = 0
Output by flow = 0.5C
Output by reaction = VCk = 105 · C · 10–2h–1(1/3600) = 0.278C
Thus,
0 – 0.5C – 0.278C = 105dC/dt
dC/C = –0.778 × 10–5dt
C = Co exp(–0.778 × 10–5 t)
Since CO is 1.0 mol/m3, after 1 day or 864000 s, C will be 0.51 mol/m3.
t = 10 days = 86400s; C = 0.0012 mol/m3
C = 0.1 when 0.778 × 10–5 t = –ln 0.1 or 2.3 or when t is 296000 s or 3.4 days
Example 2.11
If the concentration of CO2 in Example 2.6 has reached steady state of 3.1 g/m3,
and then the internal source is reduced to 100 g/h, deduce the equation expressing
the time course of CO2 concentration decay and the new steady-state value.
Answer 
New steady-state 1.1 g/m3 and C = 1.1 + 2.0 exp(–0.08 t)
Example 2.12
A lake of volume 106 m3 has an outflow of 500 m3/h. It is to be treated with a
piscicide, the concentration of which must be kept above 1 mg/m3. It is decided to
add 3 kg, thus achieving a concentration of 3 mg/m3, and to allow the concentration
to decay to 1 mg/m3 before adding another 2 kg to bring the concentration back to
3 mg/m3. If the piscicide has a degradation half-life of 693 hours (29 days), what
will be the interval before the second (and subsequent) applications are required?
Answer
30 days
Mr. MacLeod, being economically and ecologically perceptive, claims that if he is
allowed to make applications every 10 days instead of 30 days, he can maintain the
concentration above 1 mg/m3 but reduce the piscicide usage by 35%. Is he correct?
Answer 
Yes
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20 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
What is the absolute minimum piscicide usage every 30 days to maintain 1 mg/m3?
Answer 
A total of 1.08 kg added continuously over a 30 day period
These unsteady-state solutions usually contain exponential terms such as
exp(–kt). The term k is a characteristic rate constant with units of reciprocal time.
It is thus somewhat difficult to grasp and remember. A quantity of 0.01 h–1 does not
convey an impression of rapidity. It is convenient to calculate its reciprocal 1/k or
100 h, which is a characteristic time. This is the time required for the process to
move exp(–1) or to within 37% of the final value, i.e., it is 63% completed. Those
working with radioisotopes prefer to use half-lives rather than k, i.e., the time for
half completion. This occurs when the term exp(–kt) is 0.5 or kt is ln2 or 0.693,
thus the half-time τ is 0.693/k. Another useful time is the 90% completion value,
which is 2.303/k.
Two common mistakes are made if rate constants are manipulated as times rather
than frequencies. A rate constant of 1 day–1 is 0.042 h–1, not 24 h–1—a common
mistake. If there are two first-order reactions, the total rate constant is the sum of
the individual rate constants. This has the effect of giving a total half-time or half-
life that is less than either individual half-time. It is a disastrous mistake to add half-
lives. Their reciprocals add.
In some cases, the differential equation can become quite complex, and there
may be several of them applying simultaneously. Setting up these equations requires
practice and care. There is a common misconception that solving the equations is
the difficult task. On the contrary, it is setting them up that is most difficult and
requires the most skill. If the equation is difficult to solve, tables of integrals can
be consulted, computer programs such as Mathematica or Matlabs can be used, or
an obliging mathematician can be sought. For many differential equations, an ana-
lytical solution is not feasible, and numerical methods must be used to generate a
solution. We discuss techniques for doing this later.
2.5 EULERIAN AND LAGRANGIAN COORDINATE SYSTEMS
It is usually best to define the mass balance envelope as being fixed in space.
This can be called the Eulerian coordinate system. When there is appreciable flow
through the envelope, it may be better to define the envelope as being around a certain
amount of material and allow that envelope of material to change position. This “fix
a parcel of material then follow it in time as it moves” approach is often applied to
rivers when we wish to examine the changing condition of a volume of water as it
flows downstream and undergoes various reactions. This can be called the Lagrangian
coordinate system. It is also applied to “parcels” of air emitted from a stack and
subject to wind drift. Both systems must give the same results, but it may be easier
to write the equations in one system than the other. The following example is an
illustration. It also demonstrates the need to convert units to the SI system.
CH02 Page 20 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 21
Worked Example 2.13
Consider a river into which the 1.8 million population of a city discharges a
detergent at a rate of 1 pound per capita per year, i.e., the discharge is 1.8 million
pounds per year. The aim is to calculate the concentrations at distances of 1 and 10
miles downstream from a knowledge of the degradation rate of the detergent and
the constant downstream flow conditions, which are given below. This can be done
in Eulerian or Lagrangian coordinate systems. The input data are first converted to
SI units.
The river flow rate is Uhw, i.e., 18270 m3/h. The rate constant k is 0.693/τ1/2, i.e.
0.096 h–1. When the detergent mixes into the river, the concentration will be CO or
E/(Uhw) or 5.1 g/m3.
Lagrangian Solution
A parcel of water that maintains its integrity, i.e., it does not diffuse or disperse,
will decay according to the equation
C = CO exp(–kt)
where t is the time from discharge. At 1 mile (1609 m), the time t will be 1609/U
or 1.47 h, and at 10 miles, it will be 14.7 h.
Substituting shows that, after 1 and 10 miles, the concentrations will be 4.4 and
1.24 g/m3. The chemical will reach half its input concentration when t is 0.693/k or
7.2 h, which corresponds to 7900 m or 4.92 miles. This Lagrangian solution is
straightforward, but it is valid only if conditions in the river remain constant and
negligible upstream-downstream dispersion occurs.
Eulerian Solution
We now simulate the river as a series of connected reaches or segments or well
mixed lakes, each being L or 200 m long. Each reach thus has a volume V of Lhw
or 3330 m3. A steady-state mass balance on the first reach gives
input rate = UhwCO = output rate = UhwC1 + VkC1
where CO and C1 are the input and output concentrations. C1 is also the concentration
in the segment. It follows that
Discharge rate 1.8 × 106 lb per year 93300 g/h (E)
River flow velocity 1 ft/s 1097 m/h (U)
River depth 3 ft 0.91 m (h)
River width 20 yards 18.3m (w)
Degradation half-life 0.3 days 7.2 h (τ1/2)
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22 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
C1 = CO/(1 + Vk/(Uhw)) = CO/(1 + kL/U)
Note the consistency of the dimensions, kL/U being dimensionless. The group (1 +
kL/U) has a value of 1.0175, thus C1 is 0.983CO. 1.7% of the chemical is lost in
each segment. The same equation applies to the second reach, thus C2 is 0.983C1
or 0.9832CO. In general, for the nth reach, Cn is (0.983)nCO or CO/(1 + kL/U)n.
One mile is reached when n is 8, and 10 miles corresponds to n of 80, thus C8
is 0.9838CO or 4.45, and C8 is 1.29. The half distance will occur when 0.983n is 0.5,
i.e., when n is log 0.5/log 0.983 or 40, corresponding to 8000 m or 5 miles.
The Eulerian answer is thus slightly different. It could be made closer to the
Lagrangian result by carrying more significant figures or by decreasing L and
increasing n. An advantage of the Eulerian system is that it is possible to have
segments with different properties such as depth, width, velocity, volume, and
temperature. There can be additional inputs. The general equation employing the
group (1 + kL/U)n will not then apply, each segment having a specific value of this
factor. The mathematical enthusiast will note that L/U is t/n, where t is the flow time
to a distance nL m. The Lagrangian factor is thus also (1 + kt/n)n, which approaches
exp(kt) when n is large. It is good practice to do the calculation in both systems
(even approximately) and check that the results are reasonable. Some water quality
models of rivers and estuaries can have several hundred segments, thus it is difficult
to grasp the entirety of the results, and mistakes can go undetected.
2.6 STEADY STATE AND EQUILIBRIUM
In the previous section, we introduced the concept of “steady state” as implying
unchanging with time, i.e., all time derivatives are zero. There is frequent confusion
between this concept and that of “equilibrium,” which can also be regarded as a
situation in which no change occurs with time. The difference is very important and,
regrettably, the terms are often used synonymously. This is entirely wrong and is
best illustrated by an example.
Consider the vessel in Figure 2.1A, which contains 100 m3 of water and 100 m3
of air. It also contains a small amount of benzene, say 1000 g. If this is allowed to
stand at constant conditions for a long time, the benzene present will equilibrate
between the water and the air and will reach unchanging but different concentrations,
possibly 8 g/m3 in the water and 2 g/m3 in the air, i.e., a factor of 4 difference in
favor of the water. There is thus 800 g in the water and 200 g in the air. In this
condition, the system is at equilibrium and at a steady state. If, somehow, the air
and its benzene were quickly removed and replaced by clean air, leaving a total of
800 g in the water, and the volumes remained constant, the concentrations would
adjust (some benzene transferring from water to air) to give a new equilibrium (and
steady state) of 6.4 g/m3 in the water (total 640 g) and 1.6 g/m3 in the air (total
160 g), again with a factor of 4 difference. This factor is a partition coefficient or
distribution coefficient or, as is discussed later, a form of Henry’s law constant.
During the adjustment period (for example, immediately after removal of the air
when the benzene concentration in air is near zero and the water is still near 8 g/m3),
CH02 Page 22 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 23
the concentrations are not at a ratio of 4, the conditions are nonequilibrium, and,
since the concentrations are changing with time, they are also of unsteady-state in
nature.
This correspondence between equilibrium and steady state does not, however,
necessarily apply when flow conditions prevail. It is possible for air and water
Figure 2.1 Illustration of the difference between equilibrium and steady-state conditions. Equi-
librium implies that the oxygen concentrations in the air and water achieve a ratio
or partition coefficient of 20. Steady state implies unchanging with time, even if
flow occurs and regardless of whether equilibrium applies.
CH02 Page 23 Monday, January 15, 2001 1:47 PM
24 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
containing nonequilibrium quantities of benzene to flow into and out of the tank at
constant rates as shown in Figure 2.1B. But equilibrium and a steady-state condition
are maintained, since the concentrations in the tank and in the outflows are at a
ratio of 1:4. It is possible for near equilibrium to apply in the vessel, even when
the inflow concentrations are not in equilibrium, if benzene transfer between air
and water is very rapid. Figure 2.1B thus illustrates a flow, equilibrium, and steady-
state conditions, whereas Figure 2.1A is a nonflow, equilibrium, and steady-state
situation.
In Figure 2.1C, there is a deficiency of benzene in the inflow water (or excess
in the air) and, although in the time available some benzene transfers from air to
water, there is insufficient time for equilibrium to be reached. Steady state applies,
because all concentrations are constant with time. This is a flow, nonequilibrium,
steady-state condition in which the continuous flow causes a constant displacement
from equilibrium.
In Figure 2.1D, the inflow water and/or air concentration or rates change with
time, but there is sufficient time for the air and water to reach equilibrium in the
vessel, thus equilibrium applies (the concentration ratio is always 4), but unsteady-
state conditions prevail. Similar behavior could occur if the tank temperature changes
with time. This represents a flow, equilibrium, and unsteady-state condition.
Finally, in Figure 2.1E, the concentrations change with time, and they are not
in equilibrium; thus, a flow, nonequilibrium, unsteady-state condition applies, which
is obviously quite complex.
The important point is that equilibrium and steady state are not synonymous;
neither, either, or both can apply. Equilibrium implies that phases have concentrations
(or temperatures or pressures) such that they experience no tendency for net transfer
of mass. Steady state merely implies constancy with time. In the real environment,
we observe a complex assembly of phases in which some are (approximately) in
steady state, others in equilibrium, and still others in both steady state and equilib-
rium. By carefully determining which applies, we can greatly simplify the mathe-
matics used to describe chemical fate in the environment.
A couple of complications are worthy of note. Chemical reactions also tend to
proceed to equilibrium but may be prevented from doing so by kinetic or activation
considerations. An unlit candle seems to be in equilibrium with air, but in reality it
is in a metastable equilibrium state. If lit, it proceeds toward a “burned” state. Thus,
some reaction equilibria are not achieved easily, or not at all. 
Second, “steady state” depends on the time frame of interest. Blood circulation
in a sleeping child is nearly in steady state; the flow rates are fairly constant, and
no change is discernible over several hours. But, over a period of years, the child
grows, and the circulation rate changes; thus, it is not a true steady state when
viewed in the long term. The child is in a “pseudo” or “short-term” steady state.
In many cases, it is useful to assume steady state to apply for short periods,
knowing that it is not valid over long periods. Mathematically, a differential
equation that truly describes the system is approximated by an algebraic equation
by setting the differential or the d(contents)/dt term to zero. This can be justified
by examining the relative magnitude of the input, output, and inventory change
terms.
CH02 Page 24 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 25
2.7 DIFFUSIVE AND NONDIFFUSIVE ENVIRONMENTAL TRANSPORT 
PROCESSES
In the air-water example, it was argued that equilibrium occurs when the ratioof the benzene concentrations in water and air is 4. Thus, if the concentration in
water is 4 mol/m3, equilibrium conditions exist when the concentration in air is 1
mol/m3. If the air concentration rises to 2 mol/m3, we expect benzene to transfer by
diffusion from air to water until the concentration in air falls, concentration in water
rises, and a new equilibrium is reached. This is easily calculated if the total amount
of benzene is known. In a nonflow system, if the initial concentrations in air and
water are CAO and CWO mol/m3, respectively, and the volumes are VA and VW, then
the total amount M is, as shown earlier,
M = CAOVA + CWOVW mol
Here, CAO is 2, and CWO is 4 mol/m3, and since the volumes are both 100 m3, M is
600 mols. This will distribute such that CW is 4CA or
M = 600 = CAVA + CWVW = CAVA + 4CAVW = CA(VA + 4VW) = CA 500
Thus, CA is 1.2 mol/m3, and CW is 4.8 mol/m3. Thus, the water concentration rises
from 4.0 to 4.8, while that of the air drops from 2.0 to 1.2 mol/m3.
Conversely, if the concentration in water is increased to 10 mol/m3, there will
be transfer from water to air until a new equilibrium state is reached.
A worrisome dilemma is, “How does the benzene in the water know the con-
centration in the air so that it can decide to start or stop diffusing?” In fact, it does
not know or care. It diffuses regardless of the condition at the destination. Equilib-
rium merely implies that there is no net diffusion, the water-to-air and air-to-water
diffusion rates being equal and opposite. Chemicals in the environment are always
striving to reach equilibrium. They may not always achieve this goal, but it is useful
to know the direction in which they are heading.
Other transport mechanisms occur that are not driven by diffusion. For example,
we could take 1 m3 of the water with its associated 1 mol of benzene and physically
convey it into the air, forcing it to evaporate, thus causing the concentration of
benzene in the air to increase. This nondiffusive, or “bulk,” or “piggyback” transfer
occurs at a rate that depends on the rate of removal of the water phase and is not
influenced by diffusion. Indeed, it may be in a direction opposite to that of diffusion.
In the environment, it transpires that there are many diffusive and nondiffusive
processes operating simultaneously. Examples of diffusive transfer processes include
1. Volatilization from soil to air
2. Volatilization from water to air
3. Absorption or adsorption by sediments from water
4. Diffusive uptake from water by fish during respiration
Some nondiffusive processes are
CH02 Page 25 Monday, January 15, 2001 1:47 PM
26 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
1. Fallout of chemical from air to water or soil in dustfall, rain, or snow
2. Deposition of chemical from water to sediments in association with suspended
matter which deposits on the bed of sediment
3. The reverse process of resuspension
4. Ingestion and egestion of food containing chemical by biota
The mathematical expressions for these rates are quite different. For diffusion,
the net rate of transfer or flux is written as the product of the departure from
equilibrium and a kinetic quantity, and the net flux becomes zero when the phases
are in equilibrium. We examine these diffusive processes in Chapter 7. For nondif-
fusive processes, the flux is the product of the volume of the phase transferred (e.g.,
quantity of sediment or rain) and the concentration. We treat nondiffusive processes
in Chapter 6.
We use the word flux as short form for transport rate. It has units such as mol/h
or g/h. Purists insist that flux should have units of mol/h·m2, i.e., it should be area
specific. We will apply it to both. It is erroneous to use the term flux rate since flux,
like velocity, already contains the “per time” term.
2.8 RESIDENCE TIMES AND PERSISTENCE
In some environments, such as lakes, it is convenient to define a residence time
or detention time. If a pond has a volume of 1000 m3 and experiences inflow and
outflow of 2 m3/h, it is apparent that, on average, the water spends 500 h (20.8 days)
(i.e., 1000 m3/2 m3/h) in the lake. This residence or detention time may not bear
much relationship to the actual time that a particular parcel of water spends in the
pond, since some water may bypass most of the pond and reside for only a short
time, and some may be trapped for years. The quantity is very useful, however,
because it gives immediate insight into the time required to flush out the contents.
Obviously, a large lake with a long residence time will be very slow to recover from
contamination. Comparison of the residence time with a chemical reaction time (e.g.,
a half-life) indicates whether a chemical is removed from a lake predominantly by
flow or by reaction.
If a well mixed lake has a volume V m3 and equal inflow and outflow rates G
m3/h, then the flow residence time τF is V/G (h). If it is contaminated by a nonreacting
(conservative) chemical at a concentration CO mol/m3 at zero time and there is no
new emission, a mass balance gives, as was shown earlier,
C = C0 exp(–Gt/V) = C0 exp (–t/τF) = CO exp(–kF t)
The residence time is thus the reciprocal of a rate constant kF with units of h–1. The
half-time for recovery occurs when t/τF or kFt is ln 2 or 0.693, i.e., when t is 0.693τ
or 0.693/k.
If the chemical also undergoes a reaction with a rate constant kR h–1, it can be
shown that
C = C0 exp[–(kF + kR)t] = C0 exp(–kTt)
CH02 Page 26 Monday, January 15, 2001 1:47 PM
SOME BASIC CONCEPTS 27
Thus, the larger (faster) rate constant dominates. The characteristic times τF and τR
(i.e., 1/kF and 1/kR) combine as reciprocals to give the total time τT, as do electrical
resistances in parallel, i.e.,
1/τF + 1/τR = 1/τT = kT + kR
Thus, the smaller (shorter) τ dominates. The term τR can be viewed as a reaction
persistence. Characteristic times such as τR and τF are conceptually easy to grasp
and are very convenient quantities to deduce when interpreting the relative impor-
tance of environmental processes. For example, if τF is 30 years and τR is 3 years,
τT is 2.73 years; thus, reaction dominates the chemical’s fate. Ten out of every 11
molecules react, and only one leaves the lake by flow.
2.9 REAL AND EVALUATIVE ENVIRONMENTS
The environmental scientist who is attempting to describe the behavior of a
pesticide in a system such as a lake soon discovers that real lakes are very complex.
Considerable effort is required to measure, analyze, and describe the lake, with the
result that little energy (or research money) remains with which to describe the
behavior of the pesticide. This is an annoying problem, because it diverts attention
from the pesticide (which is important) to the condition of the lake (which may be
relatively unimportant). A related problem also arises when a new chemical is being
considered. Into which lake should it be placed (hypothetically) for evaluation? A
significant advance in environmental science was made in 1978, when Baughman
and Lassiter (1978) proposed that chemicals may be assessed in “evaluative envi-
ronments” that have fictitious but realistic properties such as volume, composition,
and temperature. Evaluative environments can be decreed to consist of a few homo-
geneous phases of specified dimensions with constant temperature and composition.
Essentially, the environmental scientist designs a “world” to desired specifications,
then explores mathematically the likely behavior of chemicals in that world. No
claim is made that the evaluative world is identical to any real environment, although
broad similarities in chemical behavior are expected. There are good precedents for
this approach. In 1824, Carnot devised an evaluative steam engine, now termed the
Carnot cycle, which leads to a satisfying explanation of entropy and the second law
of thermodynamics. The kinetic theory of gases uses an evaluative assumption of
gas molecule behavior.
The principal advantage of evaluative environments is that theyact as an intel-
lectual stepping stone when tackling the difficult task of describing both chemical
behavior and an environment. The task is simplified by sidestepping the effort needed
to describe a real environment. The disadvantage is that results of evaluative envi-
ronment calculations cannot be validated directly, so they are suspect and possibly
quite wrong. Some validation can be sought by making the evaluative environment
similar to a simple real environment, such as a small pond or a laboratory microcosm.
Later, we construct evaluative environments or “unit worlds” and use them to
explore the likely behavior of chemicals. In doing so, we use equations that can be
CH02 Page 27 Monday, January 15, 2001 1:47 PM
28 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
validated using real environments. A somewhat different assembly of equations
proves to be convenient for real environments, but the underlying principles are the
same.
2.10 SUMMARY
In this chapter, we have introduced the system of units and dimensions. A view
of the environment has been presented as an assembly of phases or compartments
that are (we hope) mostly homogeneous rather than heterogeneous in properties,
and that vary greatly in volume and composition. We can define these phases or
parts of them as “envelopes” about which we can write mass, mole, and, if necessary,
energy balance equations. Steady-state conditions will yield algebraic equations, and
unsteady-state conditions will yield differential equations. These equations may
contain terms for discharges, flow (diffusive and nondiffusive) of material between
phases, and for reaction or formation of a chemical. We have discriminated between
equilibrium and steady state and introduced the concepts of residence time and
persistence. Finally, the use of both real and evaluative environments has been
suggested.
Having established these basic concepts, or working tools, our next task is to
develop the capability of quantifying the rates of the various flow, transport, and
reaction processes.
CH02 Page 28 Monday, January 15, 2001 1:47 PM
 
29
 
CHAPTER
 
 3
Environmental Chemicals and
Their Properties
 
3.1 INTRODUCTION AND DATA SOURCES
 
In this book, we focus on techniques for building mass balance models of
chemical fate in the environment, rather than on the detailed chemistry that controls
transport and transformation, as well as toxic interactions. For a fuller account of
the basic chemistry, the reader is referred to the excellent texts by Crosby (1988),
Tinsley (1979), Stumm and Morgan (1981), Pankow (1991), Schwarzenbach et al.
(1993), Seinfeld and Pandis (1997), Findlayson-Pitts and Pitts (1986), Thibodeaux
(1996), and Valsaraj (1995).
There is a formidable and growing literature on the nature and properties of
chemicals of environmental concern. Numerous handbooks list relevant physical-
chemical and toxicological properties. Especially extensive are compilations on
pesticides, chemicals of potential occupational exposure, and carcinogens. Govern-
ment agencies such as the U.S. Environmental Protection Agency (EPA), Environ-
ment Canada, scientific organizations such as the Society of Environmental Toxi-
cology and Chemistry (SETAC), industry groups, and individual authors have
published numerous reports and books on specific chemicals or classes of chemicals.
Conferences are regularly held and proceedings published on specific chemicals
such as the “dioxins.” Computer-accessible databases are now widely available for
consultation. Table 3.1 lists some of the more widely used texts and scientific
journals. Most are available in good reference libraries.
Most of the chemicals that we treat in this book are organic, but the mass
balancing principles also apply to metals, organometallic chemicals, gases such as
oxygen and freons, inorganic compounds, and ions containing elements such as
phosphorus and arsenic. Metals and other inorganic compounds tend to require
individual treatment, because they usually possess a unique set of properties. Organic
compounds, on the other hand, tend to fall into certain well defined classes. We are
often able to estimate the properties and behavior of one organic chemical from that
 
CH03 Page 29 Tuesday, January 16, 2001 9:52 AM
 
30 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Table 3.1 Sources of information on chemical properties and estimation methods (See 
Chapter 1.5 of Mackay, et al., 
 
Illustrated Handbooks of Physical Chemical 
Properties and Environmental Fate for Organic Chemicals, 
 
cited below, for 
 
more details)
 
The Merck Index: An Encyclopedia of Chemicals, Drugs, and Biologicals (Annual), 
 
S. 
Budavarie, ed. Whitehouse Station, NJ: Merck & Co., 1996.
 
Handbook of Chemistry and Physics,
 
 D. R. Lide, ed., 81/e. Boca Raton, FL: CRC Press.
 
Verschueren’s Handbook of Environmental Data on Organic Chemicals.
 
 New York: John Wiley 
& Sons, 1997.
 
Illustrated Handbook of Physical Chemical Properties and Environmental Fate for Organic 
Chemicals
 
 (in 5 volumes). D. Mackay, W. Y Shiu, and K. C. Ma. Boca Raton, FL: CRC Press, 
1991–1997. Also available as a CD ROM.
 
Handbook of Environmental Fate and Exposure Data for Organic Chemicals
 
 (several volumes), 
P. H. Howard, ed. Boca Raton, FL: Lewis Publications.
 
Handbook of Environmental Degradation Rates,
 
 P. H. Howard et al. Boca Raton, FL: Lewis 
Publications.
 
Lange’s Handbook of Chemistry
 
, 15/e, J. A. Dean, ed. New York: McGraw-Hill, 1998.
 
Dreisbach’s Physical Properties of Chemical Compounds,
 
 Vol I to III. Washington, DC, Amer. 
Chem. Soc.
Technical Reports, European Chemical Industry Ecology and Toxicology Centre (ECETOC). 
Brussels, Belgium.
 
Sax’s Dangerous Properties of Industrial Materials,
 
 10/e. R. J. Lewis, ed. New York: John 
Wiley & Sons.
 
Groundwater Chemicals Desk Reference,
 
 J. J. Montgomery. Boca Raton, FL: Lewis 
Publications, 1996.
 
Genium Materials Safety Data Sheets Collection.
 
 Amsterdam, NY: Genium Publishing Corp.
 
The Properties of Gases and Liquids,
 
 R. C. Reid, J. M. Prausnitz, and B. E. Poling. New York: 
McGraw-Hill, 1987.
 
NIOSH/OSHA Occupational Health Guidelines for Chemical Hazards.
 
 Washington, DC: U.S. 
Government Printing Office.
 
The Pesticide Manual,
 
 12/e. C. D. S. Tomlin, ed. Loughborough, UK: British Crop Protection 
Council.
 
The Agrochemicals Handbook,
 
 H. Kidd and D. R. James, eds. London: Royal Society of 
Chemistry.
 
Agrochemicals Desk Reference,
 
 2/e, J. H. Montgomery. Boca Raton, FL: Lewis Publications.
ARS Pesticide Properties Database, R. Nash, A. Herner, and D. Wauchope. Beltsville, MD: 
U.S. Department of Agriculture, www.arsusda.gov/rsml/ppdb.html.
 
Substitution Constants for Correlation Analysis in Chemistry and Biology,
 
 C. H. Hansch 
(currently out of print). New York: Wiley-Interscience.
 
Handbook of Chemical Property Estimation Methods,
 
 W. J. Lyman, W. F. Reehl, D. H. 
Rosenblatt (currently out of print). New York: McGraw-Hill.
 
Handbook of Property Estimation Methods for Chemicals,
 
 R. S. Boethling and D. Mackay. 
Boca Raton, FL: CRC Press, 2000.
 
Chemical Property Estimation: Theory and Practice,
 
 E. J. Baum. Boca Raton, FL: Lewis 
Publications, 1997.
Toolkit for Estimating Physiochemical Properties of Organic Compounds, M. Reinhard and A. 
Drefahl. New York: John Wiley & Sons, 1999.
IUPAC Handbook. Research Triangle Park, NC: International Union of Pure and Applied 
Chemistry.
Website for database and EPIWIN estimation methods, Syracuse, NY: Syracuse Research 
Corporation 
 
(http://www.syrres.com). 
 
CH03 Page 30 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 31
 
of other, somewhat similar or homologous chemicals. An example is the series of
chlorinated benzenes that vary systematically in properties from benzene to
hexachlorobenzene.
It is believed that some 50,000to 80,000 chemicals are used in commerce. The
number of chemicals of environmental concern runs to a few thousand. There are
now numerous lists of “priority” chemicals of concern, but there is considerable
variation between lists. It is not possible, or even useful, to specify an exact number
of chemicals. Some inorganic chemicals ionize in contact with water and thus lose
their initial identity. Some lists name PCBs (polychlorinated biphenyls) as one
chemical and others as six groups of chemicals whereas, in reality, the PCBs consist
of 209 possible individual congeners. Many chemicals, such as surfactants and
solvents, are complex mixtures that are difficult to identify and analyze. One des-
ignation, such as 
 
naphtha
 
, may represent 1000 chemicals. There is a multitude of
pesticides, dyes, pigments, polymeric substances, drugs, and silicones that have
valuable social and commercial applications. These are in addition to the numerous
“natural” chemicals, many of which are very toxic.
For legislative purposes, most jurisdictions have compiled lists of chemicals that
are, or may be, encountered in the environment, and from these “raw” lists of
chemicals of potential concern they have established smaller lists of “priority”
chemicals. These chemicals, which are usually observed in the environment, are
known to have the potential to cause adverse ecological and/or biological effects
and are thus believed to be worthy of control and regulation. In practice, a chemical
that fails to reach the “priority” list is usually ignored and receives 
 
no
 
 priority rather
than 
 
less
 
 priority.
These lists should be regarded as dynamic. New chemicals are being added as
enthusiastic analytical chemists detect them in unexpected locations or toxicologists
discover subtle new effects. Examples are brominated flame retardants, chlorinated
alkanes, and certain very stable fluorinated substances (e.g., trifluoroacetic acid) that
have only recently been detected and identified. In recent years, concern has grown
about the presence of endocrine modulating substances such as nonylphenol, which
can disrupt sex ratios and generally interfere with reproductive processes. The
popular book 
 
Our Stolen Future,
 
 by Colborn et al. (1996) brought this issue to public
attention. Some of these have industrial or domestic sources, but there is increasing
concern about the general contamination by drugs used by humans or in agriculture.
Table 3.2 lists about 200 chemicals by class and contains many of the chemicals of
current concern.
 
3.2 IDENTIFYING PRIORITY CHEMICALS
 
It is a challenging task to identify from “raw lists” of chemicals a smaller, more
manageable number of “priority” chemicals. Such chemicals receive intense scrutiny,
analytical protocols are developed, properties and toxicity are measured, and reviews
are conducted of sources, fate, and effects. This selection contains an element of
judgement and is approached by different groups in different ways. A common thread
among many of the selection processes is the consideration of six factors: quantity,
 
CH03 Page 31 Tuesday, January 16, 2001 9:52 AM
 
32 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Table 3.2 List of Chemicals Commonly Found on Priority Chemical Lists
 
Volatile Halogentated Hydrocarbons Monoaromatic Hydrocarbons
 
Chloromethane Benzene
Methylene chloride Toluene
Chloroform o-Xylene
Carbontetrachloride m-Xylene
Chloroethane p-Xylene
1,1-Dichloroethane Ethylbenzene
1,2-Dichloroethane Styrene
cis-1,2-Dichloroethene
trans-1,2-Dichloroethene
 
Polycyclic Aromatic Hydrocarbons
 
Vinyl chloride Naphthalene
1,1,1-Trichloroethane 1-Methylnaphthalene
1,1,2-Trichloroethane 2-Methylnaphthalene
Trichloroethylene Trimethylnaphthalene
Tetrachloroethylene Biphenyl
Hexachloroethane Acenaphthene
1,2-Dichloropropane Acenaphthylene
1,3-Dichloropropane Fluorene
cis-1,3-Dichloropropylene Anthracene
trans-1,3-Dichloropropylene Fluoranthene
Chloroprene Phenanthrene
Bromomethane Pyrene
Bromoform Chrysene
Ethylenedibromide Benzo(a)anthracene
Chlorodibromomethane Dibenz(a,h)anthracene
Dichlorobromomethane Benzo(b)fluoranthene
Dichlorodibromomethane Benzo(k)fluoranthene
Freons (chlorofluoro-hydrocarbons) Benzo(a)pyrene
Dichlorodifluoromethane Perylene
Trichlorofluoromethane Benzo(g,h,i)perylene
Indeno(1,2,3)pyrene
 
Halogenated Monoaromatics
 
Chlorobenzene
1,2-Dichlorobenzene
 
Dienes
 
1,3-Dichlorobenzene 1,3-Butadiene
1,4-Dichlorobenzene Cyclopentadiene
1,2,3-Trichlorobenzene Hexachlorobutadiene
1,2,4-Trichlorobenzene Hexachlorocyclopentadiene
1,2,3,4-Tetrachlorobenzene
1,2,3,5-Tetrachlorobenzene
 
Alcohols and Phenols
 
Benzyl alcohol
Phenol
o-Cresol
m-Cresol
p-Cresol
2-Hydroxybiphenyl
4-Hydroxybiphenyl
Eugenol
 
CH03 Page 32 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 33
1,2,4,5-Tetrachlorobenzene
 
Halogenated Phenols
 
Pentachlorobenzene 2-Chlorophenol
Hexachlorobenzene 2,4-Dichlorophenol
2,4,5-Trichlorotoluene 2,6-Dichlorophenol
Octachlorostyrene 2,3,4-Trichlorophenol
2,3,5-Trichlorophenol
 
Halogenated Biphenyls and Naphthalenes
 
2,4,5-Trichlorophenol
Polychlorinated Biphenyls (PCBs) 2,4,6-Trichlorophenol
Polybrominated Biphenyls (PBBs) 2,3,4,5-Tetrachlorophenol
1-Chloronaphthalene 2,3,4,6-Tetrachlorophenol
2-Chloronaphthalene 2,3,5,6-Tetrachlorophenol
Polychlorinated Naphthalenes (PCNs) Pentachlorophenol
4-Chloro-3-methylphenol
 
Aroclor Mixtures (PCBs)
 
2,4-Dimethylphenol
Aroclor 1016 2,6-Di-t-butyl-4-methylphenol
Aroclor 1221 Tetrachloroguaiacol
Aroclor 1232
Aroclor 1242
 
Nitrophenols, Nitrotoluenes
 
Aroclor 1248
 
and Related Compounds
 
Aroclor 1254 2-Nitrophenol
Aroclor 1260 4-Nitrophenol
2,4-Dinitrophenol
 
Chlorinated Dibenzo-p-dioxins
 
4,6-Dinitro-o-cresol
2,3,7,8-Tetrachlorodibenzo-p-dioxin Nitrobenzene
Tetrachlorinated dibenzo-p-dioxins 2,4-Dinitrotoluene
Pentachlorinated dibenzo-p-dioxins 2,6-Dinitrotoluene
Hexachlorinated dibenzo-p-dioxins
Heptachlorinated dibenzo-p-dioxins 1-Nitronaphthalene
Octachlorinated dibenzo-p-dioxin 2-Nitronaphthalene
Brominated dibenzo-p-dioxins 5-Nitroacenaphthalene
 
Chlorinated Dibenzofurans Fluorinated Compounds
 
Tetrachlorinated dibenzofurans Polyfluorinated alkanes
Pentachlorinated dibenzofurans Trifluoroacetic acid
Hexachlorinated dibenzofurans Fluoro-chloro acids
Heptachlorinated dibenzofurans Polyfluorinated chemicals
Octachlorodibenzofuran
 
Phthalate Esters
Nitrosamines and Other Nitrogen Compounds
 
Dimethylphthalate
N-Nitrosodimethylamine Diethylphthalate
N-Nitrosodiethylamine Di-n-butylphthalate
N-Nitrosodiphenylamine Di-n-octylphthalate
N-Nitrosodi-n-propylamine Di(2-ethylhexyl) phthalate
Diphenylamine Benzylbutylphthalate
Indole
4-aminoazobenzene
 
Chlorinated longer chain alkanes
Pesticides, including biocides, fungicides, rodenticides, insecticides and herbicides
 
Table 3.2 List of Chemicals Commonly Found on Priority Chemical Lists
 
CH03 Page 33 Tuesday, January 16, 2001 9:52 AM
 
34 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
persistence, bioaccumulation, potential for transport to distant locations, toxicity,
and a miscellaneous group of other adverse effects.
 
3.2.1 Quantity
 
The first factor is the quantity produced, used, formed or transported, including
consideration of the fraction of the chemical that may be discharged to the environ-
ment during use. Some chemicals, such as benzene, are used in very large quantities
in fuels, but only a small fraction (possibly less than a fraction of a percent) is
emitted into the environment through incomplete combustion or leakage during
storage. Other chemicals, such as pesticides, are used in much smaller quantities
but are discharged completely and directly into the environment; i.e., 100% is
emitted. At the other extreme, there are chemical intermediates that may be produced
in large quantities but are emitted in only minuscule amounts (except during an
industrial accident). Itis difficult to compare the amounts emitted from these various
categories, because they are highly variable and episodic. It is essential, however,
to consider this factor; many toxic chemicals have no significant adverse impact,
because they enter the environment in negligible quantities.
Central to the importance of quantity is the adage first stated by Paracelsus,
nearly five centuries ago, that the dose makes the poison. This can be restated in
the form that all chemicals are toxic if administered to the victim in sufficient
quantities. A corollary is that, in sufficiently small doses, all chemicals are safe.
Indeed, certain metals and vitamins are essential to survival. The general objective
of environmental regulation or “management” must therefore be to ensure that the
quantity of a specific substance entering the environment is not excessive. It need
not be zero; indeed, it is impossible to achieve zero. Apart from cleaning up past
mistakes, the most useful regulatory action is to reduce emissions to acceptable
levels and thus ensure that concentrations and exposures are tolerable. Not even the
EPA can reduce the toxicity of benzene. It can only reduce emissions. This implies
knowing what the emissions are and where they come from. This is the focus of
programs such as the Toxics Release Inventory (TRI) in the U.S.A. or the National
Pollutant Release Inventory (NPRI) system in Canada. There are similar programs
in Europe, Australia, and Japan. Regrettably, the data are often incomplete. A major
purpose of this book is to give the reader the ability to translate emission rates into
environmental concentrations so that the risk resulting from exposure to these con-
centrations can be assessed. When this can be done, it provides an incentive to
improve release inventories. 
 
3.2.2 Persistence
 
The second factor is the chemical’s environmental persistence, which may also
be expressed as a 
 
lifetime, half-life,
 
 or 
 
residence time
 
. Some chemicals, such as DDT
or the PCBs, may persist in the environment for several years by virtue of their
resistance to transformation by degrading processes of biological and physical origin.
They may have the opportunity to migrate widely throughout the environment and
reach vulnerable organisms. Their persistence results in the possibility of establishing
 
CH03 Page 34 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 35
 
relatively high concentrations. This arises because, in principle, the amount in the
environment (kilograms) can be expressed as the product of the emission rate into
the environment (kilograms per year) and the residence time of the chemical in the
environment (years). Persistence also retards removal from the environment once
emissions are stopped. A legacy of “in place” contamination remains.
This is the same equation that controls a human population. For example, the
number of Canadians (about 30 million) is determined by the product or the rate at
which Canadians are born (about 0.4 million per year) and the lifetime of Canadians
(about 75 years). If Canadians were less persistent and lived for only 30 years, the
population would drop to 12 million.
Intuitively, the amount (and hence the concentration) of a chemical in the
environment must control the exposure and effects of that chemical on ecosystems,
because toxic and other adverse effects, such as ozone depletion, are generally a
response to concentration. Unfortunately, it is difficult to estimate the environmen-
tal persistence of a chemical. This is because the rate at which chemicals degrade
depends on which environmental media they reside in, on temperature (which
varies diurnally and seasonally), on incidence of sunlight (which varies similarly),
on the nature and number of degrading microorganisms that may be present, and
on other factors such as acidity and the presence of reactants and catalysts. This
variable persistence contrasts with radioisotopes, which have a half-life that is
fixed and unaffected by the media in which they reside. In reality, a substance
experiences a distribution of half-lives, not a single value, and this distribution
varies spatially and temporally. Obviously, long-lived chemicals, such as PCBs,
are of much greater concern than those, such as phenol, that may persist in the
aquatic environment for only a few days as a result of susceptibility to biodegra-
dation. Some estimate of persistence or residence time is thus necessary for priority
setting purposes. Organo-halogen chemicals tend to be persistent and are therefore
frequently found on priority lists. Later in this book, we develop methods of
calculating persistence.
 
3.2.3 Bioaccumulation
 
The third factor is potential for bioaccumulation (i.e., uptake of the chemical by
organisms). This is a phenomenon, not an effect; thus bioaccumulation 
 
per se
 
 is not
necessarily of concern. It is of concern that bioaccumulation may cause toxicity to
the affected organism or to a predator or consumer of that organism. Historically, it
was the observation of pesticide bioaccumulation in birds that prompted Rachel
Carson to write 
 
Silent Spring 
 
in 1962, thus greatly increasing public awareness of
environmental contamination.
As we discuss later, some chemicals, notably the hydrophobic or “water-hating”
organic chemicals, partition appreciably into organic media and establish high con-
centrations in fatty tissue. PCBs may achieve concentrations (i.e., they bioconcen-
trate) in fish at factors of 100,000 times the concentrations that exist in the water in
which the fish dwell. For some chemicals (notably PCBs, mercury, and DDT), there
is also a food chain effect. Small fish are consumed by larger fish, at higher trophic
levels, and by other animals such as gulls, otters, mink, and humans. These chemicals
 
CH03 Page 35 Tuesday, January 16, 2001 9:52 AM
 
36 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
may be transmitted up the food chain, and this may result in a further increase in
concentration such that they are biomagnified.
Bioaccumulation tendency is normally estimated using an organic phase-water
partition coefficient and, more specifically, the octanol-water partition coefficient.
This, in turn, can be related to the solubility of the chemical in the water. Clearly,
chemicals that bioaccumulate, bioconcentrate, and biomagnify have the potential to
travel down unexpected pathways, and they can exert severe toxic effects, especially
on organisms at higher trophic levels.
The importance of bioaccumulation may be illustrated by noting that, in water
containing 1 ng/L of PCB, the fish may contain 10
 
5
 
 ng/kg. A human may consume
1000 L of water annually (containing 1000 ng of PCB) and 10 kg of fish (containing
10
 
6
 
 ng of PCB), thus exposure from fish consumption is 1000 times greater than
that from water. Particularly vulnerable are organisms such as certain birds and
mammals that rely heavily on fish as a food source.
 
3.2.4 Toxicity
 
The fourth factor is the toxicity of the chemical. The simplest manifestation of
toxicity is acute toxicity. This is most easily measured as a concentration that will
kill 50% of a population of an aquatic organism, such as fish or an invertebrate (e.g.,
 
Daphnia magna
 
), in a period of 24–96 hours, depending on test conditions. When
the concentration that kills (or is lethal to) 50% (the LC50) is small, this corresponds
to high toxicity. The toxic chemical may also be administered to laboratory animals
such as mice or rats, orally or dermally. The results are then expressed as a lethal
dose to kill 50% (LD50) in units of mg chemical/kg body weight of the animal.
Again, a low LD50 corresponds to high toxicity.
More difficult, expensive, and contentious are 
 
chronic
 
, or 
 
sublethal
 
, tests that
assess the susceptibility of the organism to adverse effects from concentrations or
doses of chemicals that do not cause immediatedeath but ultimately may lead to
death. For example, the animal may cease to feed, grow more slowly, be unable to
reproduce, become more susceptible to predation, or display some abnormal behav-
ior that ultimately affects its life span or performance. The concentrations or doses
at which these effects occur are often about 1/10th to 1/100th of those that cause
acute effects. Ironically, in many cases, the toxic agent is also an essential nutrient,
so too much or too little may cause adverse effects.
Although most toxicology is applied to animals, there is also a body of knowledge
on phytotoxicity, i.e., toxicity to plants. Plants are much easier to manage, and killing
them is less controversial. Tests also exist for assessing toxicity to microorganisms.
It is important to emphasise that toxicity alone is not a sufficient cause for concern
about a chemical. Arsenic in a bottle is harmless. Disinfectants, biocides, and pesti-
cides are inherently useful because they are toxic. The extent to which the organism
is injured depends on the inherent properties of the chemical, the condition of the
organism, and the dose or amount that the organism experiences. It is thus misleading
to classify or prioritize chemicals solely on the basis of their inherent toxicity, or on
the basis of the concentrations in the environment or exposures. Both must be
considered. A major task of this book is to estimate exposure. A healthy tension often
 
CH03 Page 36 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 37
 
exists between toxicologists and chemists about the relative importance of toxicity
and exposure, but fundamentally this argument is about as purposeful as squabbling
over whether tea leaves or water are the more important constituents of tea.
Most difficult is the issue of genotoxicity, including carcinogenicity, and terato-
genicity. In recent years, a battery of tests has been developed in which organisms
ranging from microorganisms to mammals are exposed to chemicals in an attempt
to determine if they can influence genetic structure or cause cancer. A major difficulty
is that these effects may have long latent periods, perhaps 20 to 30 years in humans.
The adverse effect may be a result of a series of biochemical events in which the
toxic chemical plays only one role. It is difficult to use the results of short-term
laboratory experiments to deduce reliably the presence and magnitude of hazard to
humans. There may be suspicions that a chemical is producing cancer in perhaps
0.1% of a large human population over a period of perhaps 30 years, an effect that
is very difficult (or probably impossible) to detect in epidemiological studies. But
this 0.1% translates into the premature death of 30,000 Canadians per year from
such a cancer, and is cause for considerable concern. Another difficulty is that
humans are voluntarily and involuntarily exposed to many toxic chemicals, including
those derived from smoking, legal and illicit drugs, domestic and occupational
exposure, as well as environmental exposure. Although research indicates that mul-
tiple toxicants act additively when they have similar modes of action, there are cases
of synergism and antagonism. Despite these difficulties, a considerable number of
chemicals have been assessed as being carcinogenic, mutagenic, or teratogenic, and
it is even possible to assign some degree of potency to each chemical. Such chemicals
usually rank high on priority lists. As was discussed earlier, endocrine modulating
substances are of more recent concern. It seems likely that ingenious toxicologists
will find other subtle toxic effects in the future.
 
3.2.5 Long-Range Transport
 
As lakes go, Lake Superior is fairly pristine, since there is relatively little industry
on its shores. In the U.S. part of this lake is an island, Isle Royale, which is a
protected park and is thus even more pristine. In this island is a lake, Siskiwit Lake,
which cannot conceivably be contaminated. No responsible funding agency would
waste money on the analysis of fish from that lake for substances such as PCBs.
Remarkably, perceptive researchers detected substantial concentrations of PCBs.
Similarly, surprisingly high concentrations have been detected in wildlife in the
Arctic and Antarctic. Clearly, certain contaminants can travel long distances through
the atmosphere and oceans and are deposited in remote regions.
This potential for long-range transport (LRT) is of concern for several reasons.
There is an ethical issue when the use of a chemical in one nation (which presumably
enjoys social or economic benefit from it) results in exposure in other downwind
nations that derive no benefit, only adverse effects. This transboundary pollution
issue also applies to gases such as SO
 
2
 
, which can cause acidification of poorly
buffered lakes at distant locations. A regulatory agency may then be in the position
of having little or no control over exposures experienced by its public. The political
implications are obvious.
 
CH03 Page 37 Tuesday, January 16, 2001 9:52 AM
 
38 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
There is therefore a compelling incentive to identify those chemicals that can
undertake long-range transport and implement international agreements to control
them. A start on this process has been made recently by the United Nations Envi-
ronment Program (UNEP), which has identified 12 substances or groups for inter-
national regulations or bans. These substances, listed in Table 3.3, are also identified
as persistent, bioaccumulative, and toxic. Others are scheduled for restriction or
reduction. They may represent merely the first group of chemicals that will be subject
to international controls. Most contentious of the 12 is DDT, which is still widely
and beneficially used for malaria control.
 
3.2.6 Other Effects
 
Finally, there is a variety of other adverse effects that are of concern, including
 
• the ability to influence atmospheric chemistry (e.g., freons)
• alteration in pH (e.g., oxides of sulfur and nitrogen causing acid rain)
• unusual chemical properties such as chelating capacity, which alters the availability
of other chemicals in the environment
• interference with visibility
• odor (e.g., from organo-sulfur compounds)
• color (e.g., from dyes)
• the ability to cause foaming in rivers (e.g., detergents or surfactants)
• formation of toxic metabolites or degradation products
 
3.2.7 Selection Procedures
 
A common selection procedure involves scoring these factors on some numeric
hazard scale. The factors then may be combined to give an overall factor and
 
Table 3.3 Substances Scheduled for Elimination, Restriction, or Reduction by UNEP
Scheduled for Elimination Scheduled for Restriction
Scheduled for 
Reduction
 
Aldrin DDT PAHs
Chlordane Hexachlorocyclohexanes Dioxins/furans
DDT Polychlorinated biphenyls Hexachlorobenzene
Dieldrin
Endrin
Heptachlor
Hexabromobiphenyl
Hexachlorobenzene
Mirex
Polychlorinated biphenyls
Toxaphene
 
CH03 Page 38 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 39
 
determine priority. This is a subjective process, and it becomes difficult for two
major reasons.
First, chemicals that are subject to quite different patterns of use are difficult to
compare. For example, chemical X may be produced in very large quantities, emitted
into the environment, and found in substantial concentrations in the environment,
but it may not be believed to be particularly toxic. Examples are solvents such as
trichloroethylene or plasticizers such as the phthalate esters. On the other hand,
chemical Y may be produced in minuscule amounts but be very toxic, an example
being the “dioxins.” Which deserves the higher priority?
Second, it appears that the adverse effects suffered by aquatic organisms and
other animals, including humans, are the result of exposure to a large number of
chemicals, not just to one or twochemicals. Thus, assessing chemicals on a case-
by-case basis may obscure the cumulative effect of a large number of chemicals.
For example, if an organism is exposed to 150 chemicals, each at a concentration
that is only 1% of the level that will cause death, then death will very likely occur,
but it cannot be attributed to any one of these chemicals. It is the cumulative effect
that causes death. The obvious prudent approach is to reduce exposure to all chem-
icals to the maximum extent possible. The issue is further complicated by the
possibility that some chemicals will act synergistically, i.e., they produce an effect
that is greater than additive; or they may act antagonistically, i.e., the combined
effect is less than additive. As a result, there will be cases in which we are unable
to prove that a specific chemical causes a toxic effect but, in reality, it does contribute
to an overall toxic effect. Indeed, some believe that this situation will be the rule
rather than the exception.
A compelling case can be made that the prudent course of action is for society
to cast a fairly wide net of suspicion (i.e., assemble a fairly large list of chemicals)
then work to elucidate sources, fate, and effects with the aim of reducing overall
exposure of humans, and our companion organisms, to a level at which there is
assurance that no significant toxic effects can exist from these chemicals. The risk
from these chemicals then becomes small as compared to other risks such as acci-
dents, disease, and exposure to natural toxic substances. This approach has been
extended and articulated as the “Precautionary Principle,” the “Substitution Princi-
ple,” and the “Principle of Prudent Avoidance.” 
One preferred approach is to undertake a risk assessment for each chemical.
Formal procedures for conducting such assessments have been published, notably
by the U.S. Environmental Protection Agency (EPA). The process involves identi-
fying the chemical, its sources, the environment in which it is present, and the
organisms that may be affected. The toxicity of the substance is evaluated and routes
of exposure quantified. Ultimately, the prevailing concentrations or doses are mea-
sured or estimated and compared with levels that are known to cause effects, and
conclusions are drawn regarding the proximity to levels at which there is a risk of
effect. This necessarily involves consideration of the chemical’s behavior in an actual
environment. Risk is thus assessed only for that environment. Risk or toxic effects
are thus not inherent properties of a chemical; they depend on the extent to which
the chemical reaches the organism.
 
CH03 Page 39 Tuesday, January 16, 2001 9:52 AM
 
40 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
3.3 KEY CHEMICAL PROPERTIES AND CLASSES
3.3.1 Key Properties
 
In Chapter 5, we discuss physicochemical properties in more detail and, in
Chapter 6, we examine reactivities. It is useful at this stage to introduce some of
these properties and identify how they apply to different classes of chemicals.
It transpires that we can learn a great deal about how a chemical partitions in
the environment from its behavior in an air-water-octanol (strictly 1-octanol) system
as shown later in Figure 3.2. There are three partition coefficients, K
 
AW
 
, K
 
OW
 
, and
K
 
OA
 
, only two of which are independent, since K
 
OA
 
 must equal K
 
OW
 
/K
 
AW
 
. These can
be measured directly or estimated from vapor pressure, solubility in water, and
solubility in octanol, but not all chemicals have measurable solubilities because of
miscibility. Octanol is an excellent surrogate for natural organic matter in soils and
sediments, lipids, or fats, and even plant waxes. It has approximately the same C:H:O
ratio as lipids. Correlations are thus developed between soil-water and octanol-water
partition coefficients, as discussed in more detail later.
An important attribute of organic chemicals is the degree to which they are
 
hydrophobic
 
. This implies that the chemical is sparingly soluble in, or “hates,” water
and prefers to partition into lipid, organic, or fat phases. A convenient descriptor of
this hydrophobic tendency is K
 
OW
 
. A high value of perhaps one million, as applies to
DDT, implies that the chemical will achieve a concentration in an organic medium
approximately a million times that of water with which it is in contact. In reality,
most organic chemicals are approximately equally soluble in lipid or fat phases, but
they vary greatly in their solubility in water. Thus, differences in hydrophobicity are
largely due to differences of behavior in, or affinity for, the water phase, not differences
in solubility in lipids. The word 
 
lipophilic
 
 is thus unfortunate and is best avoided. 
The chemical’s tendency to evaporate or partition into the atmosphere is primarily
controlled by its vapor pressure, which is essentially the maximum pressure that a
pure chemical can exert in the gas phase or atmosphere. It can be viewed as the
 
solubility
 
 of the chemical in the gas phase. Indeed, if the vapor pressure in units of
Pa is divided by the gas constant, temperature group RT, where R is the gas constant
(8.314 Pa m
 
3
 
/mol K), and T is absolute temperature (K), then vapor pressure can
be converted into a solubility with units of mol/m
 
3
 
. Organic chemicals vary enor-
mously in their vapor pressure and correspondingly in their boiling point. Some
(e.g., the lower alkanes) that are present in gasoline are very volatile, whereas others
(e.g., DDT) have exceedingly low vapor pressures.
Partitioning from a pure chemical phase to the atmosphere is controlled by vapor
pressure. Partitioning from aqueous solution to the atmosphere is controlled by K
 
AW
 
,
a joint function of vapor pressure and solubility in water. A substance may have a
high K
 
AW
 
, because its solubility in water is low. Partitioning from soils and other
organic media to the atmosphere is controlled by K
 
AO
 
 (air/octanol), which is con-
ventionally reported as its reciprocal, K
 
OA
 
. Partitioning from water to organic media,
including fish, is controlled by K
 
OW
 
. Substances that display a significant tendency
to partition into the air phase over other phases are termed 
 
volatile organic chemicals
 
or VOCs. They have high vapor pressures.
 
CH03 Page 40 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 41
 
Another important classification of organic chemicals is according to their dis-
sociating tendencies in water solution. Some organic acids, notably the phenols, will
form ionic species (phenolates) at high pH. The tendency to ionize is characterized
by the acid dissociation constant K
 
A
 
, often expressed as pK
 
A
 
, its negative base ten
logarithm. 
In concert with partitioning characteristics, the other set of properties that deter-
mine environmental behavior is reactivity or persistence, usually expressed as a half-
life. It is misleading to assign a single number to a half-life, because it depends on
the intrinsic properties of the chemical and on the nature of the environment. Factors
such as sunlight intensity, hydroxyl radical concentration, the nature of the microbial
community, as well as temperature vary considerably from place to place and time
to time. Here, we use a semiquantitative classification of half-lives into classes,
assuming that average environmental conditions apply. Different classes are defined
for air, water, soils, and sediments. The classification is that used in a series of
“Illustrated Handbooks” by Mackay, Shiu, and Ma is shown below in Table 3.4.
The half-lives are on a logarithmic scale with a factor of approximately 3 between
adjacent classes. It is probably misleading to divide the classes into finer groupings;
indeed, a single chemical may experience half-lives ranging over three classes,
depending on environmental conditionssuch as season.
We examine, in the following sections, a number of classes of compounds that
are of concern environmentally. In doing so, we note their partitioning and persis-
tence properties. The structures of many of these chemicals are given in Figure 3.1.
Table 3.5 gives suggested values of these properties for selected chemicals.
Figure 3.2 is a plot of log K
 
AW
 
 versus log K
 
OW
 
 for the chemicals in Table 3.5
on which lines of constant K
 
OA
 
 lie on the 45° diagonal. This graph shows the wide
variation in properties. Volatile compounds tend to lie to the upper left, water-soluble
compounds to the lower left, and hydrophobic compounds to the lower right. Assum-
ing reasonable relative volumes of air (650,000), water (1300), and octanol (1), the
percentages in each phase at equilibrium can be calculated. The lines of constant
percentages are also shown. Lee and Mackay (1995) have used equilateral triangular
diagrams to display the variation in partitioning properties in a format similar to
that of Figure 3.2.
 
Table 3.4 Classes of Chemical Half-Life or Persistence, Adapted from 
 
the Handbooks of Mackay et al., 2000
Class Mean half–life (hours) Range (hours)
 
1 5 <10
2 17 (~ 1 day) 10–30
3 55 (~ 2 day) 30–100
4 170 (~1 week) 100–300
5 550 (~3 weeks) 300–1000
6 1700 (~2 months) 1000–3000
7 5500 (~8 months) 3000–10,000
8 17000 (~2 years) 10,000–30,000
9 55000 (~6 years) >30,000
 
CH03 Page 41 Tuesday, January 16, 2001 9:52 AM
 
42 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Figure 3.1 Structures of selected chemicals of environmental interest (continues).
 
CH03 Page 42 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 43
Figure 3.1 (continued)
 
CH03 Page 43 Tuesday, January 16, 2001 9:52 AM
 
44 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
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CH03 Page 44 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 45
 
Ta
b
le
 3
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(c
o
n
ti
n
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)
 
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ch
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CH03 Page 45 Tuesday, January 16, 2001 9:52 AM
 
46 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Ta
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(c
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ti
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CH03 Page 46 Tuesday, January 16, 2001 9:52 AM
 
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 47
 
Ta
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CH03 Page 47 Tuesday, January 16, 2001 9:52 AM
48 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
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CH03 Page 48 Tuesday, January 16, 2001 9:52 AM
ENVIRONMENTAL CHEMICALS ANDTHEIR PROPERTIES 49
Figure 3.2 Plot of log KAW vs. log KOW for the chemicals in Table 3.5 on which dotted lines of
constant KOA line on the 45° diagonal. This graph shows the wide variation in
properties. Volatile compounds tend to lie to the upper left, water-soluble com-
pounds to the lower left, and hydrophobic compounds to the lower right. The thicker
lines represent constant percentages present at equilibrium in air, water, and
octanol phases, assuming a volume ratio of 656,000:1300:1, respectively. Modified
from Gouin et al. (2000).
CH03 Page 49 Tuesday, January 16, 2001 9:52 AM
50 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
3.3.2 Chemical Classes (see Fig. 3.1 for structures and Table 3.5 for 
properties)
3.3.2.1 Hydrocarbons
Hydrocarbons are naturally occurring chemicals present in crude oil and natural
gas. Some are formed by biogenic processes in vegetation, but most contamination
comes from oil spills, effluents from petroleum and petrochemical refineries, and
the use of fuels for transportation purposes.
The alkanes can be separated into classes of normal, branched (or iso) species
and cyclic alkanes, which range in molar mass from methane or natural gas to waxes.
They are usually sparingly soluble in water. For example, hexane has a solubility
of approximately 10 g/m3. This solubility falls by a factor of about 3 or 4 for every
carbon added. The branched and cyclic alkanes tend to be more soluble in water,
apparently because they have smaller molecular areas and volumes.
Highly branched or cyclic alkanes such as terpenes are produced by vegetation.
They are often sweet smelling and tend to be very resistant to biodegradation.
The alkenes or olefins are not naturally occurring to any significant extent. They
are mainly used as petrochemical intermediates. The alkynes, of which ethyne or
acetylene is the first member, are also chemical intermediates that are rarely found
in the environment. These unsaturated hydrocarbons tend to be fairly reactive and
short-lived in the environment, whereas the alkanes are more stable and persistent.
Of particular environmental interest are the aromatics, the simplest of which is
benzene. The aromatics are relatively soluble in water, for example, benzene has a
solubility of 1780 g/m3. They are regarded as fairly toxic and often troublesome
compounds. A variety of substituted aromatics can be obtained by substituting
various alkyl groups. For example, methyl benzene is toluene.
When two benzene rings are fused, the result is naphthalene, which is also a
chemical of considerable environmental interest. Subsequent fusing of benzene rings
to naphthalene leads to a variety of chemicals referred to as the polycyclic aromatic
hydrocarbons or polynuclear aromatic hydrocarbons (PAHs). These compounds tend
to be formed when a fuel is burned with insufficient oxygen. They are thus present
in exhaust from engines and are of interest because many are carcinogenic.
Biphenyl is a hydrocarbon that is not of much importance as such, but it forms
an interesting series of chlorinated compounds, the PCBs or polychlorinated biphe-
nyls, which are discussed later.
3.3.2.2 Halogenated Hydrocarbons
If the hydrogen in a hydrocarbon is substituted by chlorine (or less frequently
by bromine, fluorine, or iodine), the resulting compound tends to be less flammable,
more stable, more hydrophobic, and more environmentally troublesome. Replacing
a hydrogen with a chlorine usually causes an increase in molar volume and area and
a corresponding decrease in solubility by a factor of about 3.
The stability of many of these compounds makes them invaluable as solvents,
examples being methylene chloride and tetrachloroethylene. The fluorinated and
CH03 Page 50 Tuesday, January 16, 2001 9:52 AM
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 51
chlorofluoro compounds are very stable and are used as refrigerants. Because these
molecules are quite small, they are fairly soluble in water and are therefore able to
penetrate the tissues of organisms quite readily. They are thus used as anaesthetics
and narcotic agents.
The chlorinated aromatics are a particularly interesting group of chemicals. The
chlorobenzenes are biologically active. 1,4 or paradichlorobenzene is widely used
as a deodorant and disinfectant. The polychlorinated biphenyls, or PCBs, and their
brominated cousins, the PBBs, are notorious environmental contaminants, as are
chlorinated terpenes such as toxaphene, which is a very potent and long-lived
insecticide. Many of the early pesticides, such as DDT, mirex, and chlordane, are
chlorinated hydrocarbons. They possess the desirable properties of stability and a
high tendency to partition out of air and water into the target organisms. Thus,
application of a pesticide results in protection for a prolonged time. As Rachel Carson
demonstrated in Silent Spring, the problem is that these chemicals persist long
enough to affect non-target organisms and to drift throughout the environment,
causing widespread contamination.
Fluorinated chemicals also possess considerable stability and, because the fluo-
rine atom is lighter than chlorine, they are generally more volatile. Polyfluorinated
substances are very stable in the environment as a result of the strong C-F bond.
Brominated chemicals are also stable, but with reduced volatility. A major use of
brominated substances is in fire retardants, specifically polybrominated diphenyl
ethers.
3.3.2.3 Oxygenated Compounds
The most common oxygenated organic compounds are the alcohols, ethanol
being among the most widely used. Others are octanol, which is a convenient
analytical surrogate for fat, and glycerol is of interest because it forms the backbone
of fat molecules by esterification with fatty acids to form glycerides.
The phenols consist of an aromatic molecule in which a hydrogen is replaced
by an OH group. They are acidic and tend to be biologically disruptive. Phenol, or
carbolic acid, was the first disinfectant. Substituting chlorines on phenol tends to
increase the toxic potency of the substance and its tendency to ionize, i.e., its pKa
is reduced. Pentachlorophenol (PCP) is a particularly toxic chemical and has been
widely used for wood preservation.
The ketones such as acetone, and aldehydes such as formaldehyde, are fairly
reactive in the environment and can be of concern as atmospheric contaminants in
regions close to sources of emission. Much of the smog problem is attributable to
aldehydes formed in combustion processes.
Organic acids such as acetic acid are also fairly reactive. They are not usually
regarded as an environmental problem, but trifluoroacetic acid, which is formed by
combustion of freons and from some pesticides, is very persistent. Some chlorinated
organic acids, e.g., 2,4-D, are potent herbicides. Longer-chain acids, such as stearic
acid, are mainly of interest because they esterify with glycerol to form fats. Humic
and fulvic acids are of considerable environmental importance. These are substances
CH03 Page 51 Tuesday, January 16, 2001 9:52 AM
52 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
of complex and variable structure that are naturally present in soils, water, and
sediments. They are the remnants of living organic materials, such as wood, that
has been subjected to prolonged microbial conversion. These acids are sparingly
soluble in water, but the solubility can be increased at high pH.
The esters or “salts” or organic acids and alcohols tend to be relatively innocuous
and short-lived in most cases. A notable exception is the phthalate esters, which are
very stable oily substances and are invaluable additives (plasticizers) for plastics,
rendering them more flexible. Notable among the phthalate esters is diethylhexy-
lphthalate (DEHP), the ester with two molecules of 2 ethylhexanol. The other esters
of interest are the glycerides—for example, glyceryl trioleate, the ester of glycerine
and oleic acid. This chemical has similar properties to fat and has been suggestedas a convenient surrogate for measuring fat to water partitioning. 
The “dioxins” and “furans” are two series of organic compounds that have
become environmentally notorious. The chlorinated dibenzo-p-dioxins were never
produced intentionally but are formed under combustion conditions when chlorine
is present. They form a series of very toxic chemicals, the most celebrated of which
is 2,3,7,8 tetrachlorodibenzo-p-dioxin (TCDD). TCDD is possibly the most toxic
chemical to mammals. A dose of 2 µg of TCDD per kg of body weight is sufficient
to kill small rodents.
A related series of chemicals is the dibenzofurans, which are similar in properties
to the dioxins. It appears that molecules that are long and flat, with chlorine atoms
strategically located at the ends, are particularly toxic. Examples are the chloronaph-
thalenes, DDT, the PCBs, and chlorinated dibenzo-p-dioxins and dibenzofurans.
Other oxygenated compounds of interest include carbohydrates, cellulose, and
lignins, which occur naturally.
3.3.2.4 Nitrogen Compounds
Nitrogen compounds of environmental interest include amines, amides,
pyridines, quinolines, and amino acids, and various nitro compounds including nitro
polycyclicaromatics and nitroso compounds. Many of these compounds occur nat-
urally, are quite toxic, and are difficult to analyze.
3.3.2.5 Sulfur Compounds
Sulfur compounds, including thiols, thiophenes, and mercaptans, are well known
because of their strong odor. One of the most prevalent classes of synthetic organic
chemicals is the alkyl benzene sulfonates, which are widely used in detergents.
3.3.2.6 Phosphorus Compounds
Phosphorus compounds play a key role in energy transfer in organisms. Organo-
phosphate compounds have been developed as pesticides (e.g., chloropyrifos), which
have the very desirable properties of high biological activity but relatively short
environmental persistence. They have therefore largely replaced organo-chlorine
compounds in agriculture.
CH03 Page 52 Tuesday, January 16, 2001 9:52 AM
ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 53
3.3.2.7 Arsenic Compounds
Arsenic, which behaves somewhat similarly to phosphorus, is inadvertently
liberated in mineral processing and has a long and celebrated history as a poison.
It usually exists in anionic and organic forms.
3.3.2.8 Metals
Most metals are essential for human life in small quantities but can be toxic if
administered in excessive dosages. The metals of primary toxicological interest here
are those that form organo-metallic molecules. Notable is mercury, which can exist
as the element in various ionic and organometallic forms. Other metals such as lead
and tin behave similarly. A formidable literature exists on the behavior, fate, and
effects of the “heavy” metals such as lead, copper, and chromium. These metals
often have a complex environmental chemistry and toxicology that vary considerably,
depending on their ionic state as influenced by acidity and redox status.
3.3.2.9 Pharmaceuticals and Personal Care Products
Considerable quantities of drugs are used by humans and for veterinary purposes
on livestock. Antibiotics and steroids are examples. These substances are excreted
and may pass through sewage treatment plants or enter soils or groundwater follow-
ing agricultural use. There is a growing concern that these substances may have
adverse effects or may cause an increase in antibiotic resistance in bacteria. Among
personal care products of concern are detergents, fabric softeners, fragrances, and
certain solvents. They may evaporate or be discharged with sewage, which may or
may not be adequately treated.
3.3.2.10 Other Chemicals
Several other chemicals are of environmental concern including ozone, radon,
chlorine, organic and inorganic sulfides and cyanides, as well as the indeterminate
broad class of “conventional” pollutants or indicators of pollution such as biochem-
ical oxygen demand (BOD) and chemical oxygen demand (COD). Finally, certain
mineral substances such as asbestos are of concern, more because of their physical
structure than their chemical composition.
3.3.2.11 The Future
It would be unwise to assume that current lists of priority chemicals are complete
and will remain static. It may be that the chemicals on the lists reflect our present
ability to detect and analyze them rather than their real environmental significance.
The prevalence of organo-chlorine chemicals on lists is in part the result of the
sensitive electron capture detector. As new analytical methods emerge, new chemi-
cals will presumably be found, and priorities will change. Happy hunting grounds
for environmental chemists include combustion gases, dyes, mine tailings, effluents
CH03 Page 53 Tuesday, January 16, 2001 9:52 AM
54 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
from pulp and paper operations (especially those involving chlorine bleaching),
landfill leachates, and a vast assortment of products of metabolic conversion in
organisms ranging from bacteria to humans.
3.4 CONCLUDING EXAMPLE
Select five substances from Table 3.5 that range in their values of vapor pressure,
aqueous solubility, and log KOW. 
Calculate KAW as 
where R is 8.314 Pa m3/mol K, and T is absolute temperature (298 K).
Calculate how 100 kg of each of these chemicals would partition at equilibrium
between three phases namely, 
1 m3 octanol (representing perhaps 100 m3 of soil)
5000 m3 water
106 m3 air
Calculate all the concentrations and amounts (which should add to 100 kg!) and
discuss briefly how each substance is behaving, i.e., its partitioning preference.
vapor pressure (Pa)
solubility (g/m3)
----------------------------------------------
molar mass (g/mol( )
R T
-------------------------------------------------
CH03 Page 54 Tuesday, January 16, 2001 9:52 AM
 
55
 
CHAPTER
 
 4
The Nature of Environmental Media
 
4.1 INTRODUCTION
 
The objective of this chapter is to present a qualitative description of environ-
mental media, highlighting some of their more important properties. This is done
because the fate of a chemical depends on two groups of properties: those of the
chemical and those of the environment in which it resides. We find it useful to
assemble “evaluative” environments, which are used in later calculations. We can
consider, for example, an area of 1 
 
×
 
 1 km, consisting of some air, water, soil, and
sediment. Volumes and properties can be assigned to these media, which are typical
but purely illustrative and will, of course, require modification if chemical fate in a
specific region is to be treated. The sequence is to treat the atmosphere, the hydro-
sphere (i.e., water), and then the lithosphere (bottom sediments and terrestrial soils),
each with its resident biotic community.
It transpires that it is convenient to define two evaluative environments. First is
a simple four-compartment system that is easily understood and illustrates the
application of the general principles of environmental partitioning. Second is a more
complex, eight-compartment system that is more representative of real environ-
ments. It is correspondingly more demanding of data and leads to more lengthy
calculations.
The environments or “unit worlds” are depicted in Figure 4.1. Details are dis-
cussed by Neely and Mackay, 1982.
 
4.2 THE ATMOSPHERE
4.2.1 Air
 
The layer of the atmosphere that is in most intimate contact with the surface of
the Earth is the troposphere, which extends to a height of about 10 km. The tem-
perature, density, and pressure of the atmosphere fall steadily with increasing height,
 
CH04 Page 55 Monday, January 15, 2001 1:49 PM
 
56 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
which is a nuisance in subsequent calculations. If we assume uniform density at a
pressure of one atmosphere, then the entire troposphere can be viewed as being
compressed into a height of about 6 km. Exchange of matter from the troposphere
Figure 4.1 Evaluative environments.
 
CH04 Page 56 Monday,January 15, 2001 1:49 PM
kimhien.hienkim
Rectangle
kimhien.hienkim
Rectangle
 
THE NATURE OF ENVIRONMENTAL MEDIA 57
 
through the tropopause to the stratosphere is a relatively slow process and is rarely
important in environmental calculations, except in the case of chemicals such as the
freons, which catalyze the destruction of stratospheric ozone, thus facilitating the
penetration of UV light to the Earth’s surface. A reasonable atmospheric volume
over our 1 km square world is thus 1000 
 
×
 
 1000 
 
×
 
 6000 or 6 
 
×
 
 10
 
9
 
 m
 
3
 
.
If our environmental model is concerned with a localized situation (e.g., a state,
province, or metropolitan region), it is unlikely that most pollutants would manage
to penetrate higher than about 500 to 2000 m during the time the air resides over
the region. It therefore may be appropriate to reduce the height of the atmosphere
to 500 to 2000 m in such cases. In extreme cases (e.g., over small ponds or fields),
the accessible mixed height of the atmosphere may be as low as 10 m. The modeler
must make a judgement as to the volume of air that is accessible to the chemical
during the time that the air resides in the region of interest.
 
4.2.2 Aerosols
 
The atmosphere contains a considerable amount of particulate matter or aerosols
 
that
 
 are important in determining the fate of certain chemicals. These particles may
range in size and composition from water in the form of fog or cloud droplets to
dust particles from soil and smoke from combustion. They vary greatly in size, but
a diameter of a few 
 
µ
 
m is typical. Larger particles tend to deposit fairly rapidly. The
concentration of these aerosols is normally reported in 
 
µ
 
g/m
 
3
 
. A rural area may have
a concentration of about 5 
 
µ
 
g/m
 
3
 
, and a fairly polluted urban area a concentration
of 100 
 
µ
 
g/m
 
3
 
. For illustrative purposes, we can assume that the particles have a
density of 1.5 g/cm
 
3
 
 and are present at a concentration of 30 
 
µ
 
g/m
 
3
 
. This corresponds
to volume fraction of particles of 2 
 
×
 
 10
 
–11
 
. The density of these particles is usually
unknown, thus the volume fractions are only estimates. It is, however, convenient
for us to calculate this amount in the form of a volume fraction. In an evaluative air
volume of 6 
 
×
 
 10
 
9
 
 m
 
3
 
, there is thus 0.12 m
 
3
 
 or 120 L of solid material.
 
These aerosols are derived from numerous sources. Some are mineral dust
particles generated from soils by wind or human activity. Some are mainly organic
in nature, being derived from combustion sources such as vehicle exhaust or wood
fires, i.e., smoke. Some are generated from oxides of sulfur and nitrogen. Some
“secondary” aerosols are formed by condensation as a result of oxidation of hydro-
carbons in the atmosphere to less volatile species. These hydrocarbons can be
generated by human activity such as fuel use, or they can be of natural origin. Forests
often generate large quantities of isoprene that oxidize to give a blue haze, hence
the terms “smokey” or “blue” mountains. These aerosols also contain quantities of
water, the amount of which depends on the prevailing humidity.
 
4.2.3 Deposition Processes
 
Aerosol particles have a very high surface area and thus absorb (or adsorb or
sorb) many pollutants, especially those of very low vapor pressure, such as the PCBs
or polyaromatic hydrocarbons. In the case of benzo(a)pyrene, almost all the chemical
present in the atmosphere is associated with particles, and very little exists in the
gas phase. This is important, because chemicals associated with aerosol particles
 
CH04 Page 57 Monday, January 15, 2001 1:49 PM
 
58 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
are subject to two important deposition processes. First is dry deposition, in which
the aerosol particle falls under the influence of gravity to the Earth’s surface. This
falling velocity, or deposition velocity, is quite slow and depends on the turbulent
condition of the atmosphere, the size and properties of the aerosol particle, and the
nature of the ground surface, but a typical velocity is about 0.3 cm/s or 10.8 m/h.
The result is deposition of 10.8 m/h 
 
×
 
 2 
 
×
 
 10
 
–11
 
 (volume fraction) 
 
×
 
 10
 
6
 
 m
 
2
 
 or
0.000216 m
 
3
 
/h or 1.89 m
 
3
 
/year. Second, the particles may be scavenged or swept
out of the air by wet deposition with raindrops. As it falls, each raindrop sweeps
through a volume of air about 200,000 times its volume prior to landing on the
surface. Thus, it has the potential to remove a considerable quantity of aerosol from
the atmosphere. Rain is therefore often highly contaminated with substances such
as PCBs and PAHs. There is a common fallacy that rain water is pure. In reality, it
is often much more contaminated than surface water. Typical rainfall rates lie in the
range 0.3 to 1 m per year but, of course, vary greatly with climate. We adopt a figure
of 0.8 m/year for illustrative purposes. This results in the scavenging of 200,000 
 
×
 
0.8 m/year 
 
×
 
 2 
 
×
 
 10
 
–11
 
 
 
×
 
 10
 
6
 
 m
 
2
 
 or 3.2 m
 
3
 
/year, about twice the dry deposition. Snow
is an even more efficient scavenger of aerosol particles. It appears that one volume
of snow (as solid ice) may scavenge about one million volumes of atmosphere, five
times more than rain, presumably because of its flaky nature with a high surface
area and a slower, more tortuous downward journey.
In the four-compartment evaluative environment, we ignore aerosols, but we
include them in the eight-compartment version.
 
4.3 THE HYDROSPHERE OR WATER
4.3.1 Water
 
Seventy percent of the Earth’s surface is covered by water. In some evaluative
models, the area of water is taken as 70% of the 1 million m
 
2
 
 or 700,000 m
 
2
 
. Similarly
to the atmosphere, only near-surface water is accessible to pollutants in the short
term. In the oceans, this depth is about 100 m but, since most situations of environ-
mental interest involve fresh or estuarine water, it is more appropriate to use a
shallower water depth of perhaps 10 m. This yields a water volume of about 7 
 
×
 
10
 
6
 
 m
 
3
 
. If the aim is to mimic the proportions of water and soil in a political
jurisdiction, such as a state or province, the area of water will normally be consid-
erably reduced to perhaps 10% of the total, or about 10
 
6
 
 m
 
3
 
. We normally regard
the water as being pure, i.e., containing no dissolved electrolytes, but we do treat
its content of suspended particles.
 
4.3.2 Particulate Matter
 
Particulate matter in the water plays a key role in influencing the behavior of
chemicals. Again, we do not normally know if the chemical is absorbed or adsorbed
to the particles. We play it safe and use the vague term 
 
sorbed
 
. A very clear natural
water may have a concentration of particles as low as 1 g/m
 
3
 
 or the equivalent 1 mg/L.
 
CH04 Page 58 Monday, January 15, 2001 1:49 PM
 
THE NATURE OF ENVIRONMENTAL MEDIA 59
 
In most cases, however, the concentration is higher, in the range of 5 to 20 g/m
 
3
 
.
Very turbid, muddy waters may have concentrations over 100 g/m
 
3
 
. Assuming a
concentration of 7.5 g/m
 
3
 
 and a density of 1.5 g/cm
 
3
 
 gives a volume fraction of
particles of about 5 
 
×
 
 10
 
-6
 
. Thus, in the 7 
 
×
 
 10
 
6
 
 m
 
3
 
 of water, there is 35 m
 
3
 
 of particles.
This particulate matter consists of a wide variety of materials. It contains mineral
matter, which may be clay or silica in nature. It also contains dead or detrital organic
matter, which is often referred to as 
 
humin, humic acids,
 
 and 
 
fulvic acids
 
 or, more
vaguely, as 
 
organic matter.
 
 It is relatively easy to measure the total concentrationof organic carbon (OC) in water or particles by converting the carbon to carbon
dioxide and measuring the amount spectroscopically. Alternatively, the solids can
be dried to remove water, then heated to ignition temperatures to burn off organic
matter. The loss is referred to as 
 
loss on ignition (LOI)
 
 or as 
 
organic matter (OM).
 
Thus, there are frequent reports of the amount of dissolved organic carbon (DOC)
or total organic carbon (TOC) in water. These humic and fulvic acids have been the
subject of intense study for many years. They are organic materials of variable
composition that probably originate from the ligneous material present in vegetation.
They contain a variety of chemical structures including substituted alkane, cycloal-
kane, and aromatic groups, and they have acidic properties imparted by phenolic or
carboxylic acids. They are, therefore, fairly soluble in alkaline solution in which
they are present in ionic form, but they may be precipitated under acidic conditions.
The operational difference between humic and fulvic acids is the pH at which
precipitation occurs.
It is important to discriminate between organic matter (OM) and organic carbon
(OC). Typically, OM contains 50 to 60% OC, thus an OM analysis of 10% may also
be 5% OC. A mass basis, i.e., g/100 g, is commonly used. For convenience in our
evaluative calculations, we will treat OM as 50% OC, and we will assume the density
of both OM and OC as being equal to that of water.
 
Concentrations of these suspended materials may be defined operationally by
using filters of various pore size, for example, 0.45 
 
µ
 
m. There is a tendency to
describe material that is smaller than this, i.e., that passes through the filter, as being
operationally “dissolved.” It is not clear how we can best discriminate between
“dissolved” and “particulate” forms of such material, since there is presumably a
continuous size spectrum ranging from molecules of a few nanometres to relatively
large particles of 100 or 1000 nm. It transpires that the organic material in the
suspended phases is of great importance, because it has a high sorptive capacity for
organic chemicals. It is therefore common to assign an organic carbon content to
these phases. In a fairly productive lake, the OM content may be as high as 50%
but, for illustrative purposes, a figure of 33% for OM or 16.7% OC is convenient.
In each cubic metre of water, there is thus 2.5 g or cm
 
3
 
 of OM and 5.0 g or 2.5 cm
 
3
 
of mineral matter, totaling 7.5 g or 5.0 cm
 
3
 
, giving an average particle density of
1.5 g/cm
 
3
 
.
 
4.3.3 Fish and Aquatic Biota
 
Fish are of particular interest, because they are of commercial and recreational
importance to users of water, and they tend to bioconcentrate or bioaccumulate
 
CH04 Page 59 Monday, January 15, 2001 1:49 PM
 
60 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
metals and organic chemicals from water. They are thus convenient monitors of the
contamination status of lakes. This raises the question, “What is the volume fraction
of fish in a lake?” Most anglers and even aquatic biologists greatly overestimate this
number. It is probably, in most cases, in the region of 10
 
–8
 
 to 10
 
–9
 
, but this is somewhat
misleading, because most of the biotic material in a lake is not fish—it is material
of lower trophic levels, on which fish feed. For illustrative purposes, we can assume
that all the biotic material in the water is fish, and the total concentration is about
1 part per million, yielding a volume of “fish” of about 7 m
 
3
 
. It proves useful later
to define a lipid or fat content of fish, a figure of 5% by volume being typical.
In summary, the water thus consists of 7 
 
×
 
 10
 
6
 
 m
 
3
 
 of water containing 35 m
 
3
 
 of
particulate matter and 7 m
 
3
 
 of “fish” or biota.
In shallow or near-shore water, there may be a considerable quantity of aquatic
plants or macrophytes. These plants provide a substrate for a thriving microbial
community, and they possess inherent sorptive capacity. Their importance is usually
underestimated. Because of the present limited ability to quantify their sorptive
properties, we ignore them here.
 
4.3.4 Deposition Processes
 
The particulate matter in water is important, because, like aerosols in the atmo-
sphere, it serves as a vehicle for the transport of chemical from the bulk of the water
to the bottom sediments. Hydrophobic substances tend to partition appreciably on
to the particles and are thus subject to fairly rapid deposition. This deposition velocity
is typically 0.5 to 2.0 m per day or 0.02 to 0.08 m/h. This velocity is sufficient to
cause removal of most of the suspended matter from most lakes during the course
of a year. Thus, under ice-covered lakes in the winter, the water may clarify. Some
of the deposited particulate matter is resuspended from the bottom sediment through
the action of currents, storms, and the disturbances caused by bottom-dwelling fish
and invertebrates. During the summer, there is considerable photosynthetic fixation
of carbon by algae, resulting in the formation of considerable quantities of organic
carbon in the water column. Much of this is destined to fall to the bottom of the
lake, but much is degraded by microorganisms within the water column.
Assuming, as discussed earlier, 5 
 
×
 
 10
 
–6
 
 m
 
3
 
 of particles per m
 
3
 
 of water and a
deposition velocity of 200 m per year, we arrive at a deposition rate of 0.001 m
 
3
 
/m
 
2
 
of sediment area per year or, for an area of 7 
 
×
 
 10
 
5
 
 m
 
2
 
, a flow of 700 m
 
3
 
/year. We
examine this rate in more detail in the next section.
 
4.4 BOTTOM SEDIMENTS
4.4.1 Sediment Solids
 
Inspection of the state of the bottom of lakes reveals that there is a fairly fluffy
or nepheloid active layer at the water–sediment interface. This layer typically con-
sists of 95% water and 5% particles and is often highly organic in nature. It may
consist of deposited particles and fecal material from the water column. It is stirred
 
CH04 Page 60 Monday, January 15, 2001 1:49 PM
 
THE NATURE OF ENVIRONMENTAL MEDIA 61
 
by currents and by the action of the various biota present in this 
 
benthic
 
 region. The
sediment becomes more consolidated at greater depths, and the water content tends
to drop toward 50%. The top few centimetres of sediment are occupied by burrowing
organisms that feed on the organic matter (and on each other) and generally turn
over (bioturbate) this entire “active layer” of sediment. Depending on the condition
of the water column above, this layer may be oxygenated (aerobic or oxic) or depleted
of oxygen (anaerobic or anoxic). This has profound implications for the fate of
inorganic substances such as metals and arsenic, but it is relatively unimportant for
organic chemicals except in that the oxygen status influences the nature of the
microbial community, which in turn influences the availability of metabolic pathways
for chemical degradation. The deeper sediments are less accessible, and ultimately
the material becomes almost completely buried and inaccessible to the aquatic
environment above. Most of the activity occurs in the top 5 cm of the sediment, but
it is misleading to assume that sediments deeper than this are not accessible. There
remains a possibility of bioturbation or diffusion reintroducing chemical to the water
column.
Bottom sediments are difficult to investigate, can be unpleasant, and have little
or no commercial value. They are therefore often ignored. This is unfortunate,
because they serve as the depositories for much of the toxic material discharged into
water. They are thus very important, are valuable as a “sink” for contaminants, and
merit more sympathy and attention.
Fast-flowing rivers are normally sufficiently turbulent that the bottom is scoured,
exposing rock or consolidated mineral matter.Thus, their sediments tend to be less
important. Sluggish rivers have appreciable sediments.
 
4.4.2 Deposition, Resuspension, and Burial
 
It is possible to estimate the rate of deposition, i.e., the amount of material that
falls annually to the bottom of the lake and is retained there. This can be done by
sediment traps, which are essentially trays that collect falling particles, or by taking
a sediment core and assigning dates to it at various depths using concentrations of
various radioactive metals such as lead. Nuclear events provide convenient dating
markers for sediment depths. The measurement of deposition is complicated by the
presence of the reverse process of resuspension caused by currents and biotic activity.
It is difficult to measure how much material is rising and falling, since much may
be merely cycling up and down in the water column. Burial or net deposition rates
vary enormously, but a figure of about 1 mm per year is typical. Much of this is
water, which is trapped in the burial process.
Chemicals present in sediments are primarily removed by degradation, burial,
or resuspension back to the water column. 
For illustrative purposes we adopt a sediment depth of 3 cm and suggest that it
consists of 67% water and 33% solids, and these solids consist of about 10% organic
matter or 5% organic carbon. Living creatures are included in this figure. Some of
this deposited material is resuspended to the water column, some of the organic
matter is degraded (i.e., used as a source of energy by benthic or bottom-living
organisms), and some is destined to be permanently buried. The low 5% organic
 
CH04 Page 61 Monday, January 15, 2001 1:49 PM
 
62 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
carbon figure for deeper sediments compared to high 17% for the depositing material
implies that about 75% of the organic carbon is degraded.
It is now possible to assemble an approximate mass balance for the sediment
mineral matter (MM) and organic matter (OM) and thus the organic carbon (OC).
This is given in Table 4.1.
On a 1 m
 
2
 
 basis, the deposition rate is 0.001 m
 
3
 
 per year or 1000 cm
 
3
 
 per year.
With a particle density of 1.7 g/cm
 
3
 
, this corresponds to 1700 g/year of which 500 g
is OM, and 1200 g is MM. We assume that 40% of this is resuspended, i.e., 200 g
of OM and 480 g of MM. Of the remaining 300 g OM, we assume that 233 g is
digested or degraded to CO
 
2
 
, and 67 g is buried along with the remaining 720 g of
MM. Total burial is thus 1420 g, which consists of 720 g of MM, 67 g of OM, and
633 g of water. The total volumetric burial rate of solids is 367 cm
 
3
 
/year. Now,
associated with these solids is 633 cm
 
3
 
 of pore water; thus, the total volumetric
burial rate of solids plus water is approximately 1000 cm
 
3
 
/year, corresponding to a
rise in the sediment-water interface of 1 mm/year. The mass percentage of OC in
the depositing and resuspending material is 15%, while in the buried material it is
4.2%. The bulk sediment density, including pore water, is 1420 kg/m
 
3
 
.
On a 7 
 
×
 
 10
 
5
 
 m
 
2
 
 basis, the deposition rate is 700 m
 
3
 
/year, resuspension is
280 m
 
3/year, burial is 257 m3/year, and degradation accounts for the remaining
163 m3/year. The organic and mineral matter balances are thus fairly complicated,
but it is important to define them, because they control the fate of many hydrophobic
chemicals.
It is noteworthy that the burial rate of 1 mm/year coupled with the sediment
depth of 3 cm indicates that, on average, it will take 30 years for sediment solids
to become buried. During this time, they may continue to release sorbed chemical
back to the water column. This is the crux of the “in-place contaminated sediments”
problem, which is unfortunately very common, especially in the Great Lakes Basin.
In the simple four-compartment environment, we treat only the solids but, in the
eight-compartment version, we include the sediment pore water. In the interests of
simplicity, we assign a density of 1500 kg/m3 to the sediment in the four-compart-
ment model.
Table 4.1 Illustrative Sediment–Water Mass Balance on a 1 m2 Area Basis
Mineral matter Organic matter Total
Organic 
carbon
cm3 g cm3 g cm3 g g
Deposition 500 1200 500 500 1000 1700 250
Resuspension 200 480 200 200 400 680 100
OM conversion – – 233 233 233 233 117
Burial (solids) 300 720 67 67 367 787 33
Total burial is 1000 cm3/year or 1420 g/year, corresponding to a “velocity” of 1 mm/year.
The sediment thus has a density of 1.42 g/cm3 or 1420 kg/m3.
Assumed densities are: mineral matter 2.4 g/cm3, organic matter 1 g/cm3.
Organic matter is 50% (mass) organic carbon.
CH04 Page 62 Monday, January 15, 2001 1:49 PM
THE NATURE OF ENVIRONMENTAL MEDIA 63
4.5 SOILS
4.5.1 The Nature of Soil
Soil is a complex organic matrix consisting of air, water, mineral matter (notably
clay and silica), and organic matter, which is similar in general nature to the organic
matter discussed earlier for the water column.
The surface soil is subject to diurnal and seasonal temperature changes and to
marked variations in water content, and thus in air content. At times it may be
completely flooded, and at other times almost completely dry. The organic matter
in the soil plays a crucial role in controlling the retention of the water and thus in
ensuring the viability of plants. The organic matter content is typically 1 to 5%, but
peat soils and forest soils can have much higher organic matter contents. Depletion
of organic matter through excessive agriculture tends to render the soil infertile,
which is an issue of great concern in agricultural regions. Soils vary enormously in
their composition and texture and consist of various layers, or horizons, of different
properties. There is transport vertically and horizontally by diffusion in air and in
water, flow, or advection in water and, of course, movement of water and nutrients
into plant roots and thence into stems and foliage. Burrowing animals such as worms
can also play an important role in mixing and transporting chemicals in soils. 
A typical soil may consist of 50% solid matter, 20% air, and 30% water, by
volume. The dry soil thus has a porosity of 50%. The solid matter may consist of
about 2% organic carbon or 4% organic matter. During and after rainfall, water flows
vertically downward through the soil and may carry chemicals with it. During periods
of dry weather, water tends to return to the surface by capillary action, again moving
the chemicals.
Later, we set up equations describing the diffusion or permeation of chemicals
in soils. When doing so, we treat the soil as having a constant porosity. In reality,
there are channels or “macroporous” areas formed by burrowing animals and
decayed roots, and these enable water and air to flow rapidly through the soil,
bypassing the more tightly packed soil matrix. This phenomenon is very difficult to
address when compiling models of transport in soils and is the source of considerable
frustration to soil scientists.
Most soils are, of course, covered with vegetation, which stabilizes the soil and
prevents it from being eroded by wind or water action. Under dry conditions, with
poor vegetation cover, considerable quantities of soil can be eroded by wind action,
carrying with it sorbed chemicals. Sand dunes are an extreme example. In populated
regions, of more concern is the loss of soil in water runoff. This water often contains
very high concentrations of soil, perhaps as much as a volume fraction of 1 part per
thousand of solid material. This serves as a vehicle for the movement of chemicals,
especially agricultural chemicals such as pesticides, from the soils into water bodies
such as lakes. 
4.5.2 Transport in Soils
In most areas, there is a net movement of water vertically from the surface soil to
greater depths into a pervious layer of rock or aquifer through which groundwater
CH04 Page 63 Monday,January 15, 2001 1:49 PM
64 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
flows. The quality of this groundwater has become of considerable concern recently,
especially to those who rely on wells for their water supply. This water tends to move
very slowly (i.e., at a velocity of metres per year) through the porous sub-surface strata.
If contaminated, it can take decades or even centuries to recover. Of particular concern
are regions in which chemical leachate from dumps or landfills has seeped into the
groundwater and has migrated some distance into rivers, wells, or lakes. It is quite
difficult and expensive to investigate, sample, and measure contaminant flow in ground-
water. It may not even be clear in which direction the water is flowing or how fast it
is flowing. Chemicals associated with groundwater generally move more slowly than
the velocity of the groundwater. They are retarded by sorption to the soil to an extent
expressed as a “retardation factor,” which is essentially the ratio of (a) the amount of
chemical that is sorbed to the solid matrix to (b) the amount that is in solution. Sorption
of organic chemicals is usually accomplished preferentially to organic matter; however,
clays also have considerable sorptive capacity, especially when dry. Polar, and espe-
cially ionic, substances may interact strongly with mineral matter. The characterization
of migration of chemicals in groundwater is difficult, and especially so when a chemical
is present in an non-aqueous phase, for example, as a bulk oil or emulsified oil phase.
Considerable effort has been devoted to understanding the fate of nonaqueous phase
liquids (NAPLs) such as oils, and dense NAPLs (DNAPLs) such as chlorinated sol-
vents that can sink in the aquifer and are very difficult to recover.
For illustrative purposes, we treat the soil as covering an area 1000 m × 300 m
× 15 cm deep, which is about the depth to which agricultural soils are plowed. This
yields a volume of 45,000 m3. This consists of about 50% solids, of which 4% is
organic matter content or 2% by mass organic carbon. The porosity of the soil, or
void space, is 50% and consists of 20% air and 30% water. Assuming a density of
the soil solids of 2400 kg/m3 and water of 1000 kg/m3 gives masses of 1200 kg
solids and 300 kg water per m3 (and a negligible 0.2 kg air), totaling 1500 kg,
corresponding to a bulk density of 1500 kg/m3. Rainwater falls on this soil at a rate
of 0.8 m per year, i.e., 0.8 m3/m2 year. Of this, perhaps 0.3 m evaporates, 0.3 m runs
off, and 0.2 m percolates to depths and contributes to groundwater flow. This results
in water flows of 90,000 m3/year by evaporation, 90,000 m3/year by runoff, and
60,000 m3/year by percolation to depths totaling 240,000 m3/year. With the runoff
is associated 90 m3/year of solids, i.e., an assumed high concentration of 0.1% by
volume. Again, it must be emphasized that these numbers are entirely illustrative.
This soil runoff rate of 90 m3/year does not correspond to the deposition rate of
700 m3/year, partly because of the contribution of organic matter generated in the
water column, but mainly because of the low ratio of soil area to water area.
4.5.3 Terrestrial Vegetation
Until recently, most environmental models have ignored terrestrial vegetation.
The reason for this is not that vegetation is unimportant, but rather that modelers
currently have enormous difficulty calculating the partitioning of chemicals into
plants. This topic is receiving more attention as a result of the realization that
consumption of contaminated vegetation, either by humans, domestic animals, or
wildlife, is a major route or vector for the transfer of toxic chemicals from one
CH04 Page 64 Monday, January 15, 2001 1:49 PM
THE NATURE OF ENVIRONMENTAL MEDIA 65
species to another, and ultimately to humans. Plants play a critical role in stabilizing
soils and in inducing water movement from soil to the atmosphere, and they may
serve as collectors and recipients of toxic chemicals deposited or absorbed from the
atmosphere. They can also degrade certain chemicals and increase the level of
microbial activity in the root zone, thus increasing the degradation rate in the soil.
Amounts of vegetation, in terms of quantity of biomass per square metre, vary
enormously from near zero in deserts to massive quantities that greatly exceed
accessible soil volumes in tropical rain forests. They also vary seasonally. If it is
desired to include vegetation, a typical “depth” of plant biomass might be 1 cm.
This, of course, consists mainly of water, cellulose, starch, and ligneous material.
There is little doubt that future, more sophisticated models will include chemical
partitioning behavior into plants. But at the present state of the art, it is convenient
(and rather unsatisfactory) to regard the plants as having a volume of 3000 m3,
containing the equivalent of 1% lipid-like material and 50% water.
We ignore vegetation in the simple four-compartment model, treating the soil as
only a simple solid phase.
4.6 SUMMARY
These evaluative volumes, areas, compositions, and flow rates are summarized
in Table 4.2. From them is derived a simple four-compartment version. Also sug-
gested is an alternative environment that is more terrestrial and less aquatic, and it
reflects more faithfully a typical political jurisdiction. It is emphasized again that
the quantities are purely illustrative, and site-specific values may be quite different.
All that is needed at this stage is a reasonable basis for calculation.
Scientists who have devoted their lives to studying the intricacies of the structure,
composition, and processes of the atmosphere, hydrosphere, or lithosphere will
undoubtedly be offended at the simplistic approach taken in this chapter. The envi-
ronment is very complex, and it is essential to probe the fine detail present in its
many compartments. But, if we are to attempt broad calculations of multimedia
chemical fate, we must suppress much of the media-specific detail. When the broad
patterns of chemical behavior are established, it may be appropriate to revisit the
media that are important for that chemical and focus on detailed behavior in a specific
medium. At that time, a more detailed and site-specific description of the medium
of interest will be justified and required.
Our philosophy is that the model should be only as complex as is required to
answer the immediate question, not every question that could be asked. As questions
are answered, new questions will surface and new, more complex models can be
developed to answer these questions.
4.7 CONCLUDING EXAMPLE
Select a region with which you are familiar; for example, a county, watershed,
state, or province. Calculate the volumes of air to a height of 1000 m; soil to a depth
CH04 Page 65 Monday, January 15, 2001 1:49 PM
66 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Table 4.2 Evaluative Environments
A. Four-compartment, 1 km2 environment
Areas (m2)
Air–water 7 × 105
Air–soil 3 × 105
Water–sediment 7 × 105
Depths (m) Volumes (m3) Densities (kg/m3) Compositions
Air 6000 6 × 109 1.2
Water 10 7 × 106 1000
Soil 0.15 4.5 × 104 1500 2% OC
Sediment 0.03 2.1 × 104 1500 5% OC
B. Eight-compartment, 1 km2 environment, areas as in A above
Volumes 
(m3)
Densities 
(kg/m3) Compositions
Air 6 × 109 1.2 Air
Water 7 × 106 1000 Water
Soil (50% solids, 
20% air, 30% 
water)
4.5 × 104 1500 Soil (50% solids, 20% air, 30% water)
Sediment (30% 
solids)
2.1 × 104 1500 Sediment (30% solids)
Suspended 
Sediment
35 1500 16.7% OC
Aerosols 0.12 1500 2 × 10–11 volume fraction or 30 µg/m3
Aquatic Biota 7 1000 5% lipid
Vegetation 3000 1000 1% lipid
Rain Rate 0.8 m/year or 800,000 m3/year
560,000 m3 to water; 240,000 m3 to soil
Aerosol Deposition Rates (total)
Dry deposition 216 × 10–6 m3 /h or 1.89 m3 /year
Wet deposition 365 × 10–6 m3 /h or 3.2 m3 /year
Sediment Deposition Rates
Deposition 700 m3 /year solids 17% OC
Resuspension 280 m3 /year solids 17% OC
Netdeposition (burial) 257 m3 /year solids 5% OC
Fate of Water in Soil
Evaporation 90,000 m3 /year
Runoff to water 90,000 m3 /year
Percolation to groundwater 60,000 m3 /year
Solids runoff 90 m3 /year
CH04 Page 66 Monday, January 15, 2001 1:49 PM
THE NATURE OF ENVIRONMENTAL MEDIA 67
of 10 cm; water and bottom sediment to a depth of 3 cm, and vegetation. Obtain
data on average temperature, rain rate, water flows, and wind velocity, and calculate
air and water residence times. Attempt to obtain information on typical concentra-
tions of aerosols, suspended solids in water, and the organic carbon contents of soils,
bottom, and suspended sediments. Prepare a summary table of these data similar to
Table 4.2.
These basic environmental data can be used in subsequent assessments of the
fate of chemicals in this region.
Table 4.2 (continued)
C. Regional, 100,000 km2 environment as used in the EQC model of Mackay et al. 
(1996b)
Volume (m3) Area (m2) Composition
Air 1014 100 × 109
Aerosols 2000 – (2 × 10–11 vol frn)
Water 2 × 1011 10 × 109
Soil 9 × 109 90 × 109 2% OC
Sediment 108 10 × 109 4% OC
Suspended sediment 106 – 20% OC
Fish 2 × 105 – 5% lipid
For details of other properties see Mackay et al. 1996b.
CH04 Page 67 Monday, January 15, 2001 1:49 PM
CH04 Page 68 Monday, January 15, 2001 1:49 PM
 
69
 
CHAPTER
 
 5
Phase Equilibrium
 
5.1 INTRODUCTION
5.1.1 The Nature of Partitioning Phenomena
 
There are two distinct tasks that must be addressed when predicting equilibrium
partitioning of chemicals in the environment. First, we must fully understand how
chemicals behave under ideal, laboratory conditions of controlled temperature and
well defined, pure phases. This is the task of physical chemistry. Second is the
translation of these partitioning data into the more complex and less defined condi-
tions of the environment where phases vary in composition and properties.
In both cases, we are concerned with the equilibrium distribution of a chemical
between phases as illustrated in the simple two-compartment system of Figure 5.1.
A small volume of nonaqueous phase (e.g., a particle of organic or mineral matter,
a fish, or an air bubble) is introduced into water that contains a dissolved chemical
such as benzene. There is a tendency for some of the benzene to migrate into this
new phase and establish a concentration that is some multiple of that in the water.
In the case of organic particles, the multiple may be 100 or, if the phase is air, the
multiple may be only 0.2. Equilibrium becomes established in hours or days between
the benzene dissolved in the water and the benzene in, or on, the nonaqueous phase.
Analytical measurements may give the total or average concentration that includes
the nonaqueous phase and may differ considerably from the actual dissolved water
concentration. The phase may subsequently settle to the lake bottom or rise to the
surface, conveying benzene with it. Clearly, it is essential to establish the capability
of calculating these concentrations and thus the fractions of the total amount of
benzene that remain in the water, and enter the second phase. In some cases, 95%
of the benzene may migrate into the phase, and in others only 5%. These systems
will behave quite differently.
The aim is to answer the question, “Given a concentration in one phase, what
will be the concentration in another phase that has been in contact with it long
enough to achieve equilibrium?” This task is part of the science of thermodynamics
 
CH05 Page 69 Monday, January 15, 2001 1:49 PM
 
70 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
that is fully described in several excellent texts such as those of Denbigh (1966),
Van Ness and Abbott (1982), Prausnitz et al. (1969), and for aquatic environmental
systems by Stumm and Morgan (1981) and Pankow (1991). It is assumed here that
the reader is familiar with the general principles of thermodynamics; therefore, no
attempt is made to derive all the equations. The aim is rather to extract from the
science of thermodynamics those parts that are pertinent to environmental chemical
equilibria and explain their source, significance, and applications.
Figure 5.1 Some principles and concept in phase equilibrium.
 
CH05 Page 70 Monday, January 15, 2001 1:49 PM
 
PHASE EQUILIBRIUM 71
 
Environmental thermodynamics or phase equilibrium physical chemistry applies
to a relatively narrow range of conditions. Tropospheric or surface temperatures
range only between –40° and 
 
+
 
40°C and usually between the narrower limits of 0°
and 25°C. Total pressures are almost invariably atmospheric but, of course, with an
additional hydrostatic pressure at lake or ocean bottoms. Concentrations of chemical
contaminants are (fortunately) usually low. Situations in which the concentration is
high (as in spills of oil or chemicals) are best treated separately. These limited ranges
are fortunate in that they simplify the equations and permit us to ignore large and
complex areas of thermodynamics that deal with high and low pressures and tem-
peratures, and with high concentrations.
The presence of a chemical in the environment rarely affects the overall dominant
structure, processes, and properties of the environment; therefore, we can take the
environment “as is” and explore the behavior of chemicals in it with little fear of
the environment being changed in the short term as a result. There are, however,
certain notable exceptions, particularly when the biosphere (which can be signifi-
cantly altered by chemicals) plays an important role in determining the landscape.
An example is the stabilizing influence of vegetation on soils. Another is the role
of depositing carbon of photosynthetic origin in lakes.
A point worth emphasizing is that thermodynamics is based on a few fundamental
“laws” or axioms from which an assembly of equations can be derived that relate
certain useful properties to each other. Examples are the relationship between vapor
pressure and enthalpy of vaporization, or concentration and partial pressure. In some
cases, the role of thermodynamics is simply to suggest suitable relationships. Ther-
modynamics never defines the actual value of a property such as the boiling point
of benzene; such data must be obtained experimentally. We thus process experimental
data using thermodynamic relationships. Despite its name, thermodynamics is not
concerned with process rates; indeed, none of the equations derived in this chapter
need contain time as a dimension. 
It transpires that two approaches can be used to develop equations relating
equilibrium concentrations to each other as shown in Figure 5.1. The simpler and
most widely used is Nernst’s Distribution law, which postulates that the concentration
ratio C
 
1
 
/C
 
2
 
 is relatively constant and is equal to a partition or distribution coefficient
K
 
12
 
. Thus, C
 
2
 
 can be calculated as C
 
1
 
K
 
12
 
. K
 
12
 
 presumably can be expressed as a
function of temperature and, if necessary, of concentration. Experimentally, mixtures
are equilibrated, and concentrations measured and plotted as in Figure 5.1. Linear
or nonlinear equations then can be fitted to the data. The second approach involves
the introduction of an intermediate quantity, a criterion of equilibrium, which can
be related separately to C
 
1
 
 and C
 
2
 
. Chemical potential, fugacity, and activity are
suitable criteria, with fugacity being preferred for most organic substances because
of the simplicity of the equations that relate fugacity to concentration. The advantage
of the equilibrium criterion approach is that properties of each phase are treated
separately using a phase-specific equation. Treating phases in pairs, as is done with
partition coefficients, can obscure the nature of the underlying phenomena. We may
detect a variability in K
 
12
 
 and not know from which phase the variability is derived.
Further complications ariseif we have 10 phases to consider. There are then 90
possible partition coefficients, of which only 9 are independent. Mistakes are less
 
CH05 Page 71 Monday, January 15, 2001 1:49 PM
 
72 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
likely using an equilibrium criterion and the 10 equations relating it to concentration,
one for each phase.
It is useful to discriminate between partition coefficients and distribution coef-
ficients. Although usage varies, a partition coefficient is strictly the ratio of the
concentrations of the same chemical species in two phases. A distribution coefficient
is a ratio of total concentrations of all species. Thus, if a chemical ionizes, the
partition coefficient may apply to the unionized species, while the distribution
coefficient applies to ionized and nonionized species in total.
 
5.1.2 Some Thermodynamic Fundamentals
 
There are four laws of thermodynamics. They are numbered 0, 1, 2, and 3,
because the need for the zeroth was not realized until after the first was postulated.
Although these laws cannot be proved mathematically, they are now universally
accepted as true, or axiomatic, because they are supported by all available experi-
mental evidence. On consideration, they are intuitively reasonable, and it now seems
inconceivable that they are ever disobeyed.
The zeroth law introduces the concept of temperature as a criterion of thermal
equilibrium by stating that, when bodies are at thermal equilibrium, i.e., there is no
net heat flow in either direction, their temperatures are equal.
The first law was discovered largely as a result of careful experiments by Joule,
and it establishes the concept of energy and its conservation. Energy takes several
forms—potential, kinetic, heat, chemical, electrical, nuclear, and electromagnetic.
There are fixed conversion rates among these forms. Furthermore, energy can neither
be formed nor destroyed; it merely changes its form. Of particular importance are
conversions between thermal energy (heat) and mechanical energy (work).
The second law is intellectually more demanding and introduces the concept of
entropy and a series of useful related properties, including chemical potential and
fugacity. It is observed that, whereas there are fixed exchange rates between heat and
work energy, it is not always possible to effect the change. The conversion of mechan-
ical energy to heat (as in an automobile brake) is always easy, but the reverse process
of converting heat to mechanical energy (as in a thermal power station) proves to be
more difficult. If a quantity of heat is available at high temperature, then only a fraction
of it, perhaps one third, can be converted into mechanical energy. The remainder is
rejected as heat, but at a lower temperature. Most thermodynamics texts introduce
hypothetical processes such as the Carnot cycle at this stage to illustrate these con-
versions. After some manipulation, it can be shown that there is a property of a system,
called its 
 
entropy,
 
 that controls these conversions. Apparently, regardless of how it is
arranged to convert heat to work, the overall entropy of the system cannot decrease.
It must increase by what is termed an 
 
irreversible process, 
 
or in the limit, it could
remain constant by what is called a 
 
reversible process. 
 
Although there may be a local
entropy decrease, this must be offset by another and greater entropy increase elsewhere.
Clausius summarized this law in the statement that the “entropy of the universe
increases.” It can be shown that entropy is related to randomness or probability. An
increase in entropy corresponds to a change to a more 
 
random
 
 or 
 
disordered
 
 or
 
probable
 
 condition. The third law is not important for our immediate purposes.
 
CH05 Page 72 Monday, January 15, 2001 1:49 PM
 
PHASE EQUILIBRIUM 73
 
We are concerned with systems in which a chemical migrates from phase to
phase. These phase changes involve input or output of energy, thus this energy
exchange can compensate for entropy loss or gain. It can be shown that, whereas
entropy maximization is the criterion of equilibrium for a system containing constant
energy at constant volume, the criterion at constant temperature and pressure (the
environmentally relevant condition) is minimization of the related function, the
Gibbs free energy, which serves to combine energy and entropy in a common
currency.
Return to the example presented in Figure 5.1, of benzene diffusing from water
into an air bubble and striving to achieve equilibrium. The basic concept is that, if
we start with a benzene concentration in the water and none in air, the free energy
of the system will decrease as benzene migrates from water to air, because the
increase in free energy associated with the rise in benzene concentration in the air
is less that of the decrease associated with benzene loss from the water. The process
is thus spontaneous and irreversible. Benzene continues to diffuse from water into
the air until it reaches a point at which the free energy increase in the air is exactly
matched by the free energy decrease in the water. At this point, the system comes
to rest or 
 
equilibrium
 
. Likewise, if the system started with a higher benzene con-
centration in the air phase and approached equilibrium, it would reach exactly the
same point of equilibrium with a particular ratio of concentrations in each phase.
The system thus seeks a minimum in free energy at which its derivative with
respect to moles of benzene is equal in both air and water phases. This derivative
is of such importance that it is called the 
 
chemical potential. 
 
The underlying principle
of phase equilibrium thermodynamics is that, when a solute such as benzene achieves
equilibrium between phases such as air, water, and fish, it seeks to establish an equal
chemical potential in all phases. The net diffusion flux will always be from high to
low chemical potential. Thus, we can use chemical potential for deductions of mass
diffusion in the same way that we use temperature in heat transfer calculations.
 
5.1.3 Fugacity
 
Unfortunately, chemical potential is logarithmically related to concentration,
thus doubling the concentration does not double the chemical potential. A further
complication is that a chemical potential cannot be measured absolutely, therefore
it is necessary to establish some standard state at which it has a reference value. It
was when addressing this problem that G.N. Lewis introduced a new equilibrium
criterion in 1901, which he termed 
 
fugacity,
 
 and which has units of pressure and is
assigned the symbol f. The term 
 
fugacity
 
 comes from the Latin root 
 
fugere,
 
 describ-
ing a “fleeing” or “escaping” tendency. It is identical to partial pressure in ideal
gases and is logarithmically related to chemical potential. It is thus linearly or nearly
linearly related to concentration. Absolute values can be established because, at low
partial pressures under ideal conditions, fugacity and partial pressure become equal.
Thus, we can replace the equilibrium criterion of chemical potential by that of
fugacity. When benzene migrates between water and air, it is seeking to establish
an equal fugacity in both phases; its escaping tendency, or pressures, are equal in
both phases.
 
CH05 Page 73 Monday, January 15, 2001 1:49 PM
 
74 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Another useful quantity is the ratio of fugacity to some reference fugacity such
as the vapor pressure of liquid benzene. This is a dimensionless quantity and is
termed 
 
activity
 
. Activity can also be used as an equilibrium criterion. This proves
to be preferable for substances such as ions, metals, or polymers that do not appre-
ciably evaporate and thus cannot establish vapor phase concentrations and partial
pressures.
Our task, then, is to start with a concentration of solute chemical in one phase,
from this deduce thechemical potential, fugacity, or activity, argue that these equi-
librium criteria will be equal in the other phase, and then calculate the corresponding
concentration in the second phase. We therefore require recipes for deducing C from
f and vice versa. This approach is depicted at the bottom of Figure 5.1.
The partition coefficient approach contains the inherent assumption that, what-
ever the factors are that are used to convert C
 
1
 
 to f
 
1
 
 and C
 
2
 
 to f
 
2
 
, the ratio of these
factors is constant over the range of concentration of interest. Thus, it is not actually
necessary to calculate the fugacities; their use is sidestepped. In the fugacity
approach, no such assumption is made, and the individual calculations are under-
taken. We can illustrate these approaches with an example.
 
Worked Example 5.1
 
Benzene is present in water at a specified temperature and a concentration C
 
1
 
of 1 mol/m
 
3
 
 (78 g/m
 
3
 
). What is the equilibrium concentration in air C
 
2
 
?
 
1. Partition coefficient approach
 
K
 
21
 
 is 0.2, i.e., C
 
2
 
/C
 
1
 
Therefore,
C
 
2
 
 = K
 
21
 
C
 
1
 
 = 0.2 
 
×
 
 1 = 0.2 mol/m
 
3
 
 = 15.6 g/m
 
3
 
1. Fugacity approach
 
Using techniques devised later, we find that, for water under these conditions,
f
 
1 
 
= C
 
1
 
/Z
 
1
 
 = C
 
1
 
/0.002 = 500 Pa = f
 
2
 
C
 
2 
 
= Z
 
2
 
f
 
2
 
 = 0.0004f
 
2
 
= 0.2 mol/m
 
3
 
 = 15.6 g/m
 
3
 
Clearly, the problem is to determine the conversion factors Z
 
2
 
 and Z
 
1
 
, or K
 
21
 
,
which is their ratio. Care must be taken to avoid confusing K
 
21
 
 with its reciprocal
K
 
12
 
 or C
 
1
 
/C
 
2
 
, which in this case has a value of 5.
We therefore face the task of developing methods of estimating Z values that
relate concentration and fugacity, and partition coefficients that are ratios of Z values.
The theoretical foundations are set out in Section 5.3 and result in a set of working
equations applicable to the air-water-octanol system. The three solubilities (or
 
CH05 Page 74 Monday, January 15, 2001 1:49 PM
 
PHASE EQUILIBRIUM 75
 
pseudo-solubilities) in these media and the three partition coefficients are then
discussed in more detail in Section 5.4. Armed with this knowledge we then address
how this “laboratory” information can be applied to environmental media such as
soils and aerosols.
 
5.2 PROPERTIES OF PURE SUBSTANCES
 
For reasons discussed later, it is important to ascertain if the substance of interest
is solid, liquid, or vapor at the environmental temperature. This is obviously done
by comparing this temperature with the melting and boiling points. Figure 5.2 is the
familiar P-T diagram that enables the state of a substance to be determined. Of
particular interest for solids is the supercooled liquid vapor pressure line, shown as
a dashed line. This is the vapor pressure that a solid (such as naphthalene, which
melts at 80°C) would have if it were liquid at 25°C. The reason it is not liquid at
25°C is that naphthalene is able to achieve a lower free energy state by forming a
crystal. Above 80°C, this lower energy state is not available, and the substance
remains liquid. Above the boiling point, the liquid state is abandoned in favor of a
vapor state. It is not possible to measure the supercooled liquid vapor pressure by
direct experiment. It can be calculated as discussed shortly, and it can be measured
Figure 5.2 P-T diagram for a pure substance.
 
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76 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
experimentally, but not directly, using gas chromatographic retention times. It is
possible to measure the vapor pressure above the boiling point by operating at high
pressures. Beyond the critical point, the vapor pressure cannot be measured, but it
can be estimated.
The triple-point temperature at which solid, liquid, and vapor phases coexist is
usually very close to the melting point at atmospheric pressure, because the solid-
liquid equilibrium line is nearly vertical; i.e., pressure has a negligible effect on
melting point. Melting point is easily measured for stable substances, and estima-
tion methods are available as reviewed by Tesconi and Yalkowsky (2000). High
melting points result from strong intermolecular bonds in the solid state and
symmetry of the molecule. Ice (H
 
2
 
0) has a high melting point compared to H
 
2
 
S
because of strong hydrogen bonding. The symmetrical three-ring compound
anthracene has a higher melting point (216°C) than the similar but unsymmetrical
phenanthrene (101°C).
The critical point temperature is of environmental interest only for gases, since
it is usually well above environmental temperatures. For example, it is 305 K for
ethane and 562 K for benzene. Its principal interest lies in its being the upper limit
for measurement of vapor pressure.
The location of the liquid-vapor equilibrium or vapor pressure line is very
important, since it establishes the volatility of the substances, as does the boiling
point, which is the temperature at which the vapor pressure equals 1 atmosphere.
Methods of estimating boiling point have been reviewed by Lyman (2000), and
methods of using boiling point to estimate vapor pressures at other temperatures
have been reviewed by Sage and Sage (2000). For many substances, correlations
exist for vapor pressure as a function of temperature. The simplest correlation is the
two-parameter Clapeyron equation,
ln P = A – B/T
 
A and B are constants, and T is absolute temperature (K). B is 
 
∆
 
H/R, where 
 
∆
 
H is
the enthalpy of vaporization (J/mol), and R is the gas constant. A better fit is obtained
with the three-parameter Antoine equation,
 
lnP = A – B/(T + C)
Care must be taken to check the units of P, whether base e or base 10 logs are used,
and whether T is K or °C in the Antoine equation. Several other equations are used
as reviewed by Reid et al. (1987).
Correlations also exist for the vapor pressure of solids and supercooled liquids.
Of particular environmental interest is the relationship between these vapor pres-
sures, which can be used to calculate the unmeasurable supercooled liquid vapor
pressure from that of the solid. The reason for this is that, when a solid such as
naphthalene is present in a dilute, subsaturated, dissolved, or sorbed state at 25°C,
the molecules do not encounter each other with sufficient frequency to form a
crystal. Thus, the low-energy crystal state is not accessible. The molecule thus
behaves as if it were a liquid at 25°C. It “thinks” it is a liquid, because it has no
 
CH05 Page 76 Monday, January 15, 2001 1:49 PM
 
PHASE EQUILIBRIUM 77
 
access to information about the stability of the crystalline state, i.e., does not
“know” its melting point. As a result, it behaves in a manner corresponding to the
liquid vapor pressure. A similar phenomenon occurs above the critical point where
a gas such as oxygen, when in solution in water, behaves as if it were a liquid at
25°C, not a gas. No liquid vapor pressure can be measured for either naphthalene
or oxygen at 25°C; it can only be calculated. Later, we term this liquid vapor
pressure the reference fugacity. We may need to know this fictitious vapor pressure
for several reasons. 
The ratio of the solid vapor pressure to the supercooled liquid vapor pressure is
termed the fugacity ratio, F. To estimate F, we need to know how much energy is
involved in the solid-liquid transition, i.e., the enthalpy of melting or fusion. The
rigorous equation for estimating F at temperature T(K) is (Prausnitz et al., 1986)
 
ln F = –
 
∆
 
S(T
 
M
 
 – T)/RT + 
 
∆
 
C
 
P
 
(T
 
M
 
 – T)/RT – 
 
∆
 
C
 
P
 
 ln(T
 
M
 
/T)/R
where 
 
∆
 
S (J/mol K) is the entropy of fusion at the melting point T
 
M 
 
(K), 
 
∆C
 
P
 
 (J/mol
K) is the difference in heat capacities between the solid and liquid substances, and
R is the gas constant. The heat capacity terms are usually small, and they tend to
cancel, so the equation can be simplified to
ln F = –
 
∆
 
S(T
 
M
 
 – T)/RT = –(
 
∆
 
H/T
 
M
 
)(T
 
M
 
 – T)/RT = –(
 
∆
 
H/R)(1/T – 1/T
 
M
 
)
where 
 
∆
 
H (J/mol) is the enthalpy of fusion and equals T
 
M
 
∆
 
S.
Note that, since T
 
M
 
 is greater than T, the right-hand side is negative, and F is
less than one, except at the melting point, when it is 1.0. F can never exceed 1.0. A
convenient method of estimating 
 
∆
 
H is to exploit Walden’s rule that the entropy of
fusion at the melting point 
 
∆
 
S, which is 
 
∆
 
H/T
 
M
 
, is often about 56.5 J/mol K. It
follows that
ln F = –(
 
∆
 
S/R)(T
 
M
 
/T – 1)
The group 
 
∆
 
S/R is often assigned a value of 56/8.314 or 6.79. Thus, F is approxi-
mated as
 
F = exp[–6.79(T
 
M
 
/T – 1)]
If base 10 logs are used and T is 298 K, this equation becomes
log F = –6.79(T
 
M
 
/298 – 1)/2.303 = –0.01(T
 
M
 
 – 298)
 
This is useful as a quick and easily remembered method of estimating F. If more
accurate data are available for 
 
∆
 
H or 
 
∆
 
S, they should be used, and if the substance
is a high melting point solid, it may be advisable to include the heat capacity terms.
 
CH05 Page 77 Monday, January 15, 2001 1:49 PM
 
78 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
5.3 PROPERTIES OF SOLUTES IN SOLUTION
5.3.1 Solution in the Gas Phase
 
Equations are needed to deduce the fugacity of a solute in solution from its
concentration. We first treat nonionizing substances that retain their structure when
in solution. It transpires that, at low concentrations, a substance’s fugacity and
concentration are linearly related, i.e., fugacity is proportional to concentration. This
suggests using a relationship of the following form:
C = Zf
where C is concentration (mol/m
 
3
 
), f is fugacity (Pa), and Z, the proportionality
constant (termed the 
 
fugacity capacity
 
) has units of mol/m3Pa. The aim is then to
deduce Z for the substance in air, water, and other phases. Later, we examine the
significance of Z in more detail, because it becomes a key quantity when assessing
environmental partitioning. 
The easiest case is a solution in a gas phase (air) in which there are usually no
interactions between molecules other than collisions.
The basic fugacity equation as presented in thermodynamics texts (Prausnitz et
al., 1986) is
f = y φ PT
where y is mole fraction, φ is a fugacity coefficient, PT is total (atmospheric) pressure,
and P is yPT, the partial pressure. If the gas law applies,
PTV = nRT or PV = ynRT
Here, n is the total number of moles present, R is the gas constant, V is volume
(m3), and T is absolute temperature (K). Now the concentration of the solute in the
gas phase CA will be yn/V or P/RT mol/m3.
CA = yPT/RT = (1/ φRT) f = ZAf 
Fortunately, the fugacity coefficient φ rarely deviates appreciably from unity
under environmental conditions. The exceptions occur at low temperatures, high
pressures, or when the solute molecules interact chemically with each other in the
gas phase. Only this last class is important environmentally. Carboxylic acids such
as formic and acetic acid tend to dimerize, as do certain gases such as NO2. The
constant ZA is thus usually (1/RT) or about 4 × 10–4 mol/m3Pa and is the same for
all noninteracting substances.
The fugacity is thus numerically equal to the partial pressure of the solute P or
yPT. This raises a question as to why we use the term fugacity in preference to partial
pressure. The answers are that (1) under conditions when φ is not unity, fugacity
CH05 Page 78 Monday, January 15, 2001 1:49 PM
PHASE EQUILIBRIUM 79
and partial pressure are not equal, and (2) there is some conceptual difficulty about
referring to a “partial pressure of DDT in a fish” when there is no vapor present for
a pressure to be present in—even partially.
5.3.2 Solution in Liquid Phases
The fugacity equation (Prausnitz et al., 1986) for solute i in solution is given in
terms of mole fraction xi activity coefficient γi and reference fugacity fR on a Raoult’s
law basis.
fi = xiγifR
Now, xi, the mole fraction of solute, can be converted to concentration C mol/m3
using molar volumes v (m3/mol), amounts n (mol), and volumes V (m3) of solute
(subscript i) and solution (subscript w for water as an example). Assuming that the
solute concentration is small, i.e., Vi<<VW, we can write
Ci = ni/(VW + Vi) = ni/VW
But 
VW = nwvw
and 
xi = ni/(ni + nw) = ni/nw
Thus,
Ci = xi/vw = xi/(18 × 10–6 m3/mol)
Thus,
fi = CivwγifR
and 
Ci = (1/vwγifR)fi = Zwf
Hence, 
ZW = (1/vWγifR)
The reference fugacity fR is by definition the fugacity that the solute will have
(or tend to) when in the pure liquid state when xi is 1.0 and γi is also (by definition)
CH05 Page 79 Monday, January 15, 2001 1:49 PM
80 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
1.0. This, then, is the fugacity or vapor pressure of pure liquid solute at the temper-
ature (and strictly the pressure) of the system. 
The activity coefficient γ is defined here on a “Raoult’s law” basis such that γ is
1.0 when x is 1.0. In most cases, γ values exceed 1.0 and, for hydrophobic chemicals,
values may be in the millions. 
An alternative convention, which we do not use here, is to define γ on a Henry’s
law basis such that γ is 1.0 when x is zero. 
The activity coefficient is thus a very important quantity. It can be viewed as the
ratio of the activity or fugacity of the solute to the activity or fugacity that the solute
would have if it were in a solution consisting entirely of its own kind. It depends
on the concentration of the solute with a dependence of the type
log γ = log γO (1 – x)2
where γO is the activity coefficient at infinite dilution, i.e., when x the mole fraction
approaches zero.
Another useful way of viewing activity coefficients is that they can be regarded
as an inverse expression of solubility, i.e., an insolubility. A solute that is sparingly
soluble in a solvent will have a high activity coefficient, an example being hexane
in water. For a liquid solute such as hexane, at the solubility limit, when excess pure
hexane is present, the fugacity equals the reference fugacity fR and
fi = fR = xiγifR
Therefore,
xi = 1/γi or γi = 1/xi
The activity coefficient is thus the reciprocal of the solubility when expressed as a
mole fraction. For solids at saturation, fi is the fugacity of the pure solid fS. Thus,
fS = xiγifR
and
xi = (fS/fR)/γi = F/γi
where F is the fugacity ratio discussed earlier. Solid solutes of high melting point
thus tend to have low solubilities, because F is small.
It is more common to express solubilities in units such as g/m3. Under dilute
conditions, the solubility Si mol/m3 is xi/vS, where vS is the molar volume of the
solution (m3/mol) and approaches the molar volume of the solvent. Si is thus 1/γivS
for liquids and F /γivS for solids. In the gas phase, the solubility is essentially the
vapor pressure in disguise, i.e.,
CH05 Page 80 Monday, January 15, 2001 1:49 PM
PHASE EQUILIBRIUM 81
Si = n/V = PS/RT
Invaluable information about how a substance will behave in the environment
can be obtained by considering its three solubilities, namely those in air, water, and
octanol. These solubilities express the substance’s relative preferences for air, water,
and organic phases.
Returning to the definition of ZW as (1/vWγifR), it is apparent that ZW also can
be expressed in terms of aqueous solubility, SW, and vapor pressure PS. For liquid
solutes, SW is 1/γivW. For solid solutes it is F/γivw. The reference fugacity fR is the
vapor pressure of the liquid, i.e., it is PS for a liquid and PS/F for a solid. Substituting
gives ZW as SW/PS in both cases, the F cancelling for solids. The ratioPS/S is the
Henry’s law constant H in units of Pa m3/mol, thus ZW is 1/H.
Polar solutes such as ethanol do not have measurable solubilities in water, because
they are miscible. This generally occurs when γ is less than about 20. We can still
use the concept of solubility and call it a “hypothetical or pseudo-solubility” if it is
defined as 1/γivS. For a liquid substance that behaves nearly ideally, i.e., γi is 1.0, the
solubility approaches 1/vS, which is the density of the solvent in units of mol/m3.
For water, this is about 55,500 mol/m3, i.e., 106 g/m3 divided by 18 g/mol. For a
solid solute under ideal conditions, the solubility approaches F/vS mol/m3.
These equations are general and apply to a nonionizing chemical in solution in
any liquid solvent, including water and octanol. The solution molar volumes and the
activity coefficients vary from solvent to solvent. The Z value for a chemical in
octanol is, by anology, 1/vOγifR, where vO is the molar volume of octanol.
5.3.3 Solutions of Ionizing Substances
Certain substances, when present in solution, adopt an equilibrium distribution
between two or more chemical forms. Examples are acetic acid, ammonia, and
pentachlorophenol, which ionize by virtue of association with water releasing H+
(strictly H3O+) or OH– ions. Some substances dimerize or form hydrates. For ionizing
substances, the distribution is pH dependent, thus the solubility and activity are also
pH dependent. This could be accommodated by defining Z as being applicable to
the total concentration, but it then becomes pH dependent. A more rigorous approach
is to define Z for each chemical species, noting that, for ionic species, Z in air must
be zero under normal conditions, because ions as such do not evaporate. In any
event, it is useful to know the relative proportions of each species, because they will
partition differently. This issue is critical for metals in which only a small fraction
may be in free ionic form.
For acids, an acid dissociation constant Ka is defined as
Ka = H+ A–/HA
where H+ is hydrogen ion concentration, A– is the dissociated anionic form, and HA
is the parent undissociated acid. The ratio of ionic to nonionic forms I is thus
I = A–/HA = Ka/H+ = 10(pH – pKa)
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82 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
where 
pH is –log H+ and pKa is –log Ka
This is the Henderson-Hasselbalch relationship. For acids, when pKa exceeds
pH by 2 units or more, ionization can be ignored. When considering substances
which have the potential to ionize it is essential to obtain pKa and determine the
relative proportions of each form. The handbook by Lyman et al. (1982) and the
text by Perrin et al. (1981) can be consulted for more details of estimation methods
for pKa, and for applications.
Dissociation can be regarded as causing an increase in the Z value of a substance
in aqueous solution. The total Z value is the sum of the nonionic and ionic contri-
butions, which will have respective fractions 1/(I + 1) and I/(I + 1). The Z value of
the nonionic form ZW can be calculated by measuring solubility, activity coefficient,
or another property under conditions when I is very small, i.e., pH << pKa. The
same ZW value applies to the nonionic form at all pH levels. The additional contri-
bution of the ionic form is then calculated at the pH of interest as IZW, and the total
effective Z value is ZW(I +1), which can be used to calculate the total concentration.
An inherent assumption here is that the presence of the ionic form does not affect ZW.
For example, phenol has a pKa of 9.90, and pentachlorophenol (PCP) has a pKa
of 4.74 (Mackay et al., 1995). At a pH of 6.0, the corresponding values of I are
phenol 0.00013 and PCP 18.2. For phenol, ionization can be ignored for pH values
up to about 8. For PCP, the dominant species in solution is the ionic form, and the
total Z value in aqueous solution is 19.2 times that of the nonionic form. As a result,
partitioning of PCP from solution in water to other media, such as air or octanol, is
very pH dependent, and the issue of whether the ionic form also partitions must be
addressed. KOW is thus pH dependent if (as is usual) total concentrations are used
to calculate it.
5.3.4 Solutions in Solids
It is not usual to regard substances as having solubilities in solids. If the structure
of the solid and the size of the solute molecule are such that the molecule can diffuse
into and out of the solid matrix in a reasonable time period, then a solubility can
be defined and measured. Organic molecules can diffuse into polymers such as
polyethylene. Indeed, plasticizers such as phthalate esters are essentially in solution
in polymers such as PVC to render the plastic flexible. Over time, they evaporate
or leach from the plastic, rendering it more brittle.
Semipermeable membrane devices (SPMDs) are increasingly used in water
analysis. They consist of organic solvent (often triolein) contained in polyethylene
bags that are submerged in the water. Hydrophobic molecules partition into the
polymer, migrate through it, and accumulate in the solvent, providing a convenient
integrated sample for analysis. To some extent, they simulate fish. In this case, the
solutes have a finite solubility in the solid polymer.
The organic matter discussed in Chapter 4 is solid. The sorption phenomenon
can be regarded as simply partitioning into solid solution in this organic matter
matrix. Solubilities and Z values can thus be calculated for solutions in solids.
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5.4 PARTITION COEFFICIENTS
5.4.1 Fugacity and Solubility Relationships
If we have two immiscible phases or media (e.g., air and water or octanol and
water), we can conduct experiments by shaking volumes of both phases with a small
amount of solute such as benzene to achieve equilibrium, then measure the concen-
trations and plot the results as was shown in Figure 5.1. It is preferable to use
identical concentration units in each phase of amount per unit volume but, when
one phase is solid, it may be more convenient to express concentration in units such
as amounts per unit mass (e.g., µg/g) to avoid estimating phase densities. The plot
of the concentration data is often linear at low concentrations; therefore, we can write
C2/C1 = K21
and the slope of the line is K21. Some nonlinear systems are considered later. Now,
since C2 is Z2f2 and C1 is Z1f1, and at equilibrium f1 equals f2, it follows that K21 is
Z2/Z1. A Z value can be regarded as “half” a partition coefficient. If we know Z for
one phase (e.g., Z1 as well as K21), we can deduce the value of Z2 as K21Z1. This
proves to be a convenient method of estimating Z values. 
The line may extend until some solubility limit or “saturation” is reached. In
water, this is the aqueous solubility, but, for some substances such as lower alcohols,
there is no “solubility,” because the solute is miscible with water. In air, the “solu-
bility” is related to the vapor pressure of the pure solute, which is the maximum
partial pressure that the solute can achieve in the air phase. 
Partition coefficients are widely available and used for systems of air-water,
aerosol-air, octanol-water, lipid-water, fat-water, hexane-water, “organic carbon”-
water, and various minerals with water. 
Applying the theory that was developed earlier and noting that, at equilibrium,
the solute fugacities will be equal in both phases, we can define partition coefficients
for air-liquid and liquid-liquid systems.
For air-water as an example at a total atmospheric pressure PT,
f = xiγifR = yiPT = Pi
Thus,
yi/xi = γifR/PT
But if we use concentrations Ci (mol/m3) instead of mole fraction, yi is CiAvA or
CiART/PT where vA is the molar volume of air. Similarly, xi is CiWvW where vW is
the molar volume of the solution and is approximately that of water. Since fR is also
PS, the partition coefficient KAW is then given by 
KAW = CiA/CiW = γi vW fR/RT = γi vWPS/RT = SiA/SiW
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84 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
In terms of solubilities, SiA and SiW, KAW is simply SiA/SiW, the ratio of the two
solubilities.
The pioneering work on air-water partitioning was done by Henry, who measured
Pi as a function of xi and discovered that the solubility in water was proportional to
the partial pressure Pi. The proportionality constant H´ is γifR and has units of pressure
(Pa). Interestingly, for super-critical gases such as oxygen, fR cannot be measured,
but γifR can be measured. If concentration is expressed as mol/m3, i.e., Ci instead of
mole fraction xi, another and more convenient Henry’s law constant H can be defined
as Pi/Ci and is γivWfR. KAW is then obviously H/RT, and it is also ZA/ZW. Note that
H is also PSL/SiW and is 1/ZW, as was shown earlier. KAW is sometimes (wrongly)
referred to as a Henry’s law constant. Atmospheric scientists, who are concerned
with partitioning from air to water (e.g., into rain) use KWA, the reciprocal of KAW ,
and often refer to it as a Henry’s law constant. Extreme care thus must be taken
when using reported values of Henry’s law constants because of these different
definitions.
For a liquid solute in a liquid-liquid system such as octanol-water,
f = xiW γiW fR = xiO γiO fR
where subscripts W and O refer to water and octanol phases.
It follows that
xiO/xiW = γiW/γiO
and 
CiO/CiW = KOW = γiW vW/γiO vO = SiO/SiW
If the solute is solid the same final equation applies because F, like fR, cancels.
Because vW and vO are relatively constant, the variation in KOW between solutes
is a reflection of variation in the ratio of activity coefficients γiW/γiO. Hydrophobic
substances such as DDT have very large values of γiW and low solubilities in water.
The solubility in octanol is usually fairly constant for organic solutes, thus KOW is
approximately inversely proportional to SiW. Numerous correlations have been pro-
posed between log KOW and log SiW, which are based on this fundamental relation-
ship.
Finally, for completeness, the octanol-air partition coefficient can be shown to be
KOA = SiO/SiA = RT /γi vO PSL
where γi applies to the octanol phase. It can be shown that ZO is 1/γi vO PSL and that
KOA is ZO/ZA.
Measurements of solubilities and partition coefficients are subject to error, as is
evident by examining the range of values reported in handbooks. An attractive
approach is to measure the three partition coefficients, KAW, KOW, and KOA, and
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PHASE EQUILIBRIUM 85
perform a consistency check that, for example, KOA is KOW/KAW. Further checks are
possible if solubilities can be measured to confirm that KAW is SA/SW or PS/SWRT.
These checks are also useful for assessing the “reasonableness” of data. For example,
if an aqueous solubility SW is reported as 1 part per million or 1 g/m3 or (say)
10–2 mol/m3, and KOW is reported to be 107, then the solubility in octanol must be
SWKOW or 100,000 mol/m3. Octanol has a solubility in itself, i.e., a density of about
820 kg/m3 or 6300 mol/m3. It is inconceivable that the solubility of the solute in
octanol exceeds the solubility of octanol in octanol by a factor of 100,000/6300 or
16; therefore, either SW or KOW or both are likely erroneous.
The relationships between the three solubilities and the partition coefficients are
shown in Figure 5.3. Two points worthy of note.
There are numerous correlations for quantities such as KAW, KOW, SW, SA as a
function of molecular structure and properties. They are generally derived indepen-
dently, so it is possible to estimate SW, SA, and KAW and obtain inconsistent results,
i.e., KAW will not equal SA/SW. It is preferable, in principle, to correlate SA, SW, and
SO independently and use the values to estimate KAW, KOW, and KOA. There is then
no possible inconsistency. It must be easier to correlate S (which depends on
interactions in only one phase) than K (which depends on interactions in two phases).
Finally, all activity coefficients, solubilities, and partition coefficients are tem-
perature dependent.
Figure 5.3 Illustration of the relationships between the three solubilities, CA, CW, and CO, and
the three partition coefficients, KAW, KOW, and KOA, with values for four substances.
Note the wide substance variation in concentrations corresponding to unit con-
centration in the air phase.
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86 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
The temperature coefficient is the enthalpy of phase transfer, e.g., pure solute to
solution for solubility or from solution to solution for partition coefficients. The
enthalpies must be consistent around the cycle air-water-octanol such that their sum
is zero. This provides another consistency check. It should be noted that the enthalpy
change refers to the solubility or partition coefficient variation when expressed in
mole fractions, not mol/m3 concentrations. This is particularly important for parti-
tioning to air, where a temperature increase causes a density decrease, thus C or S
will fall while x remains constant. For details of the merits of applying the “three
solubility” approach, the reader is referred to Cole and Mackay (2000). We discuss
these partition coefficients individually in more detail in the following sections. 
5.4.2 Air-Water Partitioning
The nature of air-water partition coefficients or Henry’s law constants has been
reviewed by Mackay and Shiu (1981), and estimation methods have been described
by Mackay et al. (2000) and Baum (1997), and only a brief summary is given here.
Several group contribution and bond contribution methods have been developed,
and estimation methods are available from websites such as the EPIWIN programs
of the Syracuse Research Corporation site at www.syrres.com. As was discussed
above, the simplest method of estimating Henry’s law constants of organic solutes
is as a ratio of vapor pressure to water solubility. It must be recognized that this
contains the inherent assumption that water is not very soluble in the organic
material, because the vapor pressure that is used is that of the pure substance
(normally the pure liquid) whereas, in the case of solubility of a liquid such as
benzene in water, the solubility is not actually that of pure benzene but is inevitably
of benzene saturated with water. When the solubility of water in a liquid exceeds a
few percent, this assumption may break down, and it is unwise to use this relation-
ship. If a solute is miscible with water (e.g., ethanol), it is preferable to determine
the Henry’s law constant directly; that is, by measuring air and water concentrations
at equilibrium. This can be done by various techniques, e.g., the EPICS method
described by Gossett (1987) or a continuous stripping technique described by
Mackay et al. (1979). A desirable strategy is to measure vapor pressure PS, solubility
CS, and H or KAW and perform an internal consistency check that H is indeed PS/CS
or close to it. KAW is, of course, ZA/ZW.
Care must be taken when calculating Henry’s law constants to ensure that the
vapor pressures and solubilities apply to the same temperature and to the same phase.
In some cases, reported vapor pressures are estimated by extrapolation from higher
temperatures. They may be of a liquid or subcooled liquid, whereas the solubility
is that of a solid. As was discussed earlier, subcooled conditions are not experimen-
tally accessible but prove to be useful for theoretical purposes.
Henry’s law constants vary over many orders of magnitude, tending to be high
for substances such as the alkanes (which have high vapor pressures, low boiling
points, and low solubilities) and very low for substances such as alcohols (which
have a high solubility in water and a low vapor pressure). There is a common
misconception that substances that are “involatile,” such as DDT, will have a low
Henry’s law constant. This is not necessarily the case,because these substances also
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PHASE EQUILIBRIUM 87
have very low solubilities in the water, i.e., they are very hydrophobic; thus, their
low vapor pressure is offset by their very low water solubility, and they have relatively
large Henry’s law constants. They may thus partition appreciably from water into
the atmosphere through evaporation from rivers and lakes.
The solubility and activity of a solute in water are affected by the presence of
electrolytes and other co-solvents; thus, the Henry’s law constant is also affected.
The magnitude of the effect is discussed later in Section 5.4.5.
Worked Example 5.2
Deduce H and KAW for benzene, DDT, and phenol given the following data at
25°C:
In each case, the solubility CS in mol/m3 is the solubility in g/m3 divided by the
molar mass, e.g., 1780/78 or 22.8 mol/m3 for benzene. H is then PS/CS or 556 Pa
m3/mol for benzene. KAW is H/RT or 556/(8.314 × 298) or 0.22.
Note that these substances have very different H and KAW values because of their
solubility and vapor pressure differences. The vapor pressure of DDT is about 600
million times less than that of benzene, but H is only 400 times less, because of
DDT’s very low water solubility. Phenol has a much higher vapor pressure than
DDT, but it has a much lower H and KAW. Benzene tends to evaporate appreciably
from water into air, and DDT less so but still to a significant extent, while phenol
does not evaporate significantly. Inherent in this calculation for phenol is the assump-
tion that it does not ionize appreciably.
5.4.3 Octanol-Water Partitioning
The dimensionless octanol-water partition coefficient (KOW) is one of the most
important and frequently used descriptors of chemical behavior in the environment.
In the pharmaceutical and biological literature, KOW is given the symbol P (for
partition coefficient), which we reserve for pressure. The use of 1-octanol has been
popularized by Hansch and Leo, who have tested its correlations with many bio-
chemical phenomena and have compiled extensive databases. Various methods are
available for calculating KOW from molecular structure, as reviewed by Lyman et al.
Molar Mass (g/mol) Solubility (g/m3) Vapor Pressure (Pa)
benzene 78 1780 12700
DDT 354.5 0.0055 0.00002
phenol 94.1 88360 47
H KAW
benzene 556 0.22
DDT 1.29 5.2 × 10–4
phenol 0.050 2 × 10–5
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88 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
(1982), Baum (1997) and Leo (2000). Extensive databases are also available as
reviewed by Baum (1997). Octanol was selected because it has a similar carbon to
oxygen ratio as lipids, is readily available in pure form, and is only sparingly soluble
in water (4.5 mol/m3). The solubility of water in octanol of 2300 mol/m3, however,
is quite large (Baum, 1997). The molar volumes of these phases are 18 × 10–6 m3/mol
and 120 × 10–6 m3/mol, a ratio of 0.15. It follows that KOW is 0.15 γW/γO. 
KOW is a measure of hydrophobicity, i.e., the tendency of a chemical to “hate”
or partition out of water. As was discussed earlier, it can be viewed as a ratio of
solubilities in octanol and water but, in most cases of liquid chemicals, there is no
real solubility, because octanol and the liquid are miscible. The “solubility” of
organic chemicals in octanol tends to be fairly constant in the range 200 to 2000
mol/m3, thus variation in KOW between chemicals is primarily due to variation in
water solubility. It is therefore misleading to assert that KOW describes lipophilicity
or “love for lipids,” because most organic chemicals “love” lipids equally, but they
“hate” water quite differently. Viewed in terms of Z values, KOW is ZO/ZW. ZO is
(relatively) constant for organic chemicals; however, ZW varies greatly and is very
small (relatively) for hydrophobic substances.
Because KOW varies over such a large range, from approximately 0.1 to 107, it
is common to express it as log KOW. It is a disastrous mistake to use log KOW in a
calculation when KOW should be used!
KOW is usually measured by equilibrating layers of water and octanol containing
the solute of interest at subsaturation conditions and analyzing both phases. If KOW
is high, the concentration in water is necessarily low, and even a small quantity of
emulsified octanol in the aqueous phase can significantly increase the apparent
concentration. A “slow stirring” method is usually adopted to avoid emulsion for-
mation. An alternative is to use a generator column in which water is flowed over
a packing containing octanol and the dissolved chemical.
5.4.4 Octanol-Air Partition Coefficients
This partition coefficient is invaluable for predicting the extent to which a
substance partitions from the atmosphere to organic media including soils, vegeta-
tion, and aerosol particles. It can be estimated as KOW/KAW or measured directly,
usually by flowing air through a column containing a packing saturated with octanol
with the solute in solution. Values of KOA can be very large, i.e., up to 1012 for
substances of very low volatility such as DDT, and values are especially high at low
temperatures. Harner et al. (2000) have reported data for this coefficient and cite
other data and measurement methods.
5.4.5 Solubility in Water
This property is of importance as a measure of the activity coefficient in aqueous
solution, which in turn affects air-water and octanol-water partitioning. It can be
regarded as a partition coefficient between the pure phase and water, but the ratio
of concentrations is not calculated. A comprehensive discussion is given in the text
by Yalkowsky and Banerjee (1992), and estimation methods are described by Mackay
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PHASE EQUILIBRIUM 89
(2000) and Baum (1997). Extensive databases are available, for example, the hand-
books by Mackay et al. (2000), which also give details of methods of experimental
determination.
It is important to appreciate that solubility in water is affected by temperature
and the presence of electrolytes and other solutes in solution. It is often convenient
to increase the solubility of a sparingly soluble organic substance by addition of a
cosolvent to the water. Methanol and acetone are common cosolvents. To a first
approximation, a “log-linear” relationship applies in that, if the solubility in water
is SW and that in pure cosolvent is SC, then the solubility in a mixture SM is given by
log SM = (1 – vC) log SW + vC log SC
where vC is the volume fraction cosolvent in the solution. 
Electrolytes generally decrease the solubility of organics in water, the principal
environmentally relevant issue being the solubility in seawater. The Setschenow
equation is usually applied for predictive purposes, namely
log (SW/SE) = kCS
where SW is solubility in water, SE is solubility in electrolyte solution, k is the
Setschenow constant specific to the ionic species, and CS is the electrolyte concen-
tration (mol/L). Values of k generally lie in the range 0.2 to 0.3 L/mol; thus, in
seawater, which is approximately 33 g NaCl/L or CS is about 0.5 mol/L, the solubility
is about 70 to 80% of that in water. Xie et al. (1997) have reviewed this literature,
especially with regard to seawater.
5.4.6 Solubility in Octanol
There are relatively few data on this solubility, and for many substances, espe-
cially liquids, the low activity coefficients render the solute-octanol system miscible;
thus, no solubility is measurable. Pinsuwan and Yalkowsky (1995) have reviewed
available solubility data and the relationships between KOW and solubilities in octanol
and water.
5.4.7 Solubility of a Substance in Itself
The fugacity of a pure solute is its vapor pressure PS, and its “concentration” is
the reciprocal of its molar volume vS (m3/mol) (typically, 10–4 m3/mol). Thus,
C = (1/vS) = Zpf = ZPPS
and
ZP = 1/PS vS
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90 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Although it may appear environmentallyirrelevant to introduce ZP, there are
situations in which it is used. If there is a spill of PCB or an oil into water of
sufficient quantity that the solubility is exceeded, at least locally, the environmental
partitioning calculations may involve the use of volumes and Z values for water, air,
sediment, biota, and a separate pure solute phase. Indeed, early in the spill history,
most of the solute will be present in this phase. The difference in behavior of this
and other phases is that the pure phase fugacity (and, of course, concentration)
remains constant, and as the chemical migrates out of the pure phase, the phase
volume decreases until it becomes zero at total dissolution or evaporation. In the
case of other phases, the concentration changes at approximately constant volume
as a result of migration.
It can be useful to compare a set of calculated Z values with ZP to gain an
impression of the degree of nonideality in each phase. Rarely does a Z value of a
chemical in a medium exceed ZP, but they may be equal when ideality applies and
activity coefficients are close to 1.0.
5.4.8 Partitioning to Interfaces
Chemicals tend to adsorb from air or water to the surface of solids. An extensive
literature exists on this subject as reviewed in texts in chemistry and environmental
processes. A good review with environmental applications is given by Valsaraj (1995)
in which the fundamental Gibbs equation is developed into the commonly used
adsorption isotherms. These isotherms relate concentration at the surface to concen-
tration in the bulk phase. Examples are the Langmuir, BET, and Freudlich isotherms.
Generally, a linear isotherm applies at low concentrations as are usually encountered
in the environment in relatively uncontaminated situations. Nonlinear behavior
occurs at high concentrations in badly contaminated systems and in process equip-
ment such as carbon adsorption units.
It is often not realized that partitioning also occurs at the air-water interface,
where an excess concentration may exist. This is exploited in the solvent sublation
process for removing solutes from water using fine bubbles.
If an area of the surface is known, a surface concentration in units of mol/m2
can be calculated, but more commonly the concentration is given in mol/mass of
sorbent, which is essentially the product of the surface concentration and a specific
area expressed in m2 surface per unit mass of sorbent. Solids such as activated carbon
have very high specific areas and are thus effective sorbents. Partitioning to the air-
water interface can become very important when the area of that interface is large
compared to the associated volume of air or water. This occurs in fog droplets and
snow where the ratio of area to water volume is very large, or in fine bubbles where
the ratio of area to air volume is large. These ratios are (6/diameter) m2/m3.
A Z value can be defined on an area basis (mol/m2 Pa) or for the bulk phase by
including the specific area. 
Schwarzenbach et al. (1993) have reviewed mechanisms of sorption and have
summarized reported data. This partitioning is important for ionizing substances
but less important for nonpolar compounds, which sorb more strongly to organic
matter.
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5.4.9 Quantitative Structure Property Relationships
An invaluable feature of many series of organic chemicals is that their properties
vary systematically, and therefore predictably, with changes in molecular structure.
This relationship is illustrated for the chlorobenzenes in Figure 5.4. Figure 5.4A is
a plot of log subcooled liquid solubility versus chlorine number from 0 (benzene)
to 6 (hexachlorobenzene), which shows the steady drop in solubility as a result of
substituting a chlorine for a hydrogen. The magnitude is a decrease in log solubility
of about 0.65 units (factor of 4.5) per chlorine. Vapor pressure (Figure 5.4B) behaves
similarly, with a drop of 0.72 units (factor of 5.2). KOW (Figure 5.4C) shows an
increase of 0.53 units (factor of 3.4). The Henry’s law constant (Figure 5.4D)
decreases by 0.16 units (factor of 1.4).
These plots are invaluable as a method of interpolating to obtain values for
unmeasured compounds. They provide a consistency check for newly reported data.
They form the basis of estimation methods in which these properties are calculated
for a variety of atomic and group fragments.
An extension of the QSPR concept is to use the same principles to correlate and
estimate toxicity. This is referred to as a quantitative structure activity relationship
or QSAR. The best environmental example is the correlation of fish toxicity data
expressed as a LC50 (µmol/L) versus KOW as obtained by Konemann (1981).
log (1/LC50) = 0.87 log KOW – 4.87
This and other correlations have been reviewed and discussed by Veith et al. (1983),
Kaiser (1984, 1987), Karcher and Devillers (1990), and Abernethy et al. (1986,
1980). The fundamental relationship expressed by this correlation is best explained
by an example.
Consider two chemicals of log KOW 3 and 5. The LC50 values will be 182 and
3.3 µmol/L, a factor of 55 different. If the target site is similar to octanol in solvent
properties, and equilibrium is reached, then the concentrations at the target site will
be the product LC50 × KOW or 182,000 and 330,000 µmol/L, only a difference of a
factor of 1.8. The chemical of lower KOW appears to be less toxic (it has a higher
LC50) when viewed from the point of view of water concentration. When viewed
from the target site concentration, it is slightly more toxic. The chemicals in the
correlation display similar toxicities when evaluated from the target site concentra-
tions. The correlation therefore expresses two processes, partitioning and toxicity,
with most of the chemical-to-chemical variation being caused by partitioning dif-
ference. Such chemicals are referred to as narcotics in which the effect seems to be
induced by high lipid concentrations.
5.4.10 Summary
The key properties of a pure substance for our purposes are its vapor pressure
(i.e., its solubility in air), its solubility in water, its solubility in octanol, and the
three partition coefficients KAW, KOW, and KOA. The magnitudes of these quantities
are controlled by vapor pressure and activity coefficients.
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92 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
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PHASE EQUILIBRIUM 93
We can also relate equilibrium concentrations in these phases using the three Z
values and fugacity. The partition coefficients are simply the ratio of the respective
Z values; e.g., KAW is ZA/ZW. The use of Z values at this stage brings little benefit,
but they become very useful when we calculate partitioning to environmental media.
We use ZA, ZW, and ZO to calculate Z in phases such as soils, sediments, fish, and
aerosol particles, and it proves useful to have ZP as a reference point when examining
the magnitude of Z values. It is enlightening to calculate all the physical chemical
properties of a solid and a liquid substance as shown in the following example.
Worked Example 5.3
Deduce all relevant thermodynamic air-water partitioning properties for benzene
(liquid) and naphthalene (a solid of melting point 80°C) at 25°C.
Benzene
Vapor pressure = 12700 Pa (PSL), molar mass = 78 g/mol, solubility in water = 
1780 g/m3.
ZA = 1/RT = 1/(8.314 × 298) = 4.04 × 10–4
Solubility CSL = 1780/78 = 22.8 mol/m3
Activity coefficient γ = 1/vwCSL = 1/(18 × 10–6 × 23.1) = 2440
H = PSL/CSL = 556, also = vWγPSL
ZW = 1/H or 1/vwγPSL = 0.0018
KAW = H/RT or ZA/ZW = 0.22
Naphthalene
Solubility= 33 g/m3, molar mass = 128 g/mol, vapor pressure = 10.9 Pa
ZA = 4.04 × 10–4 as before
F = exp(–6.79(353/298–1)) = 0.286 (mp = 353 K)
PSL = PSS/F = 38.1
CSS = 33/128 = 0.26 mol/m3, CSL = 0.90 mol/m3
γ = 1/vwCSL = 61700
H = PSS /CSS or PSL/CSL = 42
ZW = 1/H or 0.024 = 1/vwγPSL
KAW = H/RT or ZA/ZW = 0.017
Note that naphthalene has a higher activity coefficient corresponding to its lower
solubility and greater hydrophobicity.
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94 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
5.5 ENVIRONMENTAL PARTITION COEFFICIENTS AND Z VALUES
5.5.1 Introduction
Our aim is now to use the physical chemical data to predict how a chemical will
partition in the environment. Information on air-water-octanol partitioning is invalu-
able and can be used directly in the case of air and pure water, but the challenge
remains of treating other media such as soils, sediments, vegetation, animals, and
fish. The general strategy is to relate partition coefficients involving these media to
partitioning involving octanol. We thus, for example, seek relationships between
KOW and soil-water or fish-water partitioning.
5.5.2 Organic Carbon-Water Partition Coefficients
Studies by agricultural chemists have revealed that hydrophobic organic chem-
icals tended to sorb primarily to the organic matter present in soils. Similar obser-
vations have been made for bottom sediments. In a definitive study, Karickhoff
(1981) showed that organic carbon was almost entirely responsible for the sorbing
capacity of sediments and that the partition coefficient between sediment and water
expressed in terms of an organic carbon partition coefficient (KOC) was closely related
to the octanol-water partition coefficient. Indeed, the simple relationship was estab-
lished to be
KOC = 0.41 KOW
This relationship is based on experiments in which a soil-water partition coeffi-
cient was measured for a variety of soils of varying organic carbon content (y) and
chemicals of varying KOW. The soil concentration was measured in units of µg/g or
mg/kg (usually of dry soil) and the water in units of µg/cm3 or mg/L. The ratio of
soil and water concentration (designated KP) thus has units of L/kg or reciprocal
density.
KP = CS/CW (mg/kg)/(mg/L) = L/kg
If a truly dimensionless partition coefficient is desired, it is necessary to multiply
KP by the soil density in kg/L (typically 2.5), or equivalently multiply CS by density
to give a concentration in units of mg/L. A plot of KP versus organic carbon content,
y (g/g), proves to be nearly linear and passes close to the origin, suggesting the
relationship
KP = y KOC
where KOC is an organic carbon-water partition coefficient.
In practice, there is usually a slight intercept, thus the relationship must be used
with caution when y is less than 0.01, and especially when less than 0.001. Since
y is dimensionless, KOC, like KP also has units of L/kg. Measurements of KOC for a
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PHASE EQUILIBRIUM 95
variety of chemicals show that KOC is related to KOW as discussed above. KOW is
dimensionless, thus the constant 0.41 has dimensions of L/kg. Care must be taken
to use consistent units in these calculations. For example, if the water concentration
has units of mol/m3, and KP is applied, the soil concentration will be in mol/Mg,
i.e., moles per 106 grams. The usual units used are mg/L in water and mg/kg in soil.
Units of either mass (g) or amount (mol) of solute can be used, but they must be
consistent in both water and soil.
The relationship between KOW and KOC has been the subject of considerable
investigation, and it appears to be variable. For example, DiToro (1985) has sug-
gested that, for suspended matter in water, KOC approximately equals KOW. Other
workers, notably Gauthier et al. (1987), have shown that the sorbing quality of the
organic carbon varies and appears to be related to its aromatic content as revealed
by NMR analysis. 
Gawlik et al. (1997) have recently reviewed some 170 correlations between KOC
and KOW, solubility in water, liquid chromatographic retention time, and various
molecular descriptors. They could not recommend a single correlation as being
applicable to all substances. Seth et al. (1999) analyzed these data and suggested
that KOC is best approximated as 0.35 KOW (a coefficient slightly lower than Karick-
hoff’s 0.41) but that the variability is up to a factor of 2.5 in either direction. It is
thus expected that, depending on the nature of the organic carbon, KOC can be as
high as 0.9 KOW and as low as 0.14 KOW. Values outside this range may occur because
of unusual combinations of chemical and organic matter. Doucette (2000) has given
a very comprehensive review of this issue, and Baum (1997) has reviewed estimation
methods.
In summary, Z values can be calculated for soils and sediments containing
organic carbon of 0.35 ZOy(ρ/1000) or 0.41 ZOy(ρ/1000), where ZO is for octanol,
y is the organic carbon content, and ρ is the solid density, typically 2500 kg/m3. If
an organic matter content is given, the organic carbon content can be estimated as
56% of the organic matter content.
These relationships provide a very convenient method of calculating the extent
of sorption of chemicals between soils or sediments and water, provided that the
organic carbon content of the soil and the chemical’s octanol-water partition coef-
ficient are known. This is illustrated in Example 5.4 below.
Worked Example 5.4
Estimate the partition coefficient between a soil containing 0.02 g/g of organic
carbon for benzene (KOW of 135) and DDT (log KOW of 6.19), and the concentrations
in soil in equilibrium with water containing 0.001 g/m3, using the Karickhoff (0.41)
correlation.
benzene KOC = 0.41 KOW = 55, KP = 0.02 KOC = 1.1 L/kg
DDT KOC = 0.41 KOW = 635000, KP = 0.02 KOC = 12700 L/kg
KP and KOC have units of L/kg or m3/Mg, i.e., reciprocal density thus when
applying the equation below CS the soil concentration will have units of g/Mg or µg/g.
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96 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
CS = KPCW benzene = 1.1 × 0.001 = 0.0011 µg/g
DDT = 10320 × 0.001 = 12.7 µg/g
Note the much higher DDT concentration because of its hydrophobic character.
The concentrations in the organic carbon are CWKOC or 0.055 µg/g for benzene
and 635 µg/g for DDT. If octanol was exposed to this water, similar concentrations
of CWKOW or 0.135 µg/cm3 for benzene and 1549 µg/cm3 for DDT would be estab-
lished in the octanol.
5.5.3 Lipid-Water and Fish-Water Partition Coefficients
Studies of fish-water partitioning by workers such as Spacie and Hamelink
(1982), Neely et al. (1974), Veith et al. (1979), and Mackay (1982) have shown that
the primary sorbing or dissolving medium in fish for hydrophobic organic chemicals
is lipid or fat. A similar approach can be taken as for soils, but there is a more
reliable relationship between KOW and KLW, the lipid-water partition coefficient. For
most purposes, they can be assumed to be equal although, for the very hydrophobic
substances, Gobas et al. (1987) suggest that this breaks down, possibly because of
the structured nature of the lipid phases. It is thus possible to calculate an approx-
imate fish-to-water bioconcentration factor or partition coefficient if the lipid content
of the fish is known. Mackay (1982) reanalyzed a considerable set of bioconcentra-
tion data and suggested the simple linear relationship,
KFW = 0.048 KOW
This can be viewed as containing the assumption that fish is about 4.8% lipid.
Lipid contents vary considerably, and it is certain that there is some sorption to non-
lipid material, but it appears that, on average, the fish behaves as if it is about 4.8%
octanol by volume.
In summary, Z for lipids can be equated to ZO for octanol. For a phase such as
a fish of lipid volume fraction L, ZF is LZO.
5.5.4 Mineral Matter-Water Partition Coefficients
Partition coefficients of hydrophobic organics between mineralmatter and water
are generally fairly low and do not appear to be simply related to KOW. Typical values
of the order of unity to 10 are observed as reviewed by Schwarzenbach et al. (1993).
A notable exception occurs when the mineral surface is dry. Dry clays display very
high sorptive capacities for organics, probably because of the activity of the inorganic
sorbing sites. This raises a problem that some soils may display highly variable
sorptive capacities as they change water content as a result of heating, cooling, and
rainfall during the course of diurnal or seasonal variations. Some pesticides are
supplied commercially in the form of the active ingredient sorbed to an inorganic
clay such as bentonite.
In the environment, most clay surfaces appear to be wet and thus of low sorptive
capacity. Most mineral surfaces that are accessible to the biosphere also appear to
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PHASE EQUILIBRIUM 97
be coated with organic matter probably of bacterial origin. They thus may be shielded
from the solute by a layer of highly sorbing organic material. It is thus a fair (and
very convenient) assumption that the sorptive capacity of clays and other mineral
surfaces can be ignored. Notable exceptions to this are subsurface environments in
which there may be extremely low organic carbon contents and when the solute
ionizes. In such cases, the inherent sorptive capacity of the mineral matter may be
controlling.
5.5.5 Aerosol-Air Partition Coefficient
One of the most difficult, and to some extent puzzling, sorption partition coef-
ficients is that between air and aerosol particles. These particles have very high
specific areas, i.e., area per unit volume. They also appear to be very effective
sorbents. The partition coefficient is normally measured experimentally by passing
a volume of air through a filter then measuring the concentrations before and after
filtration, and also the concentrations of the trapped particles. Relationships can then
be established between the ratio of gaseous to aerosol material and the concentration
of total suspended particulates (TSP). 
There has been a profound change over the years in our appreciation of this
partitioning phenomenon. The pioneering work was done by Junge and later by
Pankow resulting in the Junge-Pankow equation, which takes the form
φ = C θ/(PSL + C θ)
where φ is the fraction on the aerosol, θ is the area of the aerosol per unit volume
of air, C is a constant, and PSL is the liquid vapor pressure. This is a Langmuir type
of equation, which implies that sorption is to a surface, and the maximum extent of
sorption is controlled by the available area.
Experimental data were better correlated by calculating KP. It can be shown that
φ = KP TSP/(1 + KP TSP)
from which
KP = φ/[TSP(1 – φ)]
The units of TSP are usually µg/m3, thus it is convenient to express KP in units
of m3/µg. KP is usually correlated against PSL for a series of structurally similar
chemicals using the relationship
log KP = m log PSL + b 
where m and b are fitted constants, and m is usually close to –1 in value. Bidleman
and Harner (2000) list 21 such correlations and present a more detailed account of
this theory.
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98 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Mackay (1986), in an attempt to simplify this correlation, forced m to be –1 and
obtained a one-parameter equation, using the liquid vapor pressure,
KQA = 6 × 106/PSL
where KQA is a dimensionless partition coefficient, i.e., a ratio of (mol/m3)/(mol/m3),
and PSL has units of Pascals. Here, we use subscript Q to designate the aerosol phase.
This enables ZQ, the Z value of the chemical in the aerosol particle, to be estimated
as KQAZA. It can be shown that KQA is 1012 KP(ρ/1000) where ρ is the density of
the aerosol (kg/m3), i.e., typically 1500 kg/m3. The 1012 derives from the conversion
of µg to Mg. The fraction on the aerosol ϕ can then be calculated as
ϕ = KQA vQ/(1 + KQA vQ)
where vQ is the volume fraction of aerosol and is 10–12 TSP/(ρ/1000) when TSP has
units of µg/m3. TSP is typically about 30 µg/m3, plus or minus a factor of 5; thus,
vQ is about 20 × 10–12, plus or minus the same factor. Equipartitioning between air
and aerosol phases occurs when ϕ is 0.5 or KP·TSP and KQAvQ equals 1.0. This
implies a chemical with a KP of 0.03 m3/µg or KQAof 0.05 × 1012 and a vapor pressure
of about 10–4 Pa.
It is noteworthy that ZQ has a value of KQAZA or about (6 × 106/PSL)(4 × 10–4)
or 2400/PSL. This is comparable to ZP, the pure substance Z value of 1/v PSL, where
v is the chemical’s molar volume and is typically 100 cm3/mol or 10–4 m3/mol, giving
a ZP of about 10,000/PSL. This implies that the solute is behaving near-ideally in the
aerosol, i.e., the solubility in the aerosol is about 24% of the solubility of a substance
in itself. This casts doubt on the surface sorption model, since it seems a remarkable
coincidence that the area is such that it gives this near-ideal behavior. It further
suggests that ZQ may correlate well with ZO for octanol.
This was explored by Finizio et al. (1997), Bidleman and Harner (2000), and
Pankow (1998), leading ultimately to a suggestion that 
KP = 10–12 KOA y (1000/820) = 10–11.91 KOA y
where 820 kg/m3 is the density of octanol, and y is the fraction organic matter in
the aerosol, which is typically 0.2.
This reduces to
KP = 10–12.61 KOA or 0.25 × 10–12 KOA m3/µg
the use of KOA is advantageous, because it eliminates the need to deduce the fugacity
ratio, F, when calculating the subcooled liquid vapor pressure. It is also possible
that, for a series of chemicals, the activity coefficients in octanol and aerosol organic
matter are similar, or at least have a fairly constant ratio.
This approach is appealing, because it mimics the Karickhoff method of calcu-
lating soil-water partitioning, except that partitioning is now to air instead of water;
thus, KOA replaces KOW.
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PHASE EQUILIBRIUM 99
KQA thus can be calculated by the analogous equation,
KQA = yKOA(ρ/1000)
where y is organic matter mass fraction. This is equivalent to ZQ = yZO (ρ/1000),
where ZO is the chemical’s Z value in octanol and ρ is the aerosol density.
In summary, ZQ can be deduced using the above equation, using the simple one-
parameter expression for KQA or one of the two-parameter equations for KP. Bidle-
man and Harner (2000) discuss the merits of these approaches in more detail.
5.5.6 Other Partition Coefficients
In principle, partition coefficients can be defined and correlated for any phase
of environmental interest, usually with respect to the fluid media air or water. For
example, vegetation or foliage-air partition coefficients KFA can be measured and
correlated against KOA. Since KFA is ZF/ZA and KOA is ZO/ZA, the correlation is
essentially of ZF versus ZO. Hiatt (1999) has suggested that, for foliage, KFA is
approximately 0.01 KOA, implying that ZF is about 0.01 ZO, or foliage has a content
of octanol-equivalent material of 1%.
It is thus possible to estimate Z values for chemicals in any phase of environ-
mental interest, provided that the appropriate partition coefficient has been measured
or can be estimated. Figure 5.5 summarizes the relationships between fugacity,
concentrations, partition coefficients, and Z values.
5.6 MULTIMEDIA PARTITIONING CALCULATIONS
5.6.1 The Partition Coefficient Method
The calculation of one phase concentration from another by the use of a simple
partition coefficient is the most direct and convenient method. Care must be taken
that the concentration units and the partition coefficient dimensions are consistent,
especially when dealing with solid phases. There may also be inadvertent inversion
of a partition coefficient, i.e., the use of K12 instead of K21. It is also possible to
deduce certain partition coefficients from others; e.g., if an air/water and a soil/water
partition coefficient are available, then the air/soilor soil/air partition coefficient can
be deduced as follows:
KAS = KAW/KSW
If we are treating 10 phases, then it is possible to define 9 independent inter-
phase partition coefficients, the 10th being dependent on the other 9. In principle,
with 10 phases, it is possible to define 90 partition coefficients, half of which are
reciprocals of the others. When dealing with very complex multicompartment envi-
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100 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
ronmental media, extreme care must be taken to avoid over- or underspecifying
partition coefficients and to ensure that the ratios are not inverted.
We have developed the capability of performing our first multimedia partitioning
calculations. If we have a series of phases of volume V1, V2, V3, and V4, and we
know the partition coefficients K12, K13, K14, and we introduce a known amount of
Definition of Fugacity Capacities
Compartment Definition of Z (mol/m3 Pa)
Air 1/RT R = 8.314 Pa m3/mol K T = temp. (K)
Water 1/H or CS/PS CS = aqueous solubility (mol/m3)
PS = vapor pressure (Pa)
H = Henry’s law constant (Pa m3/mol)
Solid sorbent (e.g., soil, 
sediment, particles)
KPρS/H KP = partition coeff. (L/kg)
ρS = density (kg/L)
Biota KBρS/H KB = bioconcentration factor (L/kg)
ρB = density (kg/L)
Pure solute 1/PSv v = solute molar volume (m3/mol)
Figure 5.5 Relationships between Z values and partition coefficients and summary of Z value
definitions.
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PHASE EQUILIBRIUM 101
chemical M mol into this hypothetical environment, then we can argue that M must
be the sum of the concentration-volume products as follows:
M = C1V1 + C2V2 + C3V3 + C4V4
= C1V1 + (K21C1)V2 + (K31C1)V3 + (K41C1)V4
= C1[V1 + K21V2 + K31V3 + K41V4]
Therefore,
C1 = M/[V1 + K21V2 + K31V3 + K41V4]
and
C2 = K21C1, C3 = K31C1 and C4 = K41C1
It is thus possible to calculate the concentrations in each phase, the amounts in each
phase, and the percentages, and obtain a tentative picture of the behavior of this
chemical at equilibrium in an evaluative environment. This is best illustrated by an
example.
Worked Example 5.5
Benzene partitions in a hypothetical environment between air, water, sediment,
and fish (subscripted A, W, S, and F). The volumes of each phase are given below.
The dimensionless partition coefficients are also given below. Calculate the concen-
trations, amounts, and percentages in each phase, assuming that a total of 10 moles
of benzene is introduced into this system.
VA = 1000 VW = 20 VS = 10 VF = 0.05 m3
KAW = 0.2 KSW = 15 KFW = 20
Using the equation developed above,
CW = 10/(20 + 0.2 × 1000 + 15 × 10 + 20 × 0.05) = 10/371 = 0.027 mol/m3
Therefore,
CA = 0.0054, CS = 0.405, CF = 0.54
The amounts in each phase are the products CV, namely,
air = 5.39, water = 0.539, sediment = 4.04, fish = 0.03
from which the percentages are, respectively, 53.9, 5.4, 40.4, and 0.3.
It is clear that, in this system, benzene partitions primarily into air, mainly
because of the large volume of air. The concentration in the air is lower than in any
other phase; thus, we must discriminate between phases where the amount is large,
which depends on the product CV, and where the concentration is large. There is a
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102 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
high concentration in the fish, but only a negligible fraction of the benzene is
associated with fish. Such calculations are invaluable, because they establish the
dominant medium into which the chemical is likely to partition, and they even give
approximate concentrations.
5.6.2 The Fugacity Method
We now repeat these calculations using the fugacity concept and replacing C by
Zf. We know that Z will depend on
1. the nature of the solute (i.e., the chemical)
2. the nature of the medium or compartment
3. temperature
4. pressure (but the effect is usually negligible)
5. concentration (but the effect is negligible at low concentrations)
We have developed procedures by which Z values can be estimated for any given
environmental situation. Equilibrium concentrations can then be deduced using f as
a common criterion of equilibrium. We can repeat the previous partitioning example
using the fugacity method and demonstrate the equivalence of the two approaches
as before, but now applying the same fugacity to each phase.
M = C1V1 + C2V2 + C3V3 + C4V4
= Z1fV1 + Z2fV2 + Z3fV3 + Z4fV4
= f (V1Z1 + V2Z2 + V3Z3 + V4Z4)
Therefore,
f = M/(V1Z1 + V2Z2 + V3Z3 + V4Z4)
from which each C can be calculated as Zf, and the amount in each phase m is CV
or VZf.
In general,
f = M/ΣViZi Ci = Zif mi = ViZif
Worked Example 5.6
Using the data in Example 5.5, recalculate the distribution using fugacity and
assuming that ZA is 4 × 10–4 mol/m3 Pa.
ZW = ZA/KAW = 0.002
ZS = KSWZW = 0.03
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PHASE EQUILIBRIUM 103
ZF = KFWZW = 0.04
f = 10/(1000 × 4 × 10–4 + 20 × 0.002 + 10 × 0.03 + 0.05 × 0.04)
= 10/0.742 = 13.48
CA = ZAf = 0.0054, CW = 0.027, CS = 0.405, CF = 0.54
And the amounts and percentages are as before.
The procedure is simply to tabulate the volumes, the Z values, calculate and sum
the VZ products, and divide this into the total amount to obtain the prevailing
fugacity. This is readily done using a computer spreadsheet or program, and there
is no increase in mathematical complexity with increasing numbers of phases.
5.6.3 A Digression: The Heat Capacity Analogy to Z
The fugacity capacity Z is at first a difficult concept to grasp, since it has
unfamiliar units of mol/(volume × pressure). Heat capacity calculations provide a
precedent for introducing Z and may help to illustrate the fundamental nature of
this quantity.
The traditional heat capacity equation is written in the form
heat content (J) = mass of phase (kg) × heat capacity (J/kg K) × temperature (K) 
For example, water has a heat capacity of 4180 J/kg K, which is more familiar as
1 cal/g°C. We can rearrange this equation in terms of volumes instead of masses to
give
heat concentration (J/m3) = heat capacity (J/m3 K) × temperature (K)
This new volumetric heat capacity for water is 4,180,000 J/m3 K. The use of mass
rather than volume in heat capacities is an “accident” resulting from the greater ease
and accuracy of mass measurements compared to volume measurements, and the
complication that volumes change on heating, while mass remains constant.
The equilibrium criterion used above is temperature (K), whereas we are con-
cerned with fugacity (Pa). The quantity that partitions above is heat (J), whereas we
are concerned with amount of matter (moles). Replacing K by Pa and J by mol leads
to the analogous fugacity equation,
C (mol/m3) = Z (mol/m3Pa) × f (Pa)
Z is thus analogous to a heat capacity. Experience with heat calculations leads to a
mental concept of heat capacity as a property describing the “capacity of a phase
to absorb heat for a certain temperature rise.” For example, if 1 g of water (heat
capacity 4.2 J/g°C) absorbs 4.2 J, its temperature will rise 1°C. Copper with a lower
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104 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
heat capacity of 0.38 J/g°C requires absorption of only 0.38 J to cause the same rise
in temperature. Hydrogen gas has a large heat capacity of 14.3 J/g°C and thus
requires a great deal of heat to raise its temperature. These substances differ markedly
in their temperature response when heat is added. If 1000 J are added to equal masses
of 1 g of these substances, the copper becomes much hotter by 263°C (or 100/0.38),
while the water only heats up by 24°C (or 100/4.2) and the hydrogen by only 7°C
(or 100/14.3). Hydrogen and water can thus absorb or “soak up” larger quantities
of heat without becoming much hotter.
The fugacity capacity is similar. Phases of high Z (possibly sediments or fish)
are able to absorb a greater quantity of solute yet retain a low fugacity.It follows
that solutes will tend to partition into these high Z phases and build up a substantial
concentration yet retain a relatively low fugacity. Conversely, phases with low Z
values will tend to experience a large increase in f following absorption of a small
quantity of solute. A substance such as DDT is readily absorbed by fish and achieves
a high concentration at low fugacity. The Z value of DDT in fish is large. On the
other hand, DDT is not readily absorbed by water; indeed, it is hydrophobic or
“water hating.” Its Z value in water is very low.
This analogy between heat and fugacity capacity is perhaps best illustrated by
the following pair of numerically identical examples, the fugacity quantities being
given in parentheses.
Worked Example 5.7
A system consists of three phases 10 g of water (10 m3 of water) of heat capacity
4.2 J/g°C (fugacity capacity 4.2 mol/m3 Pa), 5g of copper (5 m3 of air) of heat
capacity 0.38 J/g°C (fugacity capacity 0.38 mol/m3 Pa), and 1 g of hydrogen (1 m3
of sediment) of heat capacity 14.3 J/g°C (i.e., fugacity capacity mol/m3 Pa). To this
system is added 582 J of heat (582 mol of solute). What is the heat (solute)
distribution at equilibrium, and what is the rise in temperature (fugacity) and heat
concentrations in J/g (concentrations in mol/m3). We assume for simplicity that the
initial temperature is 0°C, and the initial concentrations are also zero. (Note that Z
for a solute in air never has the above value.)
When approaching equilibrium, the temperatures (fugacities) will rise equally
to a new common value at T °C, (f) such that the amount of heat (solute) in each
phase will be
mass (g) × heat capacity × T or [volume (m3) × Zf]
Thus, the total will be the summation over the three phases, i.e.,
582 = 10 × 4.2 × T + 5 × 0.38 × T + 1 × 14.3 × T
Thus,
T = 582/(10 × 4.2 + 5 × 0.38 + 1 × 14.3) = 10°C (Pa)
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PHASE EQUILIBRIUM 105
1. Heat (moles) in water = 10 × 4.2 × 10 = 420 J (moles) 72%
2. Heat (moles) in copper (air) = 5 × 0.38 × 10 = 19 J (moles) 33%
3. Heat (moles) in hydrogen (sediment) = 1 × 14.3 × 10 = 143 J (moles) 25%
Total = 582 J and 100%
The concentrations are
The distribution of heat (moles) is influenced by the relative phase masses
(volumes) and the heat capacities (Z values). Despite the fact that the third phase is
small, its much larger heat capacity (Z) results in accumulation of a substantial
fraction of the total (25%), and its concentration is a factor of 3.4 and 38 greater
than the other two phases—which is, of course, the ratio of the heat capacities (the
ratio of Z values, this ratio being the partition coefficient).
This example could have been solved using heat capacity partition coefficients
but, of course, no such quantities are tabulated in handbooks. Indeed, any suggestion
that heat partition coefficients are useful would be treated with derision. In environ-
mental calculations, on the other hand, the use of Z is less conventional, and the
use of partition coefficients is routine. In essence, the use of fugacity capacities is
an attempt to bring to environmental calculations some of the procedural benefits
that are routinely enjoyed by the use of heat capacities.
Worked Example 5.8
A three-phase system has Z values Z1 = 5 × 10–4, Z2 = 1.0, and Z3 = 20 (all
mol/m3 Pa), and volumes V1 = 1000, V2 = 10, and V3 = 0.1 (all m3). Calculate the
distributions, concentrations, and fugacity when 1 mol of solute is distributed at
equilibrium between these phases. It is suggested that the calculations be done in
tabular form.
M = V1Z1f + V2Z2f + V3Z3f = f ΣViZi
Water 4.2 × 10 = 42 J/g (mol/m3)
Copper (air) 0.38 × 10 = 3.8 J/g (mol/m3)
Hydrogen (sediment) 14.3 × 10 = 143 J/g (mol/m3)
Phase Z V VZ C = Zf VC %
1 5 × 10–4 1000 0.5 4 × 10–5 0.04 4
2 1.0 10 10 0.08 0.80 80
3 20 0.1 2 1.6 0.16 16
Total 12.5 1.0 100
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106 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Therefore,
f = M/ΣViZi = 1.0/12.5 = 0.08
Again, a large value of Z or C does not necessarily imply a large quantity. Quantity
is controlled by VZ. Concentration is controlled by Z.
5.6.4 Sorption by Dispersed Phases
A frequently encountered environmental calculation is the estimation of the
fraction of a chemical that is present in a fluid that is sorbed to some dispersed
sorbing phases within that fluid. This is a special case of multimedia partitioning
involving only two phases. Examples are the estimation of the fraction of material
attached to aerosols in air or associated with suspended solids or with biotic matter
in water. The reason for this calculation is that the measured concentration is often
of the total (i.e., dissolved and sorbed) chemical, and it is useful to know what
fractions are in each phase. This is particularly useful when subsequently calculating
uptake of chemical by fish from water in which the partitioning may be only from
the dissolved solute.
It is useful to establish the general equations describing sorption in such cases
as follows. We designate the continuous phase by subscript A and the dispersed
phase by subscript B. The dispersed phase volume is typically a factor of 10–5 or
less, as compared to that of the continuous phase.
• The volumes (m3) are denoted VA and VB, and usually VA is much greater than VB.
• The equilibrium concentrations are denoted CA and CB mol/m3.
• The dimensionless partition coefficient KBA is CB/CA.
• The total amount of solute M moles is distributed between the two phases.
M = VACA + VBCB = VTCT
where CT is the total concentration. It can be assumed that VT, the total volume is
approximately VA. Now, 
CB = KBACA
Therefore,
M = CA(VA + VBKBA) = CTVA
Therefore,
CA = CT/(1 + KBAVB/VA) = CT/(1 + KBAvB)
where vB is the volume fraction of phase B and is approximately VB/VA. The fraction
dissolved (i.e., in the continuous phase) is
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PHASE EQUILIBRIUM 107
ϕA = CA /CT = 1/(1 + KBAvB)
and that sorbed is
ϕB = (1–CA/CT) = KBAvB/(1 + KBAvB)
The key quantity is thus KBAvB or the product of the dimensionless partition coeffi-
cient and the volume fraction of the dispersed sorbing phase. When this product is
1.0, half the solute is in each state. When it is smaller than 1.0, most is dissolved,
and when it exceeds 1.0, more is sorbed. When the phase B is solid, it is usual to
express the concentration CB in units of moles or grams per unit mass of B in which
case KBA has units of volume/mass or reciprocal density. For example, it is common
to use mg/L for CA, mg/kg for CB, and L/kg for KP; then, with M in mg and VA in
L, then it can be shown that
M = CA(VA + mBKP) = CTVA
where mB is the mass of sorbing phase (kg) from which
CA = CT/(1 + KPmB/VA) = CT/(1 + KPXB)
where XB is the concentration of sorbent in kg/L. The units of the partition coefficient
KBA or KP and concentration of sorbent vB or XB do not matter as long as their
product is dimensionless and consistent, i.e., the amounts of sorbing phase, contin-
uous phase, and chemical are the same in the definition of both the partition coef-
ficient and the sorbent concentration.
Care must be taken when interpreting sorbed concentrations to ascertain if they
represent the amount of chemical per unit volume or mass of sorbent, or the per
unit volume of the environmental phase such as water.
The analogous fugacity equations are simply
ϕA = VAZA/(VAZA + VBZB)
ϕB = VBZB/(VAZA + VBZB)
In some cases, it is preferable to calculate a Z value for a bulk phase consisting
of other phases in equilibrium. Examples are air plus aerosols; water plus suspended
solids; and soils consisting of solids, air, and water. If the total volume is VT, the
effective bulk Z value is ZT, and equilibrium applies, then the total amount of
chemical must be 
VTZTf = ΣviZif
Thus,
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108 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
ZT = Σ(Vi/VT)Zi =ΣviZi
where viis the volume fraction of each phase. The key point is that the component
Z values add in proportion to their volume fractions.
The use of bulk Z values helps to simplify calculations by reducing the number
of compartments, but it does assume that equilibrium exists within the bulk com-
partment.
Worked Example 5.9
An aquarium contains 10 m3 of water and 200 fish, each of volume 1 cm3. How
will 0.01 g (i.e., 10 mg) of benzene and the same mass of DDT partition between
water and fish, given that the fish are 5% lipids, and log KOW is 2.13 for benzene
and 6.19 for DDT?
KFW will be 0.05 KOW or 6.7 for benzene and 77440 for DDT.
CT is 0.001 g/m3 or 1 mg/m3 in both cases.
The fraction dissolved ϕ2 is 1/(1 + KFWvF), where vF is the volume fraction of fish,
i.e., 200 × 10–6/10 = 2 × 10–5.
For benzene, KFWvF is 0.00013.
For DDT, KFWvF is 1.55.
The fractions dissolved are 0.99987, essentially 1.0, for benzene and 0.39 for DDT.
The dissolved concentrations are thus 0.00099987 g/m3 or 0.99987 mg/m3 (benzene)
and 0.00039 g/m3 or 0.39 mg/m3 (DDT), and the sorbed concentrations (per m3 of
water) are 0.00013 mg/m3 and 0.61 mg/m3, respectively. The sorbed concentrations
per m3 of fish are 0.0067 g/m3 and 30 g/m3, respectively.
Example 5.10
A lake of volume 106 m3 contains 15 mg/L of sorbing material. The total
concentration of a chemical of KP equal to 105 L/kg is 1 mg/L. What are the dissolved
and sorbed concentrations?
Answer
0.4 mg/L dissolved and 0.6 mg/L sorbed.
5.6.5 Maximum Fugacity
When fugacities are calculated, it is advisable to check that the value deduced
is lower than the fugacity of the pure phase, i.e., the solid or liquid fugacity or, in
the case of gases, of atmospheric pressure. If these fugacities are exceeded, super-
saturation has occurred, a “maximum permissible fugacity” has been exceeded, and
the system will automatically “precipitate” a pure solute phase until the fugacity
drops to the saturation value. It is possible to calculate inadvertently and use (i.e.,
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PHASE EQUILIBRIUM 109
misuse) these “over-maximum” fugacities. For example, a chemical may be spilled
into a lake. The fugacity can be calculated as the amount spilled divided by VZ for
water. If the resulting fugacity exceeds the vapor pressure, the water has insufficient
capacity to dissolve all the chemical, and a separate pure chemical phase must be
present. A similar situation can apply when a pesticide is applied to soils.
It is likely that the maximum Z value that a solute can ever achieve is that of
the pure phase Zp. It may be useful to calculate ZP to ensure that no mistakes have
been made by grossly overestimating other Z values.
5.6.6 Solutes of Negligible Volatility
A problem arises when calculating values of the fugacity and fugacity capacity
of solutes that have a negligible or zero vapor pressure. Thermodynamically, the
problem is that of determining the reference fugacity. The practical problem may
be that no values of vapor pressure or air-water partition coefficients are published
or even exist. Examples are ionic substances, inorganic materials such as calcium
carbonate or silica and polymeric, or high-molecular-weight substances including
carbohydrates and proteins. Intuitively, no vapor pressure determination is needed
(or may be possible), because the substance does not partition into the atmosphere,
i.e., its “solubility” in air is effectively zero. Ironically, its air fugacity capacity can
still be calculated as (1/RT), but all the other (and the only useful) Z values cannot
be calculated, since H cannot be determined and indeed may be zero. Apparently,
the other Z values are infinite or at least are indeterminably large.
This difficulty is more apparent than real and is a consequence of the selection
of fugacity rather than activity as an equilibrium criterion. There are two remedies.
The first method, which is convenient but somewhat dishonest, is to assume a
fictitious and reasonable, but small, value for vapor pressure (such as 10–6 Pa) and
proceed through the calculations using this value. The result will be that Z for air
will be very small compared to the other phases, and negligible concentrations will
result in the air. It is obviously essential to recognize that these air concentrations
are fictitious and erroneous. The relative values of the other concentrations and Z
values will be correct, but the absolute fugacity will be meaningless.
The second method, which is less convenient but more honest, is to select a new
equilibrium criterion. We can illustrate this for air, water, and another phase(s) by
equating fugacities as follows:
f = CA/ZA = CW/ZW = CS/ZS
f = CART = CwPS/CS = CSPS/(CSKSW)
We can divide through by PS to give
f/PS = a = CART/PS = CW/CS = CS/CSKSW
The equilibrium criterion is now a, an activity that is dimensionless and is the
ratio of fugacity to vapor pressure. The new Z values with units of mol/m3 can be
defined as
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110 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
air Z = PS/RT, water Z = CS, sorbed Z = CSKSW
A saturated solution thus has an activity of 1.0. A zero or near-zero vapor pressure
can be used to calculate Z for air as zero or near zero.
In some cases, we may have to go farther, because we are uncertain about the
solubility CS. The simple expedient is then to multiply through by CS to give a new
equilibrium criterion A as
fCS/PS = A = CARTCS/PS = CW = CS/KSW
or, for air,
Z = PS/RTCS, water Z = 1.0, sorbed phase Z = KSW.
We are now using the water concentration or the equivalent equilibrium water
concentration as the criterion of equilibrium. This has been termed the aquivalent
concentration (Mackay and Diamond, 1989) and can be used for metals in ionic
form when the solubility is meaningless. 
The essential procedure is that, for most organic substances, Z is defined in air
as 1/RT, then all other Z values are deduced from it. In the “aquivalence” approach,
Z is arbitrarily set to 1.0 in water, and all other Z values are deduced from this basis
using partition coefficients. This approach is used in the EQC model for involatile
substances (Mackay et al., 1996).
5.6.7 Some Environmental Implications
Viewing the behavior of a solute in the environment in terms of Z introduces
new and valuable insights. A solute tends to migrate into (or stay in) the phase of
largest Z. Thus, SO2 and phenol tend to migrate into water, freons into air, and DDT
into sediment or biota. The phenomenon of bioconcentration is merely a manifes-
tation of Z in biota, which is much higher (by the bioconcentration factor) than Z
in the water. Occasionally, a solute such as inorganic mercury changes its chemical
form becoming organometallic (e.g., methylmercury). Its Z values change, and the
mercury now sets out on a new environmental journey with a destination of the new
phase in which Z is now large. In the case of mercury, the ionic form will sorb to
sediments or dissolve in water but will not appreciably bioconcentrate. The organic
form experiences a large Z in biota and will bioconcentrate. The metallic form tends
to evaporate.
Some solutes, such as DDT or PCBs, have very low Z values in water because
of their highly hydrophobic nature; i.e., they exert a high fugacity even at low
concentration, reflecting a large “escaping tendency.” They will therefore migrate
readily into any neighboring phase such as sediment, biota, or the atmosphere.
Atmospheric transport should thus be no surprise, and the contamination of biota
in areas remote from sites of use is expected. With this hindsight, it is not surprising
that these substances are found in the tissues of Arctic bears and Antarctic penguins!
From the environmental monitoring and analysis viewpoint, it is preferable to
sample and analyze phases in which Z is large, because it is in these phases that
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PHASE EQUILIBRIUM 111
concentrations are likelyto be large and thus easier to determine accurately. When
monitoring for PCBs in lakes, it is thus common to sample sediment or fish rather
than water, since the expected concentrations in water are very low. Likewise, those
concerned with PCB behavior in the atmosphere may measure the PCBs on aerosols
or in rainfall containing aerosols, since concentrations are higher than in the air.
In general, when assessing the likely environmental behavior of a new chemical,
it is useful to calculate the various Z values and from them identify the larger ones,
since it is likely that the high Z compartments are the most important. It is no
coincidence that solutes such as halogenated hydrocarbons, about which there is
great public concern, have high Z values in humans!
It should be borne in mind that, when calculating the environmental behavior of
a solute, Z values are needed only for the phases of concern. For example, if no
atmospheric partitioning is considered, it is not necessary to know the air-water
partition coefficient or H. An arbitrary value of H can be used to define Z for water
and other phases, because H cancels. Intuitively, it is obvious that H, or vapor
pressure, play no role in influencing water-fish-sediment equilibria.
In summary, in this chapter we have introduced the concept of equilibrium
existing between phases and have shown that this concept is essentially dictated by
the laws of thermodynamics. Fortunately, we do not need to use or even understand
the thermodynamic equations on which equilibrium relationships are based. How-
ever, it is useful to use these relationships for purposes such as correlation of partition
coefficients. It transpires that there are two approaches that can be used to conduct
equilibrium calculations. First is to develop and use empirical correlations for par-
tition coefficients. Using these coefficients, it is possible to calculate the partitioning
of the chemical in a multimedia environment.
The second approach, which we prefer, is to use an equilibrium criterion such
as fugacity or, in the case of involatile chemicals, an aquivalent concentration. The
criterion can be related to concentration for each chemical and for each medium
using a proportionality constant or Z value. The Z value can be calculated from
fundamental equations or from partition coefficients. We have established recipes
for the various Z values in these media using information on the nature of the media
and the physical chemical properties of the substance of interest. This enables us to
undertake simple multimedia partitioning calculations.
5.7 LEVEL I CALCULATIONS
Calculation of the equilibrium Level I distribution of a chemical is simple, but
it can be tedious. It is ideal for implementation on a computer. The obvious steps are
1. Definition of the environment, i.e., volumes and compositions
2. Input of relevant physical chemical properties
3. Calculation of Z values for each medium (see Table 5.1)
4. Input of chemical amount
5. Calculation of fugacity, and hence concentrations, amounts, and percent distribu-
tion
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112 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Table 5.1 Table 5.1 Summary of Definitions of Z Values and Equations Used in Level 
I Calculations
Definitions of Z values
ZA = 1/RT
ZW = 1/H = CS/PS = ZA/KAW
ZO = ZW KOW (octanol)
ZP = 1/vPPS (pure phase)
ZS = yOCKOCZW (ρS/1000) (soils, sediments)
 KOC = 0.41 KOW
ZQ = ZAKQA (aerosols)
 KQA = 6 × 106/PSL or yOMKOA (ρQ/1000)
ZB = LZO (fish, biota)
where R is the gas constant (8.314 Pa m3/mol K)
T is absolute temperature (K)
H is Henry’s law constant (Pa m3/mol)
CS is solubility in water (mol/m3)
PS is vapor pressure (Pa)
KAW is air–water partition coefficient
KOW is octanol–water partition coefficient
KOC is organic carbon–water partition coefficient
vP is molar volume of pure chemical (m3/mol)
yOC is mass fraction organic carbon
yOM is mass fraction organic matter
ρS is density of soil, etc., (kg/m3)
ρQ is density of aerosols (kg/m3)
KQA is aerosol–air partition coefficient
PSL is vapor pressure of liquid or subcooled liquid
L is lipid content (volume fraction)
Note that the Z value of a bulk phase consisting of continuous and dispersed material, e.g., 
water plus suspended solids, is given by the volume fraction weighted Z values.
ZT = Σ viZi
where vi is the volume fraction of phase i.
Fugacity equation
f = M/ΣViZi
where f is fugacity (Pa)
M is total amount of chemical (mol)
V is volume (m3)
Ci = Zif mi = CiVi = ViZif
mi is amount in phase i (mol)
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PHASE EQUILIBRIUM 113
Programs to accomplish this calculation are available on the website
www.trentu.ca/envmodel. The “Level I” calculation (Figure 5.6) is the simplest
multimedia environmental calculation possible. The EQC model contains a Level I
calculation for a regional environment as well as other, more advanced, calculations.
To assist the reader to understand the nature of this calculation, two “fugacity
forms” (Figures 5.7 and 5.8) are included at the end of this chapter. They contain a
worked example for a hypothetical chemical. Blank forms are provided in the
Appendix that may be reproduced for use in other examples. The results of the
computer calculations should be consistent with these hand calculations.
5.8 CONCLUDING EXAMPLES
The concepts presented in this chapter are best grasped by working through
examples. A chemical can be selected from those listed in Chapter 3 and the
Figure 5.6 Specimen output of a Level I calculation.
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114 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
properties used with assumed media volumes to deduce the distribution of a defined
quantity of the substance between these media.
Worked Example 5.11
A chemical has Z values in air of 4 × 10–4 mol/m3Pa, in water of 10–3 mol/m3Pa,
and in sediment of 5 mol/m3Pa. What are the air and sediment concentrations in
equilibrium with a water concentration of 2 mol/m3?
Using subscripts W for water, A for air, and S for sediment,
CW = 2.0 ZW = 10–3 f = CW/ZW = 2000 Pa
CA = f ZA = 2000 × 4 × 10–4 = 0.8 mol/m3
CS = f ZS = 2000 × 5 = 10000 mol/m3
Figure 5.7 Fugacity form 1 for deducing Z values.
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PHASE EQUILIBRIUM 115
This example illustrates the fundamental simplicity of the equilibrium calculation
and shows that the Z values are intimately related to the partition coefficients. The
dimensionless air-water partition coefficient KAW, which is defined as CA/CW, is
clearly 0.8/2.0 or 0.4. Likewise, CS/CW is 10000/2.0 or 5000. These ratios are also
ratios of Z values. 
5.9 CONCLUDING EXAMPLE
Select two nonionizing substances from Table 3.5, one a liquid and the other a
solid, preferably with a melting point exceeding 100°C, and with log KOW in the
range 3 to 6.
Figure 5.8 Fugacity form 2 for deducing Z values.
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116 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Calculate the following physical-chemical properties for these substances at
25°C: fugacity ratio (1.0 for the liquid); Henry’s law constant; solubilities (mol/m3)
in air, water, and octanol; the three partition coefficients between these phases and
the activity coefficients in water and octanol. Assume a molar volume of 18 cm3/mol
for water and 120 cm3/mol for octanol. In the case of the solid, calculate both the
solid and supercooled liquid solubilities.
Calculate the Z values in air, water, octanol, and in the pure chemical phase
(assuming a density of 1 g/cm3 if the chemical’s density is not readily available from
a handbook).
Using Fugacity Form 1 as a template, calculate Z values in the following media:
• soil solids containing 1% organic carbon
• suspended sediment solids containing 15% organic carbon
• bottom sediment solids containing 5% organic carbon
• aerosol particles
• fish containing 5% lipidAssume KOC to be 0.41 KOW and all solid densities to be 2000 kg/m3.
Using Fugacity Form 2, calculate the fugacity, concentrations, and distribution
of 100 kg of each chemical in an environment consisting of these volumes:
air 109 m3
water 106 m3
soil solids 104 m3
suspended sediment solids 50 m3
bottom sediment solids 103 m3
aerosols 1 m3
fish 5 m3
Alternatively, use the environment that was deduced in the concluding example from
Chapter 4.
Write a short account of the partitioning behavior of these substances. Where
would you analyze for monitoring purposes? Why? At what fraction of the saturation
conditions is the chemical present, i.e., the ratio of fugacity to vapor pressure? In
the water and atmosphere, what fractions of the total concentration are present in
each of the dispersed phases of aerosols, suspended sediment, and fish?
CH05 Page 116 Monday, January 15, 2001 1:49 PM
 
117
 
CHAPTER
 
 6
Advection and Reactions
 
6.1 INTRODUCTION
 
In Level I calculations, it is assumed that the chemical is conserved; i.e., it is
neither destroyed by reactions nor conveyed out of the evaluative environment by
flows in phases such as air and water. These assumptions can be quite misleading
when determining of the impact of a given discharge or emission of chemical.
First, if a chemical, such as glucose, is reactive and survives for only 10 hours
as a result of its susceptibility to rapid biodegradation, it must pose less of a threat
than PCBs, which may survive for over 10 years. But the Level I calculation treats
them identically. Second, some chemical may leave the area of discharge rapidly as
a result of evaporation into air, to be removed by advection in winds. The contam-
ination problem is solved locally, but only by shifting it to another location. It is
important to know if this will occur. Indeed, recently, considerable attention is being
paid to substances that are susceptible to long-range transport. Third, it is possible
that, in a given region, local contamination is largely a result of inflow of chemical
from upwind or upstream regions. Local efforts to reduce contamination by control-
ling local sources may therefore be frustrated, because most of the chemical is
inadvertently imported. This problem is at the heart of the Canada–U.S., and Scan-
dinavia–Germany–U.K. squabbles over acid precipitation. It is also a concern in
relatively pristine areas such as the Arctic and Antarctic, where residents have little
or no control over the contamination of their environments. 
In this chapter, we address these issues and devise methods of calculating the
effect of advective inflow and outflow and degrading reactions on local chemical
fate and subsequent exposures. It must be emphasized that, once a chemical is
degraded, this does not necessarily solve the problem. Toxicologists rarely miss an
opportunity to point out reactions, such as mercury methylation or benzo(a)pyrene
oxidation, in which the product of the reaction is more harmful than the parent
compound. For our immediate purposes, we will be content to treat only the parent
compound. Assessment of degradation products is best done separately by having
the degradation product of one chemical treated as formation of another.
 
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118 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
A key concept in this discussion that was introduced earlier, and is variously
termed 
 
persistence, lifetime, residence time,
 
 or 
 
detention time 
 
of the chemical.
In a steady-state system, as shown in Figure 6.1a, if chemical is introduced at a
rate of E mol/h, then the rate of removal must also be E mol/h. Otherwise, net
accumulation or depletion will occur. If the amount in the system is M mol, then,
on average, the amount of time each molecule spends in the steady-state system will
be M/E hours. This time, 
 
τ
 
, is a 
 
residence time 
 
and is also called a 
 
detention time
 
or 
 
persistence
 
. Clearly, if a chemical persists longer, there will be more of it in the
system. The key equation is
 
τ
 
 = 
 
M/E or M
 
 = 
 
τ
 
 E
 
This concept is routinely applied to retention time in lakes. If a lake has a volume
of 100,000 m
 
3
 
, and if it receives an inflow of 1000 m
 
3
 
 per day, then the retention
time is 100,000/1000 or 100 days. A mean retention time of 100 days does not imply
Figure 6.1 Diagram of a steady-state evaluative environment subject to (a) advective flow, (b)
degrading reactions, (c) both, and (d) the time course to steady state.
 
CH06 Page 118 Monday, January 15, 2001 1:50 PM
 
ADVECTION AND REACTIONS 119
 
that all water will spend 100 days in the lake. Some will bypass in only 10 days,
and some will persist in backwaters for 1000 days but, on average, the residence
time will be 100 days.
The reason that this concept is so important is that chemicals exhibit variable
lifetimes, ranging from hours to decades. As a result, the amount of chemical present
in the environment, i.e., the inventory of chemical, varies greatly between chemicals.
We tend to be most concerned about persistent and toxic chemicals, because rela-
tively small emission rates (E) can result in large amounts (M) present in the
environment. This translates into high concentrations and possibly severe adverse
effects. A further consideration is that chemicals that survive for prolonged periods
in the environment have the opportunity to make long and often tortuous journeys.
If applied to soil, they may evaporate, migrate onto atmospheric particles, deposit
on vegetation, be eaten by cows, be transferred to milk, and then consumed by
humans. Chemicals may migrate up the food chain from water to plankton to fish
to eagles, seals, and bears. Short-lived chemicals rarely survive long enough to
undertake such adventures (or misadventures).
This lengthy justification leads to the conclusion that, if we are going to discharge
a chemical into the environment, it is prudent to know
 
1.
 
how long the chemical will survive, i.e., 
 
τ
 
, and 
 
2. what causes its removal or “death”
 
This latter knowledge is useful, because it is likely that situations will occur in
which a common removal mechanism does not apply. For example, a chemical may
be potentially subject to rapid photolysis, but this is not of much relevance in long,
dark arctic winters or in deep, murky sediments.
In the process of quantifying this effect, we will introduce rate constants, advec-
tive flow rates and, ultimately, using the fugacity concept, quantities called D values,
which prove to be immensely convenient. Indeed, armed with Z values and D values,
the environmental scientist has a powerful set of tools for calculation and interpre-
tation.
It transpires that there are two primary mechanisms by which a chemical is
removed from our environment: advection and reaction, which we discuss individ-
ually and then in combination.
 
6.2 ADVECTION
 
Strangely, “advection” is a word rarely found in dictionaries, so a definition is
appropriate. It means
 
 the directed movement of chemical by virtue of its presence
in a medium that happens to be flowing.
 
 A lazy canoeist is advected down a river.
PCBs are advected from Chicago to Buffalo in a westerly wind. The rate of advection
N (mol/h) is simply the product of the flowrate of the advecting medium, G (m
 
3
 
/h),
and the concentration of chemical in that medium, C (mol/m
 
3
 
), namely,
N = GC mol/h 
 
CH06 Page 119 Monday, January 15, 2001 1:50 PM
 
120 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
Thus, if there is river flow of 1000 m
 
3
 
/h (G) from A to B of water containing 0.3
mol/m
 
3
 
 (C) of chemical, then the corresponding flow of chemical is 300 mol/h (N).
Turning to the evaluative environment, it is apparent that the primary candidate
advective phases are air and water. If, for example, there was air flow into the 1
squarekilometre evaluative environment at 10
 
9
 
 m
 
3
 
/h, and the volume of the air in
the evaluative environment is 6 
 
×
 
 10
 
9
 
 m
 
3
 
, then the residence time will be 6 hours,
or 0.25 days. Likewise, the flow of 100 m
 
3
 
/h of water into 70,000 m
 
3
 
 of water results
in a residence time of 700 hours, or 29 days. It is easier to remember residence
times than flow rates; therefore, we usually set a residence time and from it deduce
the corresponding flow rate.
Burial of bottom sediments can also be regarded as an advective loss, as can
leaching of water from soils to groundwater. Advection of freons from the tropo-
sphere to the stratosphere is also of concern in that it contributes to ozone depletion.
 
6.2.1 Level II Advection Algebra Using Partition Coefficients
 
If we decree that our evaluative environment is at steady state, then air and water
inflows must equal outflows; therefore, these inflow rates, designated G m
 
3
 
/h, must
also be outflow rates. If the concentrations of chemical in the phase of the evaluative
environment is C mol/m
 
3
 
, then the outflow rate will be G C mol/h. This concept is
often termed the 
 
continuously stirred tank reactor,
 
 or CSTR, assumption. The basic
concept is that, if a volume of phase, for example air, is well stirred, then, if some
of that phase is removed, that air must have a concentration equal to that of the
phase as a whole. If chemical is introduced to the phase at a different concentration,
it experiences an immediate change in concentration to that of the well mixed, or
CSTR, value. The concentration experienced by the chemical then remains constant
until the chemical is removed. The key point is that the outflow concentration equals
the prevailing concentration. This concept greatly simplifies the algebra of steady-
state systems. Essentially, we treat air, water, and other phases as being well mixed
CSTRs in which the outflow concentration equals the prevailing concentration. We
can now consider an evaluative environment in which there is inflow and outflow
of chemical in air and water. It is convenient at this stage to ignore the particles in
the water, fish, and aerosols, and assume that the material flowing into the evaluative
environment is pure air and pure water. Since the steady-state condition applies, as
shown in Figure 6.1a, the inflow and outflow rates are equal, and a mass balance
can be assembled. The total influx of chemical is at a rate G
 
A
 
C
 
BA
 
 in air, and G
 
W
 
C
 
BW
 
in water, these concentrations being the “background” values. There may also be
emissions into the evaluative environment at a rate E. The total influx I is thus
I = E + G
 
A
 
C
 
BA
 
 + G
 
W
 
C
 
BW
 
 mol/h
Now, the concentrations within the environment adjust instantly to values C
 
A
 
and C
 
W
 
 in air and water. Thus, the outflow rates must be G
 
A
 
C
 
A
 
 and G
 
W
 
C
 
W
 
. These
outflow concentrations could be constrained by equilibrium considerations; for
example, they may be related through partition coefficients or through Z values to
a common fugacity.
 
CH06 Page 120 Monday, January 15, 2001 1:50 PM
 
ADVECTION AND REACTIONS 121
 
This enables us to conceive of, and define, our first Level II calculation in which
we assume equilibrium and steady state to apply, inputs by emission and advection
are balanced exactly by advective emissions, and equilibrium exists throughout the
evaluative environment. All the phases are behaving like individual CSTRs.
Of course, starting with a clean environment and introducing these inflows, it
would take the system some time to reach steady-state conditions, as shown in Figure
6.1d. At this stage, we are not concerned with how long it takes to reach a steady
state, but only the conditions that ultimately apply at steady state. We can therefore
develop the following equations, using partition coefficients and later fugacities.
I = E + G
 
A
 
C
 
BA
 
 + G
 
W
 
C
 
BW
 
 = G
 
A
 
C
 
A
 
 + G
 
W
 
C
 
W
 
But 
C
 
A
 
 = K
 
AW
 
C
 
W
 
Therefore,
I = C
 
W
 
[G
 
A
 
K
 
AW
 
 + G
 
W
 
] and C
 
W
 
 = I/[G
 
A
 
K
 
AW
 
 + G
 
W
 
]
Other concentrations, amounts (m), and the total amount (M) can be deduced
from C
 
W
 
. The extension to multiple compartment systems is obvious. For example,
if soil is included, the concentration in soil will be in equilibrium with both C
 
A
 
 and
C
 
W
 
.
 
6.2.2 Level II Advection Algebra Using Fugacity
 
We assume a constant fugacity f to apply within the environment and to the
outflowing media, thus,
I = G
 
A
 
Z
 
A
 
f + G
 
W
 
Z
 
W
 
f = f(G
 
A
 
Z
 
A
 
 + G
 
W
 
Z
 
W
 
)
f = I/(G
 
A
 
Z
 
A
 
 + G
 
W
 
Z
 
W
 
)
 
or, in general,
f = I/
 
Σ
 
G
 
i
 
Z
 
i
 
from which the fugacity and all concentrations and amounts can be deduced.
 
Worked Example 6.1
 
An evaluative environment consists of 10
 
4
 
 m
 
3
 
 air, 100 m
 
3
 
 water, and 1.0 m
 
3
 
 soil.
There is air inflow of 1000 m
 
3
 
/h and water inflow of 1 m
 
3
 
/h at respective chemical
concentrations of 0.01 mol/m
 
3
 
 and 1 mol/m
 
3
 
. The Z values are air 4 
 
× 
 
10
 
–4
 
, water
 
CH06 Page 121 Monday, January 15, 2001 1:50 PM
 
122 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
0.1, and soil 1.0. There is also an emission of 4 mol/h. Calculate the fugacity
concentrations, persistence amounts and outflow rates.
I = E + G
 
A
 
C
 
BA
 
 + G
 
W
 
C
 
BW
 
 = 4 + 1000 
 
×
 
 0.01 + 1 
 
×
 
 1 = 15 mol/h
 
Σ
 
GZ = 1000 
 
×
 
 4 
 
×
 
 10
 
–4
 
 +1 
 
×
 
 10
 
–1
 
 = 0.5 f = I/
 
Σ
 
GZ = 30 Pa
C
 
A
 
 = 0.012 C
 
W
 
 = 3 C
 
S
 
 = 30 mol/m
 
3
 
m
 
A
 
 = 120 m
 
W
 
 = 300 m
 
S
 
 = 30 M (total) = 450 mol
G
 
A
 
C
 
A
 
 = 12 G
 
W
 
CW = 3 GSCS = 0 Total = 15 = I mol/h
τ = 450/15 =30 h
In this example, the total amount of material in the system, M, is 450 mol. The
inflow rate is 15 mol/h, thus the residence time or the persistence of the chemical
is 30 hours. This proves to be a very useful time. Note that the air residence time
is 10 hours, and the water residence time is 100 hours; thus, the overall residence
time of the chemical is a weighted average, influenced by the extent to which the
chemical partitions into the various phases. The soil has no effect on the fugacity
or the outflow rates, but it acts as a “reservoir” to influence the total amount present
M and therefore the residence time or persistence.
6.2.3 D values
The group G Z, and other groups like it, appear so frequently in later calculations
that it is convenient to designate them as D values, 
i.e.,
G Z = D mol/Pa h 
The rate, N mol/h, then equals D f. These D values are transport parameters, with
units of mol/Pa h. When multiplied by a fugacity, they give rates of transport. They
are thus similar in principle to rate constants, which, when multiplied by a mass of
chemical, give a rate of reaction. Fast processes have large D values. We can write
the fugacity equation for the evaluative environment in more compact form, as shown
below:
f = I/(DAA + DAW) = I/ΣDAi
where DAA = GAZA, DAW = GWZW, and the first subscript A refers to advection.
Recalculating Example 6.1,
DAA = 0.4 and DAW = 0.1 and ΣDAi = 0.5
Therefore,
CH06 Page 122 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 123
f = 15/0.5 = 30 
and the rates of output, Df, are 12 and 3 mol/h, totaling 15 mol/h as before.
It is apparent that the air D value is larger and most significant. D values can be
added when they are multiplied by a common fugacity. Therefore, it becomes
obvious which D value, and hence which process, is most important. We can arriveat the same conclusion using partition coefficients, but the algebra is less elegant. 
Note that how the chemical enters the environment is unimportant, all sources
being combined or lumped in I, the overall input. This is because, once in the
environment, the chemical immediately achieves an equilibrium distribution, and it
“forgets” its origin.
6.2.4 Advective Processes
In an evaluative environment, there are several advective flows that convey
chemical to and from the environment, namely,
1. inflow and outflow of air 
2. inflow and outflow of water
3. inflow and outflow of aerosol particles present in air
4. inflow and outflow of particles and biota present in water
5. transport of air from the troposphere to the stratosphere, i.e., vertical movement
of air out of the environment 
6. sediment burial, i.e., sediment being conveyed out of the well mixed layer to depths
sufficient that it is essentially inaccessible
7. flow of water from surface soils to groundwater (recharge)
It also transpires that there are several advective processes which can apply to
chemical movement within the evaluative environment. Notable are rainfall, water
runoff from soil, sedimentation, and food consumption, but we delay their treatment
until later.
In situations 1 through 4, there is no difficulty in deducing the rate as GC or Df,
where G is the flowrate of the phase in question, C is the concentration of chemical
in that phase, and the Z value applies to the chemical in the phase in which it is
dissolved or sorbed.
For example, aerosol may be transported to an evaluative world in association
with the inflow of 1012 m3/h of air. If the aerosol concentration is 10–11 volume
fraction, then the flowrate of aerosol GQ is 10 m3/h. The relevant concentration of
chemical is that in the aerosol, not in the air, and is normally quite high, for example,
100 mol/m3. Therefore, the rate of chemical input in the aerosol is 1000 mol/h. This
can be calculated using the D and f route as follows, giving the same result.
If ZQ = 108, then 
f = CQ/ZQ = 100/108 = 10–6 Pa
DAQ = GQZQ = 10 × 108 = 109
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124 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Therefore,
N = Df = 109 × 10–6 = 1000 mol/h
Treatment of transport to the stratosphere is somewhat more difficult. We can
conceive of parcels of air that migrate from the troposphere to the stratosphere at
an average, continuous rate, G m3/h, being replaced by clean stratospheric air that
migrates downward at the same rate. We can thus calculate the D value. As discussed
by Neely and Mackay (1982), this rate should correspond to a residence time of the
troposphere of about 60 years, i.e., G is V/τ. Thus, if V is 6 × 109 and τ is 5.25 ×
105 h, G is 11400 m3/h. This rate is very slow and is usually insignificant, but there
are situations in which it is important. 
We may be interested in calculating the amount of chemical that actually reaches
the stratosphere, for example, freons that catalyze the decomposition of ozone. This
slow rate is thus important from the viewpoint of the receiving stratospheric phase,
but is not an important loss from the delivering, or tropospheric, phase. Second, if
a chemical is very stable and is only slowly removed from the atmosphere by reaction
or deposition processes, then transfer to the troposphere may be a significant mech-
anism of removal. Certain volatile halogenated hydrocarbons tend to be in this class.
If we emit a chemical into the evaluative world at a steady rate by emissions and
allow for no removal mechanisms whatsoever, its concentrations will continue to
build up indefinitely. Such situations are likely to arise if we view the evaluative
world as merely a scaled-down version of the entire global environment. There is
certainly advective flow of chemical from, for example, the United States to Canada,
but there is no advective flow of chemical out of the entire global atmospheric
environment, except for the small amounts that transfer to the stratosphere. Whether
advection is included depends upon the system being simulated. In general, the
smaller the system, the shorter the advection residence time, and the more important
advection becomes.
Sediment burial is the process by which chemical is conveyed from the active
mixed layer of accessible sediment into inaccessible buried layers. As was discussed
earlier, this is a rather naive picture of a complex process, but at least it is a starting
point for calculations. The reality is that the mixed surface sediment layer is rising,
eventually filling the lake. Typical burial rates are 1 mm/year, the material being
buried being typically 25% solids, 75% water. But as it “moves” to greater depths,
water becomes squeezed out. Mathematically, the D value consists of two terms,
the burial rate of solids and that of water. 
For example, if a lake has an area of 107 m2 and has a burial rate of 1 mm/year,
the total rate of burial is 10,000 m3/year or 1.14 m3/h, consisting of perhaps 25%
solids, i.e., 0.29 m3/h of solids (GS) and 0.85 m3/h of water (GW). The rate of loss
of chemical is then
GSCS + GWCW = GSZSf + GWZWf = f(DAS + DAW)
Usually, there is a large solid to pore water partition coefficient; therefore, CS
greatly exceeds CW or, alternatively, ZS is very much greater than ZW, and the term
CH06 Page 124 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 125
DAS dominates. A residence time of solids in the mixed layer can be calculated as
the volume of solids in the mixed layer divided by GS. For example, if the depth of
the mixed layer is 3 cm, and the solids concentration is 25%, then the volume of
solids is 75,000 m3 and the residence time is 260,000 hours, or 30 years. The
residence time of water is probably longer, because the water content is likely to be
higher in the active sediment than in the buried sediment. In reality, the water would
exchange diffusively with the overlaying water during that time period.
As discussed in Chapter 5, there are occasions in which it is convenient to
calculate a “bulk” Z value for a medium containing a dispersed phase such as an
aerosol. This can be used to calculate a “bulk” Z value, thus expressing two loss
processes as one. D is then GZ where G is the total flow and Z is the bulk value.
6.3 DEGRADING REACTIONS
The word reaction requires definition. We regard reactions as processes that alter
the chemical nature of the solute, i.e., change its chemical abstract system (CAS)
number. For example, hydrolysis of ethyl acetate to ethanol and acetic acid is
definitely a reaction, as is conversion of 1,2-dichlorobenzene to 1,3-dichlorobenzene,
or even conversion of cis butene 2 to trans butene 2. In contrast, processes that
merely convey the chemical from one phase to another, or store it in inaccessible
form, are not reactions. Uptake by biota, sorption to suspended material, or even
uptake by enzymes are not reactions. A reaction may subsequently occur in these
locations, but it is not until the chemical structure is actually changed that we
consider reaction to have occurred. In the literature, the word reaction is occasionally,
and wrongly, applied to these processes, especially to sorption.
We have two tasks. The first is to assemble the necessary mathematical frame-
work for treating reaction rates using rate constants, and the second is to devise
methods of obtaining information on values of these rate constants.
6.3.1 Reaction Rate Expressions
We prefer, when possible, to use a simple first-order kinetic expression for all
reactions. The basic rate equation is
rate N = VCk = Mk mol/h
where V is the volume of the phase (m3), C is the concentration of the chemical
(mol/m3), M is the amount of chemical, and k is the first-order rate constant with
units of reciprocal time. The group VCk thus has units of mol/h.
The classical application of this equation is to radioactive decay, which is usually
expressed in the forms
dM/dt = –kM or dC/dt = –Ck
The use of C insteadof M implies that V does not change with time.
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126 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Integrating from an initial condition of CO at zero time gives the following
equations:
ln(C/CO) = –kt or C = CO exp(–kt)
Rate constants have units of frequency or reciprocal time and are therefore not
easily grasped or remembered. A favorite trick question of examiners is to ask a
student to convert a rate constant of 24 h–1 into reciprocal days. The correct answer
is 576 days–1, so beware of this conversion! It is more convenient to store and
remember half-lives, i.e., the time, τ1/2, which is the time required for C to decrease
to half of CO. This can be related to the rate constant as follows.
When C = 0.5 CO, then t = τ1/2
ln (0.5) = –kτ1/2, therefore, τ1/2 = 0.693/k
For example, an isotope with a half-life of 10 hours has a rate constant, k, of
0.0693 h–1.
6.3.2 Non-First-Order Kinetics
Unfortunately, there are many situations in which the real reaction rate is not a
first-order reaction. Second-order rate reactions occur when the reaction rate is
dependent on the concentration of two chemicals or reactants. For example, if
A + B → D + E
then the rate of the reaction is dependent on the concentration of both A and B.
Therefore, the reaction rate is as follows:
N = Vk CA CB
Reactant “B” is often another chemical, but it could be another environmental
reactant such as a microbial population or solar radiation intensity. Third-order
reaction rates, when the rate of reaction is dependent on the concentration of three
reactants (N = Vk CA CB CC), are very rare and are unlikely to occur under environ-
mental conditions.
We can often circumvent these complex reaction rate equations by expressing
them in terms of a pseudo first-order rate reaction. The primary assumption is that
the concentration of reactant “B” is effectively constant and will not change appre-
ciably as the reaction proceeds. Thus, the constant k and concentration of reactant
“B” can be lumped into a new rate constant, kP, and the second-order reaction
becomes a pseudo first-order reaction. Therefore,
N = Vk CACB
CH06 Page 126 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 127
and
kP = k CB
Therefore,
N = VkP CA
which has the form of a simple first-order reaction. Examples of pseudo first-order
reactions include photolysis reactions where reactant “B” is the solar radiation
intensity (I, in photons/s) or microbial degradations processes where “B” is the
populations of microorganisms. Reactions between two chemicals can also be con-
sidered a pseudo first-order reaction when CA << CB, so the concentration of B does
not change as the reaction proceeds.
Second-order rate expressions also arise when a chemical reacts with itself,
giving rise to a messy quadratic equation. Thus, if A + A → D, the rate equation is
N = Vk CACA = Vk CA2
Fortunately, most pollutants are present at low concentrations and tend not to react
with themselves, so these types of reactions are rare.
Zero-order expressions occasionally occur in which the rate is independent of
the concentration of the chemical and is thus proportional to concentration to the
power zero. Including zero-order expressions in mass balance models is potentially
dangerous, because the equations can now predict a positive rate of reaction, even
when there is no chemical present. It is embarrassing when computer models cal-
culate negative concentrations of chemicals.
Our strategy is to use every reasonable excuse to force first-order kinetics on
systems by lumping parameters in k. The dividends that arise are worth the effort,
because subsequent calculations are much easier.
Perhaps most worrisome are situations in which we treat the kinetics of microbial
degradation of chemicals. It is possible that, at very low concentrations, there is a
slower or even no reaction, because the required enzyme systems are not “turned
on.” At very high concentrations, the enzyme may be saturated; therefore, the rate
of degradation ceases to be controlled by the availability of the chemical and becomes
controlled by the availability of enzyme. In other cases, the rate of conversion may
be influenced by the toxicity of the chemical to the organism or by the presence of
co-metabolites, chemicals that the enzyme recognizes as being similar to that of the
chemical of interest. Microbiologists have no difficulty conceiving of a multitude
of situations in which chemical kinetics become very complicated and very difficult
to predict and express. They seem to obtain a certain perverse delight in finding
these situations.
Saturation kinetics is usually treated by the Michaelis–Menten equation, which
can be derived from first principles or, more simply, by writing down the basic first-
order equation and multiplying the rate expression by the group shown below.
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128 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Basic expression N = VCk
Group CM/(C + CM)
Combined expression N = VCCMk/(C + CM)
When C is small compared to CM, the rate reduces to VCk. When C is large compared
to CM, it reduces to VCMk, which is independent of C, is constant, and corresponds
to the maximum, or zero-order rate. The concentration, CM, therefore corresponds
to the concentration that gives the maximum rate using the basic expression. When
C equals CM, the rate is half the maximum value. This can be (and usually is)
expressed in terms of other rate constants for describing the kinetics of the associ-
ation of the chemical with the enzyme.
The rate expression is usually written in biochemistry texts in the form
N/V = C vM/(C + kM)
where vM is a maximum rate or velocity equivalent to kCM, and kM is equivalent to
CM and is viewed as a ratio of rate constants. A somewhat similar expression, the
Monod equation, is used to describe cell growth.
If kinetics are not of the first order, it may be necessary to write the appropriate
equations and accept the increased difficulty of solution. A somewhat cunning but
unethical alternative is to guess the concentration, calculate the rate N using the
non-first-order expression, then calculate the pseudo first-order rate constant in the
expression. For example, if a reaction is second order and C is expected to be about
2 mol/m3, V is 100 m3, and the second-order rate constant, k2, is 0.01 m3/mol·h,
then N equals 4 mol/h. We can set this equal to VCk; then, k is 0.02 h–1. Essentially,
we have lumped Ck2 as a first-order rate constant. This approach must be used, of
course, with extreme caution, because k depends on C.
6.3.3 Additivity of Rate Constants
A major advantage of forcing first-order kinetics on all reactions is that, if a
chemical is susceptible to several reactions in the same phase, with rate constants
kA, kB, kC, etc., then the total rate constant for reaction is (kA + kB + kC), i.e., the
rate constants are simply added. Another favorite trick of perverse examiners is to
inform a student that a chemical reacts by one mechanism with a half-life of 10
hours, and by another mechanism with a half-life of 20 hours, and asks for the total
half-life. The correct answer is 6.7 hours, not 30 hours. Half-lives are summed as
reciprocals, not directly.
6.3.4 Level II Reaction Algebra Using Partition Coefficients
We can now perform certain calculations describing the behavior of chemicals
in evaluative environments. The simplest is a Level II equilibrium steady-state
CH06 Page 128 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 129
reaction situation in which there is no advection, and there is a constant inflow of
chemical in the form of an emission, as depicted in Figure 6.1b. When a steady state
is reached, there must be an equivalent loss in the form of reactions. Starting from
a clean environment, the concentrations would build up until they reach a level such
that the rates of degradation or loss equal the total rate of input. Wefurther assume
that the phases are in equilibrium, i.e., transfer between them is very rapid. As a
result, the concentrations are related through partition coefficients, or a common
fugacity applies. The equations are as follows:
E = V1C1k1 + V2C2k2 etc. = ΣViCiki
Using partition coefficients,
E = ΣViCwKiwki = CwΣViKiwki
from which Cw can be deduced, followed by other concentrations, amounts, rates
of reaction, and the persistence. In the general expression, KWW, the water-water
partition coefficient is unity.
Worked Example 6.2
The evaluative environment in Example 6.1 is subject to emission of 10 mol/h
of chemical, but no advection. The reaction half-lives are air, 69.3 hours; water, 6.93
hours; and soil, 693 hours. Calculate the concentrations. Recall that KAW = 0.004
and KSW = 10.
The rate constants are 0.693/half-lives or air, 0.01; water, 0.1; soil, 0.001; h–1.
E = VACAkA + VWCWkW + VSCSkS
= CW(VAKAWkA + VWkW + VSKSWkS)
= CW(0.4 + 10 + 0.01) = CW(10.41) = 10
Therefore,
CW = 0.9606 mol/m3, CA = 0.0038, CS = 9.606
The rates of reaction then are
air = 0.38
water = 9.61
soil = 0.01
which add to the emission of 10.
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130 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
It is important to note that the reaction rate is controlled by the product V, C,
and k. A large value of any one of these quantities may convey the wrong impression
that the reaction is important. 
6.3.5 Level II Using Fugacity and D Values for Reaction
We can now follow the same process as used when treating advection and define
D values for reactions. If the rate is V C k or V Z f k, it is also DRf, where DR is V
Z k. Note that DR has units of mol/m3 Pa identical to those of DA or G Z, discussed
earlier. If there are several reactions occurring to the same chemical in the same
phase, then each reaction can be assigned a D value, and these D values can be
added to give a total D value. This is equivalent to adding the rate constants. The
Level II mass balance becomes
E = ΣViCiki = ΣViZifki = fΣViZik = fΣDR
Thus, f can be deduced, followed by concentrations, amounts, the total amount M,
and the rates of individual reactions as V C k or D f. We can repeat Example 6.2
in fugacity format.
Worked Example 6.3
An evaluative environment consists of 10000 m3 air, 100 m3 water, and 10 m3
soil. There is input of 25 mol/h of chemical, which reacts with half-lives of 100
hours in air, 75 hours in water, and 50 hours in soil. Calculate the concentrations
and amounts given the Z values below:
Air VA = 104 ZA = 4 × 10–4 kA = 0.01 DRA = 0.04
Water VW = 100 ZW = 0.1 kW = 0.1 DRW = 1.0
Sediment VS = 1.0 ZS = 1.0 kS = 0.001 DRS = 0.001
Total = 1.041
f = E/ΣDRi = 10/1.041 = 9.606
CA = 0.0038 rate = D f = 0.384
CW = 0.9606 = 9.606
CS = 9.6060 = 0.010
Phase
Volume
V (m3) Z k
VZk
or D
C
(mol/m3)
m
(mol)
Rate
(mol/h)
Air 10000 4 × 10–4 0.00693 0.0277 0.0386 386 2.68
Water 100 0.1 0.00924 0.0924 9.66 966 8.93
Soil 10 1.0 0.0139 0.1386 96.6 966 13.39
Total 2318 25.0
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ADVECTION AND REACTIONS 131
The rate constants in each case are 0.693/half-life. The sum of the V Z k terms
or D values is 0.2587, thus,
f = E/ΣD = 96.6 Pa
Thus, each C is Z f and each amount m is VC, totaling 2318 mol. Each rate is V C
k or D f, totaling 25 mol/h.
It is clear that the D value V Z k controls the overall importance of each process.
Despite its low volume and relatively slow reaction rate, the soil provides a fairly
fast-reacting medium because of its large Z value. It is not until the calculation is
completed that it becomes obvious where most reaction occurs. The overall residence
time is 2318/25 or 93 hours.
Note that the persistence or M/E is a weighted mean of the persistence or
reciprocal rate constants in each phase. It is also ΣVZ/ΣD.
6.4 COMBINED ADVECTION AND REACTION
Advective and reaction processes can be included in the same calculation as
shown in the example below, which is similar to those presented earlier for reaction.
We now have inflow and outflow of air and water at rates given below and with
background concentrations as shown in Figure 6.1c. The mass balance equation now
becomes
I = E + GACBA + GWCBW = GACA + GWCW + ΣViCiki
This can be solved either by substituting KiWCW for all concentrations and solving
for CW, or calculating the advective D values as GZ and adding them to the reaction
D values. The equivalence of these routes can be demonstrated by performing both
calculations. 
Worked Example 6.4
The environment in Example 6.3 has advective flows of 1000 m3/h in air and
1 m3/h in water as in Example 6.1 and reaction D values as in Example 6.3, with a
total input by advection and emission of 40 mol/h. Calculate the fugacity concen-
trations, amounts, and chemical residence time.
Phase
Volume
(m3) Z
DA
(advection)
DR
(reaction)
C
(mol/m3)
m
(mol)
Rate
(mol/h)
f(DA + DR)
Air 10000 4 × 10–4 0.4 0.0277 0.021 210 22.55
Water 100 0.1 0.1 0.0924 5.27 527 10.14
Soil 10 1.0 0.0 0.1386 52.7 527 7.31
Total 0.5 0.2587 1264 40
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132 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
The total of all D values is 0.7587.
E = 40
Therefore,
f = 40/ΣD = 52.7
The total amount is 1264 mols, giving a mean residence time of 31.6 hours. The
most important loss process is advection in air, which accounts for 21.08 mol/h.
Next is soil reaction at 7.31 mol/h, the water advection at 5.27 mol/h, etc. Each
individual rate is D f mol/h.
6.4.1 Advection as a Pseudo Reaction
Examination of these equations shows that the group G/V plays the same role
as a rate constant having identical units of h–1. It may, indeed, be convenient to
regard advective loss as a pseudo reaction with this rate constant and applicable to
the phase volume of V. Note that the group V/G is the residence time of the phase
in the system. Frequently, this is the most accessible and readily remembered quan-
tity. For example, it may be known that the retention time of water in a lake is 10
days, or 240 hours. The advective rate constant, k, is thus 1/240 h–1, and the D value
is V Z k, which is, of course, also G Z.
It is noteworthy that this residence time is not equivalent to a reaction half-time,
which is related to the rate constant through the constant 0.693 or ln 2. Residence
time is equivalent to 1/k.
6.4.2 Residence Times and Persistence
Confusion may arise when calculating the residence time or persistence of a
chemical in a system in which advection and reaction occur simultaneously. The
overall residence time in Example 6.4 is 31.6 hours and is a combination of the
advective residence time and the reaction time. The presence of advection does not
influence the rate constant of the reaction; therefore, it cannot affect the persistence
of the chemical. But, by removing the chemical, it does affect the amount of chemical
that is available for reaction, and thus it affects the rate of reaction. It would be
useful if we could establish a method of breaking down the overall persistence or
residence time into the time attributable to reaction and the time attributable to
advection. This is best done by modifying the fugacity equations as shown below
for total input I.
I = ΣDAif + ΣDRif
But I = M/τO, where M is the amount of chemical and τO is the overall residence
time. Furthermore, M = ΣVZf or fΣVZ. Thus, dividing both sides by M and can-
celling f gives
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ADVECTION AND REACTIONS 133
1/τO = ΣDAi/ΣVZ + ΣDRi/ΣVZ
= 1/τA + 1/τR
The key point is that the advective and reactive residence times τA and τR add as
reciprocals to give the reciprocal overall time. These are the residence times that
would apply to the chemical if only that process applied. Clearly, the shorter resi-
dence time dominates, corresponding, of course, to the faster rate constant. It can
be shown that the ratio of the amounts removed by reaction and by advection are inthe ratio of the overall rate constants or the reciprocal residence times.
Example 6.5
Calculate the individual and overall residence times in Example 6.4. Each resi-
dence time is VZ/D and the rate constant is D/VZ.
Adding the reciprocals, i.e., the rate constants, gives
1/60 + 1/240 + 1/866 + 1/260 + 1/∞ + 1/173
= 0.0167 + 0.0042 + 0.0012 + 0.0038 + 0 + 0.0058 = 0.0209 + 0.0108
= 0.0317 = 1/31.5
The advection residence time is 1/0.0209 or 47.8 h, and for reaction it is 1/0.0108
or 92.6 h. Each residence time (e.g., 60, 866, etc.) contributes to give the overall
residence time of 31.5 hours, reciprocally.
In mass balance models of this type, it is desirable to calculate the advection,
reaction, and overall residence times. An important observation is that these resi-
dence times are independent of the quantity of chemical introduced; in other words,
they are intensive properties of the system. Concentrations, amounts, and fluxes are
dependent on emissions and are extensive properties.
These concepts are useful, because they convey an impression of the relative
importance of advective flow (which merely moves the problem from one region to
another) versus reaction (which may help solve the problem). These are of particular
interest to those who live downwind or downstream of a polluted area.
6.5 UNSTEADY-STATE CALCULATIONS
A related calculation can be done in unsteady-state mode in which we introduce
an amount of chemical, M, into the evaluative environment at zero time, then allow
VZ ΣVZ/D (advection) VZ/D (reaction)
Air 4 60 866
Water 10 240 260
Soil 10 ∞ 173
Total 24
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134 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
it to decay in concentration with time, but maintain equilibrium between all phases
at the same time. This is analogous to a batch chemical reaction system. Although
it is possible to include emissions or advective inflow, we prefer to treat first the
case in which only reaction occurs to an initial mass M. We assume that all volumes
and Z values are constant with time.
dM/dt = –ΣViCiki = –fΣViZiki = –fΣDRi
But,
M = ΣViZif = fΣViZi
df/dt = –fΣViZiki/ΣViZi = –fΣDRi/ΣViZi
Solving gives
f = fO exp(–kOt)
where kO = ΣViZiki/ΣViZi = ΣDRi/ΣViZi, and fO is the initial fugacity. Note that kO,
the overall rate constant, is the reciprocal of the overall residence time.
Worked Example 6.6
Calculate the time necessary for the environment in Example 6.3 to recover to
50%, 36.7%, 10%, and 1% of the steady-state level of contamination after all
emissions cease.
Here, ΣVZ is 24 and ΣD is 0.2587. Thus,
f = fO exp (–0.2587t/24) = fO exp (–0.01078t)
Since M is proportional to f, and fO is 96.6 Pa, we wish to calculate t at which f is
48.3, 35.4, 9.66, and 0.966 Pa. Substituting and rearranging gives t = –1/0.01078 ln
(48.3/96.6), etc., or t is, respectively, 64 h, 93 h, 214 h, and 427 h. The 93-hour time
is significant as both the steady-state residence time and the time of decay to 36.7%
or exp(–1) of the initial concentration.
It is possible to include advection and emissions with only slight complications
to the integration. The input terms may no longer be zero.
This example raises an important point, which we will address later in more
detail. The steady-state situations in the Level II calculations are somewhat artificial
and contrived. Rarely is the environment at a steady state; things are usually getting
worse or better. A valid criticism of Level II calculations is that steady-state analysis
does not convey information about the rate at which systems will respond to changes.
For example, a steady-state analysis of salt emission into Lake Superior may dem-
onstrate what the ultimate concentration of salt will be, but it will take 200 years
for this steady state to be achieved. In a much smaller lake, this steady state may
CH06 Page 134 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 135
be achieved in 10 days. Detractors of steady-state models point with glee to situations
in which the modeler will be dead long before steady state is achieved.
Proponents of steady-state models respond that, although they have not specif-
ically treated the unsteady-state situation, their equations do contain much of the
key “response time” information, which can be extracted with the use of some
intelligence. The response time in the unsteady-state Example 6.5 was 93 hours,
which was ΣVZ/ΣD. This is identical to the overall residence time, t, in Example
6.2. The response time of an unsteady-state Level II system is equivalent to the
residence time in a steady-state Level II system. By inspection of the magnitude of
groups, VZ/D, or the reciprocal rate constants that occur in steady-state analysis, it
is possible to determine the likely unsteady-state behavior. This is bad news to those
who enjoy setting up and solving differential equations, because “back-of-the-
envelope” calculations often show that it is not necessary to undertake a complicated
unsteady-state analysis.
Indeed, when calculating D values for loss from a medium, it is good practice
to calculate the ratio VZ/D, where VZ refers to the source medium. This is the
characteristic time of loss, or specifically the time required for that process to reduce
the concentration to e–1 of its initial value if it were the only loss process. In some
cases, we have an intuitive feeling for what that time should be. We can then check
that the D value is reasonable.
6.6 THE NATURE OF ENVIRONMENTAL REACTIONS
The most important environmental reaction processes are biodegradation, hydrol-
ysis, oxidation, and photolysis. We treat each process briefly below with the view
to establishing methods by which the rate of the reaction can be characterized, and
giving references to authoritative reviews.
6.6.1 Biodegradation
Microbiologists are usually quick to point out that the process of microbial
conversion of chemicals in the environment is exceedingly complex. The rate of
conversion depends on the nature of the chemical compound; on the amount and
condition of enzymes that may be present in various organisms in various states of
activation and availability to perform the chemical conversion; on the availability
of nutrients such as nitrogen, phosphorus, and oxygen; as well as pH, temperature,
and the presence of other substances that may help or hinder the conversion process.
Virtually all organic chemicals are susceptible to microbial conversion or biodegra-
dation. Notable among the slowly degrading or recalcitrant compounds are high-
molecular-weight compounds such as the humic acids, certain terpenes that appear
to have structures that are too difficult for enzymes to attack, and many organo-
halogen substances. Generally, water-soluble organic chemicals are fairly readily
biodegraded. Over evolutionary time, enzymes have adapted and evolved the capa-
bility of handling most naturally occurring organic compounds. When presented
with certain synthetic organic compounds that do not occur in nature (notably the
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136 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
halogentated hydrocarbons), they experience considerable difficulty, and they may
or may not be able to perform useful chemical conversions. In such cases, if envi-
ronmental degradation does take place, it is often the result of abiotic processes such
as photolysis or reaction with free radicals.
Our aim is to be able to define a half-life or rate constant for microbial conversion
of the chemical, usually in water but often also in soil and in sediments. These rate
constants may be measured by introducing the chemical into the medium of interest
and following its decay in concentration. If first-order behavior is observed, a rate
constant and half-life may be established. Care must be taken to ensure that the
decay is truly attributable to biodegradation and not to other processes such as
volatilization.
In many cases, non-first-order behavior occurs.For example, it is suspected that,
in some situations, the concentration of chemical is so low that the enzymes neces-
sary for conversion do not become adequately activated, and the chemical is essen-
tially ignored. At high concentrations, the presence of the chemical may result in
toxicity to the microorganisms, and therefore the conversion process ceases. The
number of active enzymatic sites may also be limited, thus the rate of conversion
of the chemical species becomes controlled not by the concentration of the species
but by the number of active sites and the rate at which chemicals can be transferred
into and out of these sites. Under these conditions of saturation, a Michaelis–Menten
type equation can be applied as described earlier.
Much to the chagrin of microbiologists, we will adopt a simple expedient assum-
ing that a first-order rate constant (or half-life) applies and that the rate constant can
be estimated by experiment or from experience. This is necessarily an approximation
to the truth and often involves merely a judgement that, in a particular type of water
or soil, this compound is subject to biodegradation with a half-life of approximately
x hours. The rate constant is therefore 0.693/x hours. Valiant efforts have been made
to devise experimental protocols in which chemicals are subjected to microbial
degradation conditions in the field or in the laboratory using, for example, innocu-
lated sewage sludge. Such estimates are of particular importance in the prediction
of chemical fate in sewage treatment plants. Even more valiant attempts are being
made to predict the rate of biodegradation of chemicals purely from a knowledge
of their molecular structure. Others have been content to categorise organic chem-
icals into various groups that have similar biodegradation rates or characteristics.
Several standard and near-standard tests exist for determining biodegradation
rates under aerobic and anaerobic conditions in water and in soils. Simplest is the
biochemical oxygen demand (BOD) test as described in various standard methods
compilations by agencies such as ASTM and APHA. More complex systems involve
the use of chemostats and continuous flow systems, which are analogous to bench-
top sewage treatment plants.
An important characterization of biodegradation relates to whether the organism
requires an oxygenated environment to thrive. All organisms require energy, which
is obtained by performing chemical reactions. The most common reaction is oxida-
tion, which is performed by aerobic organisms when oxygen is present. Oxidation
of ethanol to acetic acid is an example. When oxygen is absent and anaerobic
conditions prevail, the organism can obtain energy by processes such as reducing
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ADVECTION AND REACTIONS 137
sulfate to sulfide or by dechlorinating a molecule. The latter is very important as a
method of degrading organo-chlorine compounds, which are recalcitrant to direct
oxidation.
Howard (2000) has reviewed the principles surrounding biodegradation pro-
cesses, the laboratory and field test methods that are employed, and a variety of
methods by which biodegradation half-lives or classes can be estimated. One of the
most popular and accessible biodegradation estimation methods is the BIODEG
program, which is available from the Syracuse Research Corp. website
(www.syrres.com). It is well established that certain groupings of atoms impart
reactivity or recalcitrance to a molecule, thus a molecular structure can be examined
to identify how fast it is likely to degrade. Computer programs such as BIODEG
can do this automatically and assign a structure to a class such as “biodegrades fast”
with a half-life of days to weeks. The half-life may be reported for primary degra-
dation, i.e., loss of the parent compound, but also if interest is the time for complete
mineralization to CO2 and water. The science of biodegradation is still a long way
from being able to estimate half-lives within an accuracy of a factor of three; indeed,
it may not be possible to estimate half-lives with greater accuracy.
In addition to the excellent review by Howard (2000), the reader will find
valuable material in the texts by Alexander (1994), Pitter and Choduba (1990), and
Schwarzenbach et al. (1993). Howard (2000) also lists databases, notably the
BIOLOG database of some 6000 chemicals.
6.6.2 Hydrolysis
In this process, the chemical species is subject to addition of water as a result
of reaction with water, hydrogen ion, or hydroxyl ion. All three mechanisms may
occur simultaneously at different rates; therefore, the overall rate can be very sen-
sitive to pH. Rates of environmental hydrolysis have been thoroughly reviewed by
Mabey and Mill (1978) and Wolfe and Jeffers (2000). For many organic compounds,
hydrolysis is not applicable.
A systematic method of testing for susceptibility to hydrolysis is to subject the
chemical to pH levels of 3, 7, and 11; observe the decay; and deduce rate constants
for acid, base, and neutral hydrolysis. These rate constants can be combined to give
an expression for the rate at any desired pH, namely,
dC/dt = –kH[H+]C – kOH[OH–]C – kW[H2O]C
Structure activity approaches can be used to correlate and predict these rate constants.
Often, the best approach is to seek data on a structurally similar substance.
Other useful references on hydrolysis include the Wolfe (1980), Pankow and
Morgan (1981), Zepp et al. (1975), Wolfe et al. (1977), and Jeffers et al. (1989).
6.6.3 Photolysis
The energy present in sunlight (photons) is often sufficient to cause chemical
reactions or the rupture of chemical bonds in molecules that are able to absorb this
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138 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
light. Sunburn and photosynthesis are examples of such reactions. This process is
primarily of interest when considering the fate of chemicals in solution in the
atmosphere and in water. The radiation that is most likely to effect chemical change
is high-energy, short-wavelength photons at the blue and near UV end of the spec-
trum, i.e., shorter than 400 nm. The relationships between energy, wavelength, and
frequency are readily deduced using the fundamental constants of the speed of light
c (3.0 × 108 m/s), Planck’s constant h (6.6 × 10–34 Js), and Avogadro’s Number N
(6.0 × 1023). The energy of a photon of wavelength λ nm (frequency c/λ Hz) is hc/λ
J/molecule or hcN/λ J/mol or Einsteins. A photon of wavelength 307 nm has a
frequency of 9.8 × 1014Hz and energy of 387,000 J/mol or Einsteins. This is approx-
imately the dissociation energy of the tertiary C-H bond in isobutane (2 methyl
propane); thus, in principle, if the energy in such a photon could be applied to that
bond, dissociation would occur. Short-wavelength photons are more energetic and
are more likely to induce chemical reactions.
There are two general concerns. Will the photon be absorbed such that reaction
will occur? Will the quantity of photons be such that the reaction rate will be
significant?
To be absorbed directly, the molecule must have a chromophore that imparts
suitable absorption characteristics. These properties can be measured using a spec-
trophotometer. As discussed later, there may be indirect absorption of the energy
from another species that absorbs the photon then passes on the energy to the
substance of interest.
The issue of quantity can be assessed by calculating the amount of energy
absorbed, recognizing that there are competitive absorbing substances such as natural
organic matter present in the environment. The extent of absorption can be calculated
from the Beer–Lambert Law such that
log I = log IO – εCL = log IO – A
where IO is the incident radiation, I is the surviving radiation at distance L, con-
centration C, extinction coefficient ε, and absorbance A. The quantity of light
absorbed is (IO – I), and the fraction that is absorbed by the chemical can be deduced
by comparingA for the chemical with A for the natural organic matter. In near-
transparent or clear water when A is small, the quantity of light absorbed approaches
2.3IOεCL Einsteins/m2·h. Note that (1 – 10–x) approaches 2.3x when x is small. If
each photon absorbed causes ϕ molecules (the quantum yield) to react, then the
reaction rate will be 2.3ϕIOεCL mol/m2·h and, in principle, the first-order rate
constant is 2.3ϕIOε, IO having units of mol/m2 h and ε units of m2/mol. In practice,
IO and ε are functions of wavelength. Not only is there direct absorption of sunlight
from the sun, but diffuse radiation from the sky also contributes. IO also depends
on latitude, time of day and year, and cloud cover. If ε is known as a function of
wavelength, computer programs can be used to integrate over the solar spectrum to
give the total photolysis rate constant. The quantum yield may be quite small, e.g.,
0.1 or, in the case of chain reactions, it can be larger than 1.0. Computer programs
such as SOLAR are available to undertake these calculations. The reader is referred
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ADVECTION AND REACTIONS 139
to Zepp and Cline (1977) for the original work in this area; to Leifer (1988) for an
update; and to Calvert and Pitts (1966), Mill (2000), and Schwarzenbach et al.
(1993) for more details and examples of photochemical reactions and computer
programs.
For our purposes, it is sufficient to appreciate that, knowing the absorbance
properties of the molecule, the quantum yield and the local insolation conditions, it
is possible to calculate a rate constant and a half-life for direct photolysis.
Relatively simple experiments can be conducted in which the chemical is dis-
solved in distilled or natural water in a suitable container and exposed to natural
sunlight or to artificial light for a period of time, and the concentration decay is
monitored. Test methods have been described by Svenson and Bjarndahl (1988),
Lemaire et al. (1982), and Dulin and Mill (1982).
The issue is complicated by the presence of photosensitizing molecules or
substances. These substances absorb light then pass on the energy to the chemical
of interest, resulting in subsequent chemical reaction. It is therefore not necessary
for the chemical to absorb the photon directly. It can receive it second hand from a
photosensitizer. This is a troublesome complication, because it raises the possibility
that chemicals may be subject to photolysis due to the unexpected presence of a
photosensitizer. Of particular interest are the naturally occuring organic matter pho-
tosensitizers that are present in water and give it its characteristic brown color,
especially in areas in which there is peat and decaying vegetation.
6.6.4 Atmospheric Oxidation Reactions
A chemical present in the atmosphere may react with oxygen, an activated form
of oxygen such as singlet oxygen, ozone, hydrogen peroxide, or with various radicals,
notably OH radicals. Fortunately, we live in a world with an abundance of oxygen,
and it is not surprising that a suite of oxygen compounds exists that are eager to
oxidize organic chemicals. The rates of these reactions can be estimated by con-
ducting conventional chemical kinetic experiments in which the substance is con-
tacted with known concentrations of the oxidant, the decay of chemical is followed,
and a kinetic law and rate constant established.
The most important oxidative process is the reaction of hydroxyl radicals with
chemical species in the atmosphere. The concentration of sunlight-induced hydroxyl
radicals is exceedingly small, averaging only about 1 million molecules per cubic
centimetre. Peak concentrations approach 8 million per cm3 in urban areas. Concen-
trations in rural or remote areas are much lower. They are extremely reactive and
are responsible for the reaction of many organic chemicals in the environment that
would otherwise be persistent. 
Ozone is produced by UV radiation in the stratosphere and by certain high-
temperature and photolytic processes in the troposphere. The average mixing ratio,
i.e., the ratio of ozone to non-ozone molecules, is in the range of 10 to 40 × 10–9.
Oxides of nitrogen produced at high temperature include NO, NO2, and the
reactive NO3 radical. The latter has an average concentration of about 500 million
molecules per cm3 and peaks in concentration at night.
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140 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
A formidable literature exists on the kinetics of gas phase organic substances,
notably hydrocarbons, with OH radicals. Quantitative structure activity relationships
have been developed in which each part of the molecule is assigned a rate constant
for abstraction of H by OH radicals, or for addition of OH radicals to unsaturated
bonds. Atkinson (2000) has reviewed these estimation methods and provides refer-
ences to compilations of rate constant data. Computer programs exist to estimate
these rate constants from molecular structure, for example from the Syracuse
Research Corporation website (www.syrres.com).
It is important to appreciate that the atmosphere is a very reactive medium in
which large quantities of chemical species are converted into oxidized products.
This is fortunate, because otherwise there would be more severe air pollution and
problems associated with the transport of these chemicals to remote regions.
6.6.5 Aqueous Oxidation and Reduction
Natural oxidizing agents include oxygen, hydrogen peroxide, ozone, and “engi-
neered” oxidants include chlorine, hypochlorite, chlorine dioxide, permanganate,
chromate, and ferrate. Natural reducing agents include sulphide, ferrous and man-
ganous ion, and organic matter, while “engineered” reductants include dithionite
and zero-valent (metal) iron. Oxidation usually involves the addition of oxygen but,
in more general terms, it is the removal of or abstraction of an electron. Reduction
involves electron addition. The potential or feasibility of such a reaction occurring
can be readily evaluated from the standard potential of the half reactions.
The kinetics are usually expressed using a second-order expression including
the concentration of the substance and the oxidant or reductant. In some cases, the
reactant is a solid (e.g., zero-valent iron), and an area-normalized value can be used.
Tratnyek and Macalady (2000) provide an excellent review of this literature and
give several examples of oxidation and reduction processes. Again, for our purposes,
a first-order rate constant can be estimated that includes the concentration of the
oxidising or reducing agent. This can be used to calculate the corresponding half-
life and D value.
6.6.6 Summary
It has been possible to provide only a brief account of the vast literature relating
to chemical reactivity in the environment. The air pollution literature is particularly
large and detailed. References have been provided to give the reader an entry to the
literature.
The susceptibility of a chemical in a specific medium to degrading reaction
depends both on the inherent properties of the molecule and on the nature of the
medium, especially temperature and the presence of candidate reacting molecules
or enzymes. In this respect, environmental chemicals are fundamentally different
from radioisotopes, which are totally unconcerned about external factors. Translation
and extrapolation of reaction rates from environment to environment and laboratory
to environment is therefore a challenging and fascinating task that will undoubtedly
keep environmental chemists busy for many more decades.
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ADVECTION AND REACTIONS 141
6.7 LEVEL II COMPUTER CALCULATIONS
As with Level I calculations, it is desirable to reduce the tedium of calculations
by using the computer. Figure 6.2 gives an illustrative fugacity form calculation,
a blank form being provided in the appendix. Computer programs that conduct
Level II calculations are availablefrom the Internet. The input data include the
properties of the environment, the chemical properties, input rates by emission
and advection, and information on reaction and advection rates. The fugacity is
calculated, followed by a complete mass balance. Since equilibrium is assumed
to apply within the environment, it is immaterial into which phase the chemical
is introduced.
The user is encouraged to test the environmental behavior of some of the chem-
icals introduced earlier, assuming or obtaining literature data on reaction rates.
Worked Example 6.7
Calculate the partitioning of the hypothetical chemical in Figure 5.6 assuming
that the rate constants for reaction are 0.001 h–1 in water, 0.01 h–1 in soil, and
0.0001 h–1 in sediment, and with no reaction in air. Assume advective inputs in air
at 10–6 mol/m3 (flow 107 m3/h) and in water at 0.01 mol/m3 (flow 1000 m3/h). The
emission rate is 100 mol/h.
The hand calculation is fairly tedious and is reproduced in Figure 6.2. It involves
calculation of the total inputs of 100 mol/h (emission), 10 mol/h (advection in air),
and 10 mol/h (advection in water) totaling 120 mol/h (I). The reaction and advection
D values are then deduced and added to give a total (ΣD) of approximately
10,390 mol/Pa h. The fugacity is then I/ΣD or 0.0115 Pa.
Concentrations, amounts, and process rates can then be deduced and added to
check the mass balance. The computed output is given in Figure 6.3. Note that it is
not possible to input an infinite half life in air to give a zero rate of reaction. A
fictitious, large value of 1011 h is used instead.
6.8 SUMMARY
In this chapter, we have learned to include advection and reaction rates in
evaluative Level II calculations. These calculations can be done using concentrations
and partition coefficients or fugacities and D values. The concepts of residence time
and persistence have been reintroduced. These are invaluable descriptors of envi-
ronmental fate. We have briefly reviewed the essential environmental chemistry of
biodegradation, photolysis, hydrolysis, and other reactions, and provided references
to studies, reviews, and estimation methods. Critics will be eager to point out a
major weakness in these calculations. Environmental media are rarely in equilibrium;
therefore, a use of a common fugacity or the use of equilibrium partition coefficients
to relate concentrations between phases or media is often not valid. Treating non-
equilibrium situations is the task of Chapter 7.
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142 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
Fugacity Form 3 Level II
Chemical: Hypothene
Direct emission rate E 100 mol/h
Advective input rates
Compartment
Volume m3 (V)
Residence time h (t)
Flow rate m3/h = V/t = G
Inflow concentration mol/m3 CB
Chemical inflow rate mol/h = GCB
Air
6 × 109
600
107
10–6
10
Water
7 × 106
7000
1000
10–2
10
Total input rate E + ΣGCB = I = 100 + 10 + 10 = 120
Compartment
Volume m3 (V)
Z
VZ
Reaction half life (h)t
Rate constant k = 0.693/t (h–1)
Advective flow G m3/h
D reaction = VZk = DR
D advection = GZ = DA
DR + DA = DT
Air
6 × 109
4 × 10–4
2.4 × 106
∞
0
107
0
4000
4000
Water
7 × 106
0.1
7 × 105
693
0.001
1000
700
100
800
Soil
45000
12.3
5.5 × 105
69.3
0.01
0
5535
0
5535
Sediment
21000
24.6
5.17 × 105
6930
0.0001
0
51.7
0
51.7
Total D value = ΣDT = 10387 Fugacity f = I/ΣD = 120/10387 = 1.15 × 10–2
C = Z f mil/m3
m = C V mol
Percent
CG g/m3, i.e., CW
Density ρ kg/m3
CU µg/g, i.e., CG × 1000/ρ
4.6 × 10–6
27731
57.5
9.2 × 10–4
1.18
0.79
1.15 × 10–3
8087
16.8
0.23
1000
0.23
0.14
6394
13.3
28.4
1500
19
0.28
5968
12.4
56.8
1500
38
Reaction rate DRf
Advection rate DAf
Total DTf
0
46.2
46.2
8.1
1.2
9.3
63.9
0
63.9
0.6
0
0.6
Total amount M = Σm =
Total reaction rate = ΣDRf =
Total advection rate = ΣDAf =
Total output rate (mol/h) = I =
48180
72.6
47.4
120
Reaction residence time (h) = M/ΣDRf
Advection residence time (h) = M/ΣDAf
Overall residence time (h) = M/I =
663
1017
401
Figure 6.2 Fugacity form for completing a Level II calculation.
CH06 Page 142 Monday, January 15, 2001 1:50 PM
ADVECTION AND REACTIONS 143
Figure 6.3 Fugacity Level II calculation corresponding to Figure 6.3.
CH06 Page 143 Monday, January 15, 2001 1:50 PM
144 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
6.9 CONCLUDING EXAMPLE
For the two substances selected from Table 3.5, which were the subject of the
concluding example in Chapter 5, perform a Level II calculation for the air, water,
soil, and bottom sediment phases either as defined in that example or using the
environment deduced in the concluding example from Chapter 4. Use Fugacity Form
3 and ignore other phases. Assume reasonable residence times in air, water, and
bottom sediment for the purpose of calculating advection rates. There is no advection
from soil. Use the degradation half-lives from Table 3.5.
Assume first a total input by emission of 100 kg/h and calculate the fugacity,
concentrations, amounts, and the three chemical residence times (overall, reaction,
and advection).
Second, recalculate this Level II example assuming that the inflow air and water
both contain the chemical at a concentration that is 20% of the air and water
concentrations calculated above.
Discuss the results and present them in a diagrammatic form. Discuss which
reaction and advection processes are most important. Are the residence times in
these two examples equal or not? Explain why they are equal or different.
CH06 Page 144 Monday, January 15, 2001 1:50 PM
 
145
 
CHAPTER
 
 7
Intermedia Transport
 
7.1 INTRODUCTION
 
The Level II calculations described in Chapter 6 contain the major weakness
that they assume environmental media to be in equilibrium. This is rarely the case
in the real environment; therefore, the use of a common fugacity (or concentrations
related by equilibrium partition coefficients) is usually, but not always, invalid.
Reasons for this are best illustrated by an example.
Suppose we have air and water media as illustrated in Figure 7.1, with emissions
of 100 mol/h of benzene into the water. There is only slow reaction in the water
(say, 20 mol/h), but there is rapid reaction (say, 80 mol/h) in the air. This implies
that benzene is evaporating from water to air at a rate of 80 mol/h. The question
arises: is benzene capable of evaporating at 80 mol/h, or will there be a resistance
to transfer that prevents evaporation at this rate? If only 40 mol/h could evaporate,
the evaporated benzene may react in the air phase at 40 mol/h, but it will tend to
build up in the water phase to a higher concentration and fugacity until the rate of
reaction in the water increases to 60 mol/h. The benzene fugacity in the air will thus
be lower than the fugacity in water, and a nonequilibrium situation will have devel-
oped. The ability to calculate how fast chemicals can migrate from one phase to
another is the challenging task of this chapter. The topic is one in which there still
remain considerable uncertainty and scope for scientific investigation and innovation.
We begin it by listing and categorizing all the transport processes that are likely to
occur.
 
7.2 DIFFUSIVE AND NONDIFFUSIVE PROCESSES
7.2.1 Nondiffusive Processes
 
The first group of processes consists of
 
 
 
nondiffusive
 
, or 
 
piggyback,
 
 or 
 
advective
 
processes. A chemical may move from one phase to another by 
 
piggybacking
 
 on
 
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146 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
material that has decided, for reasons unrelated to the presence of the chemical, to
make this journey. Examples include advective flows in air, water, or particulate
phases, as discussed in Chapter 6; deposition of chemical in rainfall or sorbed to
aerosols from the atmosphere to soil or water; and sedimentation of chemical in
association withparticles that fall from the water column to the bottom sediments.
These are usually one-way processes. The rate of chemical transfer is simply
the product of the concentration C mol/m
 
3
 
 of chemical in the moving medium, and
the flowrate of that medium, G, m
 
3
 
/h. We can thus treat all these processes as
advection and calculate the D value and rate as follows:
N = GC = GZf = Df mol/h
The usual problem is to measure or estimate G and the corresponding Z value
or partition coefficient. We examine these rates in more detail later, when we focus
on individual intermedia transfer processes. 
 
7.2.2 Diffusive Processes
 
The second group of processes are 
 
diffusive
 
 in nature. If we have water containing
1 mol/m
 
3
 
 of benzene and add some octanol to it as a second phase, the benzene will
Figure 7.1 Illustration of nonequilibrium behavior in an air-water system. In the lower diagram,
the rate of reaction in air is constrained by the rate of evaporation.
 
CH07 Page 146 Monday, January 15, 2001 1:51 PM
 
INTERMEDIA TRANSPORT 147
 
diffuse from the water to the octanol until it reaches a concentration in octanol that
is K
 
OW
 
, or 135, times that in the water. We could rephrase this by stating that, initially,
the fugacity of benzene in the water was (say) 500 Pa, and the fugacity in the octanol
was zero. The benzene then migrates from water to octanol until both fugacities
reach a common value of (say) 200 Pa. At this common fugacity, the ratio C
 
O
 
/C
 
W
 
is, of course, Z
 
O
 
/Z
 
W
 
 or K
 
OW
 
. We argue that diffusion will always occur from high
fugacity (for example, f
 
W
 
 in water) to low fugacity (f
 
O
 
 in octanol). Therefore, it is
tempting to write the transfer rate equation from water to octanol as
N = D(f
 
W
 
 – f
 
O
 
) mol/h
This equation has the correct property that, when f
 
W
 
 and f
 
O
 
 are equal, there is no
net diffusion. It also correctly describes the direction of diffusion.
In reality, when the fugacities are equal, there is still active diffusion between
octanol and water. Benzene molecules in the water phase do not know the fugacity
in the octanol phase. At equilibrium, they diffuse at a rate, Df
 
W
 
, from water to
benzene, and this is balanced by an equal rate, Df
 
O
 
, from octanol to water. The
escaping tendencies have become equal, and N is zero. The term (f
 
W
 
 – f
 
O
 
) is termed
a 
 
departure from equilibrium
 
 group, just as a temperature difference represents a
departure from thermal equilibrium. It quantifies the diffusive 
 
driving force.
 
Other areas of science provide good precedents for using this approach. Ohm’s
law states that current flows at a rate proportional to voltage difference times
electrical conductivity. Electricians prefer to use resistance, which is simply the
reciprocal of conductivity. The rate of heat transfer is expressed by Fourier’s law as
a thermal conductivity times a difference in temperature. Again, it is occasionally
convenient to think in terms of a thermal resistance (the reciprocal of thermal
conductivity), especially when buying insulation. These equations have the general
form
rate = (conductivity)
 
 × 
 
(departure from equilibrium)
or 
rate = (departure from equilibrium)/(resistance)
Our task is to devise recipes for calculating D as an expression of conductivity or
reciprocal resistance for a number of processes involving diffusive interphase trans-
fer. These include the following:
 
1. Evaporation of chemical from water to air and the reverse process of absorption.
Note that we consider the chemical to be in solution in water and not present as
a film or oil slick, or in sorbed form.
2. Sorption from water to suspended matter in the water column, and the reverse
desorption. 
3. Sorption from the atmosphere to aerosol particles, and the reverse desorption.
4. Sorption of chemical from water to bottom sediment, and the reverse desorption.
 
CH07 Page 147 Monday, January 15, 2001 1:51 PM
 
148 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
5. Diffusion within soils, and from soil to air.
6. Absorption of chemical by fish and other organisms by diffusion through the gills,
following the same route traveled by oxygen.
7. Transfer of chemical across other membranes in organisms, for example, from air
through lung surfaces to blood, or from gut contents to blood through the walls of
the gastrointestinal tract, or from blood to organs in the body.
 
Armed with these D values, we can set up mass balance equations that are similar
to the Level II calculations but allow for unequal fugacities between media.
To address these tasks, we return to first principles, quantify diffusion processes
in a single phase, then extend this capability to more complex situations involving
two phases. Chemical engineers have discovered that it is possible to make a great
deal of money by inducing chemicals to diffuse from one phase to another. Examples
are the separation of alcohol from fermented liquors to make spirits, the separation
of gasoline from crude oil, the removal of salt from sea water, and the removal of
metals from solutions of dissolved ores. They have thus devoted considerable effort
to quantifying diffusion rates, and especially to accomplishing diffusion processes
inexpensively in chemical plants. We therefore exploit this body of profit-oriented
information for the nobler purpose of environmental betterment.
 
7.3 MOLECULAR DIFFUSION WITHIN A PHASE
7.3.1 Diffusion As a Mixing Process
 
In liquids and gases, molecules are in a continuous state of relative motion. If
a group of molecules in a particular location is labeled at a point in time, as shown
in the upper part of Figure 7.2, then at some time later it will be observed that they
have distributed themselves randomly throughout the available volume of fluid.
Mixing has occurred.
Since the number of molecules is large, it is exceedingly unlikely that they will
ever return to their initial condition. This process is merely a manifestation of mixing
in which one specific distribution of molecules gives way to one of many other
statistically more likely mixed distributions. This phenomenon is easily demonstrated
by combining salt and pepper in a jar, then shaking it to obtain a homogeneous
mixture. It is the rate of this mixing process that is at issue.
We approach this issue from two points of view. First is a purely mathematical
approach in which we postulate an equation that describes this mixing, or diffusion,
process. Second is a more fundamental approach in which we seek to understand
the basic determinants of diffusion in terms of molecular velocities.
Most texts follow the mathematical approach and introduce a quantity termed
 
diffusivity
 
 or 
 
diffusion coefficient, 
 
which has dimensions of m
 
2
 
/h, to characterize this
process. It appears as the proportionality constant, B, in the equation expressing
Fick’s first law of diffusion, namely
N = –B A dC/dy
 
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INTERMEDIA TRANSPORT 149
 
Here, N is the flux of chemical (mol/h), B is the diffusivity (m
 
2
 
/h), A is area (m
 
2
 
),
C is concentration of the diffusing chemical (for example, benzene in water)
(mol/m
 
3
 
), and y is distance (m) in the direction of diffusion. The group dC/dy is
thus the concentration gradient and is characteristic of the degree to which the
solution is unmixed or heterogeneous. The negative sign arises because the direction
of diffusion is from high to low concentration, i.e., it is positive when dC/dy is
negative. Here, we use the symbol B for diffusivity to avoid confusion with D values.
Most texts sensibly use the symbol D. The equation is really a statement that the
rate of diffusion is proportional to the concentration gradient and the proportionality
constant is diffusivity. When the equation is apparentlynot obeyed, we attribute this
misbehavior to deviations or changes in the diffusivity, not to failure of the equation.
As was discussed earlier, there are differences of opinion about the word 
 
flux
 
.
We use it here to denote a transfer rate in units such as mol/h. Others insist that it
should be area specific and have units of mol/m
 
2
 
h. We ignore their advice. Occa-
Figure 7.2 The fundamental nature of molecular diffusion.
 
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150 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
 
sionally, the term 
 
flux rate 
 
is used in the literature. This is definitely wrong, because
flux contains the concept of rate just as does speed. 
 
Flux rate
 
 is as sensible as 
 
speed
rate.
 
It is worthwhile digressing to examine how the mixing process leads to diffusion
and eventually to Fick’s first law. This elucidates the fundamental nature of diffu-
sivity and the reason for its rather strange units of m
 
2
 
/h. Much of the pioneering
work in this area was done by Einstein in the early part of this century and arose
from an interest in Brownian movement—the erratic, slow, but observable motion
of microscopic solid particles in liquids, which is believed to be due to multiple
collisions with liquid molecules.
 
7.3.2 Fick’s Law and Diffusion at Steady State
 
We consider a square tunnel of cross-sectional area A m
 
2
 
 containing a nonuniform
solution, as shown in the middle of Figure 7.2, having volumes V
 
1
 
, V
 
2
 
, etc., separated
by planes 1–2, 2–3, 3–4, etc., each y metres apart.
We assume that the solution consists of identical dissolved particles that move
erratically, but on the average travel a horizontal distance of y metres in t hours. In
time t, half the particles in volume V
 
3
 
 will cross the plane 2–3, and half the plane
3–4. They will be replaced by (different) particles that enter volume V
 
3
 
 by crossing
these planes in the opposite direction from volumes V
 
2
 
 and V
 
4
 
. Let the concentration
of particles in V
 
3
 
 and V
 
4
 
 be C
 
3
 
 and C
 
4
 
 mol/m
 
3
 
 such that C
 
3
 
 exceeds C
 
4
 
. The net
transfer across plane 3–4 will be the sum of the two processes: C
 
3
 
 yA/2 moles from
left to right, and C
 
4
 
 yA/2 moles from right to left. The net amount transferred in
time t is then
C
 
3
 
yA/2 – C
 
4
 
 yA/2 = (C
 
3
 
 – C
 
4
 
) yA/2 mol
Note that CyA is the product of concentration and volume and is thus an amount
(moles).
The concentration gradient that is causing this net diffusion from left to right is
(C
 
3
 
 – C
 
4
 
)/y or, in differential form, dC/dy. The negative sign below is necessary,
because C decreases in the direction in which y increases. It follows that 
(C
 
3
 
 – C
 
4
 
) = –ydC/dy
The flux or diffusion rate is then N or
N = (C
 
3
 
 – C
 
4
 
) yA/2t = –(y
 
2
 
A/2t) dC/dy = –BAdC/dy mol/h
which is referred to as Fick’s first law. The diffusivity B is thus (y
 
2
 
/2t), where y is
the molecular displacement that occurs in time t.
In a typical gas at atmospheric pressure, the molecules are moving at a velocity
of some 500 m/s, but they collide after traveling only some 10
 
–7
 
 m, i.e., after 10
 
–7
 
/500
or 2 
 
×
 
 10
 
–10
 
 s. It can be argued that y is 10
 
–7
 
 m, and t is 2 
 
×
 
 10
 
–10
 
; therefore, we
 
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INTERMEDIA TRANSPORT 151
 
expect a diffusivity of approximately 0.25 
 
×
 
 10
 
–4
 
 m
 
2
 
/s or 0.25 cm
 
2
 
/s or 0.1 m
 
2
 
/h,
which is borne out experimentally. The kinetic theory of gases can be used to
calculate B theoretically but, more usefully, the theory gives a suggested structure
for equations that can be used to correlate diffusivity as a function of molecular
properties, temperature, and pressure.
 
In liquids, molecular motion is more restricted, collisions occur almost every
molecular diameter, and the friction experienced by a molecule as it attempts to
“slide” between adjacent molecules becomes important. This frictional resistance is
related to the liquid viscosity 
 
µ
 
 (Pa s). It can be shown that, for a liquid, the group
(B
 
µ
 
/T) should be relatively constant and (by the Stokes-Einstein equation) approx-
imately equal to R/(6
 
π
 
Nr), where N is Avogadro’s number, R is the gas constant,
and r is the molecular radius (typically 10
 
–10
 
 m). B is therefore T R/(
 
µ
 
6
 
π
 
Nr), where
the viscosity of water 
 
µ
 
 is typically 10
 
–3
 
 Pa s. Substituting values of R, T, µ, and r
suggests that B will be approximately 2 
 
× 
 
10
 
–9
 
 m
 
2
 
/s or 2 
 
×
 
 10
 
–5
 
 cm
 
2
 
/s or 7 
 
×
 
 10
 
–6
 
m
 
2
 
/h, which is also borne out experimentally. Again, this equation forms the foun-
dation of correlation equations.
The important conclusion is that, during its diffusion journey, a molecule does
not move with a constant velocity related to the molecular velocity. On average, it
spends as much time moving backward as forward, thus its net progress in one
direction in a given time interval is not simply velocity/time. In t seconds, the distance
traveled (y) will be m. Taking typical gas and liquid diffusivities of 0.25 
 
×
 
10
 
–4
 
 m
 
2
 
/s and 2 
 
×
 
 10
 
–9
 
 m
 
2
 
/s respectively, a molecule will travel distances of 7 mm
in a gas and 0.06 mm in a liquid in one second. To double these distances will
require four seconds, not two seconds. It thus may take a considerable time for a
molecule to diffuse a “long” distance, since the time taken is proportional to the
square of the distance. The most significant environmental implication is that, for a
molecule to diffuse through, for example, a 1 m depth of still water requires (in
principle) a time on the order of 3000 days. A layer of still water 1 m deep can thus
effectively act as an impermeable barrier to chemical movement. In practice, of
course, it is unlikely that the water would remain still for such a period of time.
 
The reader who is interested in a fuller account of molecular diffusion is referred
to the texts by Reid et al. (1987), Sherwood et al. (1975), Thibodeaux (1996), and
Bird et al. (1960). Diffusion processes occur in a large number of geometric con-
figurations from CO
 
2
 
 diffusion through the stomata of leaves to large-scale diffusion
in ocean currents. There is thus a considerable literature on the mathematics of
diffusion in these situations. The classic text on the subject is by Crank (1975), and
Choy and Reible (2000) have summarized some of the more environmentally useful
equations.
 
7.3.3 Mass Transfer Coefficients
 
Diffusivity is a quantity with some characteristics of a velocity but, dimension-
ally, it is the product of velocity and the distance to which that velocity applies. In
many environmental situations, B is not known accurately, nor is y or 
 
∆
 
y; therefore,
the flux equation in finite difference form contains two unknowns, B and 
 
∆
 
y. Ignoring
the negative sign,
2tB
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152 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
N = AB∆C/∆y mol/h
Combining B and ∆y in one term kM, equal to B/∆y, with dimensions of velocity
thus appears to decrease our ignorance, since we now do not know one quantity
instead of two. Hence we write
N = AkM∆C mol/h
Term kM is termed a mass transfer coefficient, has units of velocity (m/h), and is
widely used in environmental transport equations. It can be viewed as the net
diffusion velocity. The flux N in one direction is then the product of the velocity,
area, and concentration.
For example, if, as in the lower section of Figure 7.2, diffusion is occurring in
an area of 1 m2 from point 1 to 2, C1 is 10 mol/m3, C2 is 8 mol/m3,and kM is 2.0
m/h, we may have diffusion from 1 to 2 at a velocity of 2.0 m/h, giving a flux of
kMAC1 of 20 mol/h. There is an opposing flux from 2 to 1 of kMAC2 or 16 mol/h.
The net flux is thus the difference or 4 mol/h from 1 to 2, which of course equals
kMA(C1 – C2). The group kMA is an effective volumetric flowrate and is equivalent
to the term G m3/h, introduced for advective flow in Chapter 6.
7.3.4 Fugacity Format, D Values for Diffusion
The concentration approach is to calculate diffusion fluxes N as ABdC/dy or
AB∆C/∆y or kMA∆C. In fugacity format, we substitute Zf for C and define D values
as BAZ/∆y or kMAZ, and the flux is then D∆f, since ∆C is Z∆f. Note that the units
of D are mol/Pa h, identical to those used for advection and reaction D values.
D = BAZ/∆y or D = kMAZ
N = Df1 – Df2 = D(f1 – f2)
Worked Example 7.1
A chemical is diffusing through a layer of still water 1 mm thick, with an area
of 200 m2 and with concentrations on either side of 15 and 5 mol/m3. If the diffusivity
is 10–5 cm2/s, what is the flux and the mass transfer coefficient?
y = 10–3 m, B = 10–5 cm2/s × 10–4 m2/cm2 = 10–9 m2/s
Thus, 
kM is B/∆y = 10–6 m/s
The flux N is thus
kMA(C1–C2) = 10–6 (200(15 – 5)) = 0.002 mol/s
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INTERMEDIA TRANSPORT 153
This flux of 0.002 mol/s can be regarded as a net flux consisting of kMAC1 or
0.003 mol/s in one direction and kMAC2 or 0.001 mol/s in the opposing direction.
Worked Example 7.2
Water is evaporating from a pan of area 1 m2 containing 1 cm depth of water.
The rate of evaporation is controlled by diffusion through a thin air film 2 mm thick
immediately above the water surface. The concentration of water in the air imme-
diately at the surface is 25 g/m3 (this having been deduced from the water vapor
pressure), and in the room the bulk air contains 10 g/m3. If the diffusivity is 0.25
cm2/s, how long will the water take to evaporate completely?
B is 0.25 cm2/s or 0.09 m2/h
∆y is 0.002 m
∆C is 15 g/m3
N = AB∆C/∆y = 675 g/h
To evaporate 10000 g will take 14.8 hours
Note that the “amount” unit in N and C need not be moles. It can be another quantity
such as grams, but it must be consistent in both. In this example, the 2 mm thick
film is controlled by the air speed over the pan. Increasing the air speed could reduce
this to 1 mm, thus doubling the evaporation rate. This ∆y is rather suspect, so it is
more honest to use a mass transfer coefficient, which, in the example above is
0.09/0.002 or 45 m/h. This is the actual net velocity with which water molecules
migrate from the water surface into the air phase.
7.3.5 Sources of Molecular Diffusivities
Many handbooks contain compilations of molecular diffusivities. The text by
Reid et al. (1987) contains data and correlations, as does the text on mass transfer
by Sherwood, Pigford, and Wilke (1975). The handbook by Lyman et al. (1982) and
the text by Schwarzenbach et al. (1994) give correlations from an environmental
perspective. The correlations for gas diffusivity are based on kinetic theory, while
those for liquids are based on the Stokes–Einstein equation. In most cases, only
approximate values are needed. In some equations, the diffusivity is expressed in
dimensionless form as the Schmidt number (Sc) where
Sc = µ/ρB
where µ is viscosity and ρ is density.
7.4 TURBULENT OR EDDY DIFFUSION WITHIN A PHASE
So far, we have assumed that diffusion is entirely due to random molecular
motion and that the medium in which diffusion occurs is immobile or stagnant, with
CH07 Page 153 Monday, January 15, 2001 1:51 PM
154 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
no currents or eddies. In practice, of course, the environment is rarely stagnant, there
being currents and eddies induced by the motion of wind, water, and biota such as
fish and worms. This turbulent motion, illustrated in Figure 7.3, also promotes mixing
by conveying an element or eddy of fluid from one region to another. The eddies
may vary in size from millimetres to kilometres, and a large eddy may contain a
fine structure of small eddies. Intuitively, it is unreasonable for an eddy to penetrate
an interface, thus in regions close to interfaces, eddies tend to be damped, and only
slippage parallel to the interface is possible. There may, therefore, be a thin layer
of relatively quiescent fluid close to the interface that can be referred to as a laminar
sublayer. In this layer, movement of solute to and from the interface may occur only
by molecular diffusion.
Under certain conditions, eddies in fluids may be severely damped, or their
generation may be prevented. This occurs in a layer of air or water when the fluid
density decreases with increasing height. This may be due to the upper layers being
warmer or, in the case of sea water, less saline. An eddy that is attempting to move
upward immediately finds itself entering a less dense fluid and experiences a hydro-
static “sinking” force. Conversely, a companion eddy moving downward experiences
a “floating” force, which also tends to restore it to its original position. This inherent
resistance to eddy movement damps out most fluid movement, and stable, stagnant
conditions prevail. Thermoclines in water and inversions in the atmosphere are
examples of this phenomenon. These stagnant or near-stagnant layers may act as
diffusion barriers in which only molecular diffusion or slight eddy diffusion can
occur. Conversely, situations in which density increases with height tend to be
unstable, and eddy movement is enhanced and accelerated by the density field.
An attractive approach is to postulate the existence of an eddy diffusivity, or a
turbulent diffusivity, BT, which is defined identically to the molecular diffusivity,
BM. The flux equation within a phase then becomes
N = –A(BM + BT)dC/dy
The task is then to devise methods of estimating BT for various environmental
conditions. We expect that, in many situations, such as in winds or fast rivers, BT
Figure 7.3 The nature of turbulent or eddy diffusion in which chemical is conveyed in eddies
within a fluid to a surface.
CH07 Page 154 Monday, January 15, 2001 1:51 PM
INTERMEDIA TRANSPORT 155
is much greater than BM, and the molecular processes can be ignored. In stagnant
regions, such as thermoclines or in deep sediments, BT may be small or zero, and
BM dominates. As we move closer to a phase boundary, BT tends to become smaller;
thus, it is possible that much of the resistance to diffusion lies in the layer close
to the interface. The roughness of the interface plays a role in determining the
thickness of this layer. For example, grass may damp out wind eddies and retard
the rate of diffusion from soil to air. Animal fur retards both diffusion of heat and
water vapor.
A complicating factor is that we have no guarantee that BT is isotropic, i.e., that
the same value applies vertically and horizontally. In Figure 7.3, we postulate that
some eddies may be constrained to form elongated “roll cells.” The horizontal BT
will therefore exceed the vertical value. In practice, this nonisotropic situation is
common and even leads to conditions in rivers where three BT values must be
considered: vertical, upstream-downstream, and cross-stream.
To give an order of magnitude appreciation of turbulent diffusivities, it is
observed that a vertical eddy diffusivity in air is typically 3600 m2/h plus or minus
a factor of 3, thus the time for moving a distance of 1 m is of the order of 1 s.
Molecular diffusion is clearly negligible in comparison. In lakes, a vertical eddy
diffusivity may be 36 m2/h near the surface, corresponding to a velocity over a
distance of 1 m of 1 cm/s. At greater depths, diffusion is much slower, possibly by
a factor of 100. To estimate eddy diffusivities, one can watch a buoyant particle and
time its transport over a given distance. The diffusivity is then that distance squared,
divided by the time.
Turbulent processes in the environment are thus quite complex and difficult to
describemathematically. The interested reader can consult Thibodeaux (1996) or
Csanady (1973) for a review of the mathematical approaches adopted. We sidestep
this complex issue here, but certain generalizations that emerge from the study of
turbulent diffusion are worth noting.
In the bulk of most fluid masses (air and water) that are in motion, turbulent
diffusion dominates. We can measure and correlate these diffusivities. Generally,
vertical diffusion is slower than horizontal diffusion. Often, diffusion is so fast that
near-homogeneous conditions exist, which is fortunate, because it eliminates the
need to calculate diffusion rates.
In the atmosphere and oceans, there is a spectrum of eddies of varying size and
velocity. The larger eddies move faster. Consequently, when a plume in the atmo-
sphere or a dye patch in an ocean expands in size, it becomes subject to dispersion
by larger, faster eddies, and the diffusivity increases. If the velocity of expansion of
the plume or patch is constant, this implies that diffusivity increases as the square
of distance.
At phase interfaces (e.g., air-water, water-bottom sediment), turbulent diffusion
is severely damped or is eliminated, thus only molecular diffusion remains. One can
even postulate the presence of a “stagnant layer” in which only molecular diffusion
occurs and calculate its diffusion resistance. This model is usually inherently wrong
in that no such layer exists. It is more honest (and less trouble) to avoid the use of
diffusivities and stagnant layer thicknesses close to the phase interfaces and invoke
mass transfer coefficients that combine the varying eddy diffusivities, the molecular
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diffusivity, and some unknown layer thickness, into one parameter, kM. We then
measure and correlate kM as a function of fluid conditions (e.g., wind speed) and
seek advice from the turbulent transport theorists as to the best form of the correlation
equations.
In some diffusion situations, such as bottom sediments, the eddy diffusion may
be induced by burrowing worms or creatures that “pump” water. This is termed
bioturbation and is difficult to quantify. Its high variability and unpredictability is
a source of delight to biologists and irritation to physical scientists.
The study of turbulent diffusion in the atmosphere includes aspects such as the
micrometeorology of diffusion near the ground as it influences evaporation of pes-
ticides, the uptake of contaminants by foliage, and the dispersion of plumes from
stacks, in which case the plume is treated by the Gaussian dispersion equations. In
lakes, rivers, and oceans it is important to calculate concentrations near sewage and
industrial outfalls and in intensively used regions such as harbors. In each case, a
body of specialized knowledge and calculation methods has evolved.
7.5 UNSTEADY-STATE DIFFUSION
Those who dislike calculus, and especially partial differential equations, can skip
this section, but the two concluding paragraphs should be noted.
In certain circumstances, we are interested in the transient or unsteady-state
situation, which exists when diffusion starts between two volumes that are brought
into contact. This is shown conceptually in Figure 7.4, in which a “shutter” is
removed, exposing a concentration discontinuity. The two regions proceed to mix
and chemical diffuses, eventually achieving homogeneity. Environmentally, this
situation is encountered when a volume of fluid (e.g., water) moves to an interface
and there contacts another phase (e.g., air) containing a solute with a different
fugacity. Volatilization may then occur over a period of time.
There are now three variables: concentration (C), position (y), and time (t). If
we consider a volume of A∆y, as shown in Figure 7.4, then the flux in is –BA dC/dy,
and the flux out is –BA(dC/dy + ∆yd2C/dy2), while the accumulation is A∆y∆C in
the time increment ∆t. It follows that
–BA dC/dy + BA(dC/dy + ∆yd2C/dy2) = A∆y∆C/∆t
or as ∆y and ∆t tend to zero,
Bd2C/dy2 = dC/dt
This is Fick’s second law. Solution of this partial differential equation requires two
boundary conditions, usually initial concentrations at specified positions. A partic-
ularly useful solution is the “penetration” equation, which describes diffusion into
a slab of fluid that is brought into contact with another slab of constant concentration
CS. The boundary conditions are
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C = CS at y = 0 at all times
C = 0 for y > 0 at t = 0
Solution is easiest if some hindsight is invoked to suggest that the dimensionless
group X or (y/ ) will occur in the solution. Interestingly, this is of the same
form as the initial definition of B as y2/2t.
It can be shown that
C = CS (1 – (2/ ) 0∫X exp(–X2) dX) = Cs [1–erf(X)]
Figure 7.4 Unsteady-state or penetration diffusion.
4Bt
π
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158 MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY
where 
X = y/ 
Unfortunately, this integral, which is known as the Gauss Error integral or
probability function or error function, cannot be solved analytically, thus tabulated
values must be used. The error function has the property that it is zero when X is
zero, and it approaches unity when X is 3 or larger. Its value can be found in tables
of mathematical functions, or it can be evaluated using built-in approximations in
spreadsheet software. A convenient approximation is
erf(X) = 1 – exp(–0.746X – 1.101 X2)
which is quite accurate when X exceeds 0.75. When X is less than 0.5, erf(X) is
approximately 1.1X. The penetration solution shown in Figure 7.4 illustrates the
very rapid initial transfer close to the interface, followed by slower penetration that
occurs later as the concentration gradient becomes smaller. Now the transfer rate at
the boundary (y = 0) can be shown to be 
B(dC/dy)y=0 = CsA 
Over a time t, the total flux (mol) becomes
CSA
The average flux is then obtained by dividing by t
CsA mol/h
But, since the average flux is CSAkM, the average mass transfer coefficient kM, which
applies over this time, must be .
The mass transfer coefficient, kM, under these transient conditions, thus depends
on the time of exposure (short exposures giving a large kM) and on the square root
of diffusivity. This contrasts with the steady-state solution, in which kM is indepen-
dent of time and proportional to diffusivity. The reason for this behavior is that kM
is apparently very large initially, because the concentration gradient is large. It falls
in inverse proportion to , thus the average also falls in this proportion. The lower
dependence on diffusivity (to the power of 0.5 instead of 1.0) arises, because not
all the transferring mass has to diffuse the total distance; much of it goes into
“storage” during the transient concentration buildup.
A problem now arises in environmental calculations: which definition of kM
applies, B/∆y or ? Contact time is the key determinant. If the contact time
between phases is long, and the amount transferred exceeds the capacity of the
phases, it is likely that a steady-state condition applies, and we should use B/∆y.
4Bt
B/π t
4Bt/π
4B/π t
4B/π t
t
4B/π t
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Conversely, if the contact time is short, we can expect to use . If we measure
the transfer rates at several temperatures, and thus different diffusivities, or measure
the transfer rate of different chemicals of different B, then plot kM versus B on log-
log paper, the slope of the line will be 1.0 if steady-state applies, and 0.5 if unsteady-
state applies. In practice, an intermediate power of about 2/3 often applies, suggesting
that we have mostly penetration diffusion followed by a period of near-steady-state
diffusion.
7.6 DIFFUSION IN POROUS MEDIA
When a solute is diffusing in air or water, its movement is restricted only by
collisions