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The Method of Continuous Variation: A Laboratory Investigation of
the Formula of a Precipitate
William R. Furlong, Miles A. Rubinski, and Ramee Indralingam*
Department of Chemistry, Stetson University, DeLand, Florida 32723, United States
*S Supporting Information
ABSTRACT: The method of continuous variation is applied to the reaction between
barium chloride and diammonium hydrogen phosphate in neutral, acidic, and basic
conditions. Depending on the medium, barium dihydrogen phosphate, barium hydrogen
phosphate, or barium phosphate is precipitated. The precipitates are washed, dried, and
weighed. Construction of a continuous variation plot for each condition leads to the
deduction of the molecular formula of the phosphate precipitated. This experiment
demonstrates the concept of stoichiometric relationships between ions in the formation of
compounds and serves to give students practice in calculations involving limiting reagents.
KEYWORDS: First-Year Undergraduate, Second-Year Undergraduate/General, Analytical Chemistry, Laboratory Instruction,
Physical Chemistry, Hands-On Learning/Manipulatives, Gravimetric Analysis, Quantitative Analysis, Stoichiometry
The method of continuous variation, or Job’s method as it
is commonly called, has been commonly used in
laboratory experiments in instrumental analysis classes to
determine metal-to-ligand ratios in complex formation
reactions.1−13 In the technique, the total amount of ligand
and metal are held constant in a series of solutions of constant
volume, whereas the individual amounts of ligand and metal are
varied continuously. A physical property, such as the
absorbance of the colored complex, is measured for every
solution. A plot of absorbance versus mole fraction of the metal
yields a curve with ascending, then descending branches whose
extrapolated sides meet at a point of maximum absorbance.
This point denotes the optimum mole fraction of the metal at
which complete complex formation occurs. Hence the formula
of the complex is deduced.
This technique may be applied to the determination of the
formula of any ionic compound, provided some characteristic
of the compound may be measured. For instance, if the
compound forms a precipitate when aqueous solutions of the
component ions are mixed, the mass of the washed and dried
precipitate may be plotted against the mole fraction of one of
the ions. The optimum mole fraction as obtained in a Job’s plot
would indicate the mole ratio of the ions and hence the formula
of the precipitate may be deduced.
Two papers have been published with details of application
of the method of continuous variation to the determination of
the stoichiometry of precipitates. One14 gave details of
measuring the height in a graduated cylinder of barium
chromate precipitate formed when various volumes of
equimolar solutions of barium chloride and potassium
chromate were mixed, keeping the total volume constant.
The second publication15 described similar experiments carried
out by combining aqueous solutions of potassium iodide,
sodium carbonate, and potassium chromate with aqueous
lead(II) nitrate and barium chloride to form the precipitates of
lead and barium, and compared the results obtained by
measuring the mass of the precipitate with the results obtained
by measuring the height of the precipitate in a graduated
cylinder. The conclusion of the author was that “better” results
were obtained when the mass of the precipitate was used to
plot the graph.
An Internet search revealed a commercially available
laboratory experiment that is used in advanced placement
chemistry classes.16 In this experiment, iron(III) hydroxide and
copper(II) phosphate were precipitated by mixing aqueous
solutions of iron(III) nitrate and sodium hydroxide, and
copper(II) chloride and sodium phosphate, respectively. The
method of continuous variation was used to determine the
formula of the precipitates by plotting the volume of the
precipitate as measured in a graduated cylinder versus the mole
fraction of one of the reagents in each experiment. The
accuracy of the results could be improved by using the mass of
the precipitate rather than the volume, but, apparently, a trade-
off was made in favor of expediency over the more labor
intensive technique of having high school students filter, wash,
and dry each precipitate.
The experiments described above do not pose any surprises
for students. They are all given the same two aqueous solutions,
and because they are aware of the identity of the two ions that
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form the precipitate, they all know what optimum mole ratio
they should obtain from a graph.
A new experiment that applies the method of continuous
variation to the determination of the formula of an ionic
compound is described here. The experiment is unusual in that,
although the same two reagents are used as the starting
materials, conditions are controlled so that one of three
different molecular formulas is obtained. This allows an
instructor to assign an unknown to each student or pair of
students.
Barium phosphate, Ba3(PO4)2, barium hydrogen phosphate,
BaHPO4, and barium dihydrogen phosphate, Ba(H2PO4)2, are
all sparingly soluble salts that are precipitated when aqueous
barium chloride is mixed with a solution of the corresponding
ammonium salt of phosphoric acid as given below:
+
→ +
3BaCl (aq) 2(NH ) PO (aq)
Ba (PO ) (s) 6NH Cl(aq)
2 4 3 4
3 4 2
barium phosphate
4
(1)
+
→ +
BaCl (aq) (NH ) HPO (aq)
BaHPO (s) 2NH Cl(aq)
2 4 2 4
4
barium hydrogen phosphate
4
(2)
+
→ +
BaCl (aq) 2(NH )H PO (aq)
Ba(H PO ) (s) 2NH Cl(aq)
2 4 2 4
2 4 2
barium dihydrogen phosphate
4
(3)
The formula of each precipitated compound consists of a
different stoichiometric ratio of the barium ion to the
phosphate group. The mole fraction of barium, calculated as
a fraction of the total amount of barium and phosphate-
containing ions, is different in each precipitate, as shown below:
Ba3(PO4)2:
Χ =
+
=
+
+ −
3 mol Ba
3 mol Ba 2 mol PO
0.600Ba
2
2
4
3
BaHPO4:
=
+
=
+
+ −X
1 mol Ba
1 mol Ba 1 mol HPO
0.500Ba
2
2
4
2
Ba(H2PO4)2:
=
+
=
+
+ −X
1 mol Ba
1 mol Ba 2 mol H PO
0.333Ba
2
2
2 4
The method of continuous variation can be employed, and
by plotting the mass of the precipitate obtained in each case
versus the mole fraction of barium, the formula of the
compound can be determined. However, although diammo-
nium hydrogen phosphate, (NH4)2HPO4, and ammonium
dihydrogen phosphate, (NH4)H2PO4, can be purchased,
ammonium phosphate, (NH4)3PO4, is not available from a
chemical supplier. Hence, pH conditions have to be controlled
to generate (NH4)3PO4 from either (NH4)2HPO4 or (NH4)-
H2PO4 The details of the conditions are given in the
Supporting Information.
This laboratory experiment is suitable for inclusion in the
general chemistry class typically taken by students in the first
year of the undergraduate chemistry curriculum. The main goal
of this experiment is to use a practical method to teach students
that formulas of ionic compounds depend on the mole ratios of
their constituent ions. A secondary objective is to give students
practice in the calculation of limiting reagents. The main
objective is met when the students calculate the mole fraction
of barium in the precipitate that they obtained and hence
deduce the formula of the compound. The concept of limiting
reagent is also reinforced when students carry out a calculation
for experimental points in each branch of the graph.
■ EXPERIMENTAL PROCEDURE
Students are provided with an unknown that consists of two
vials, one containing an aqueous solution of 0.3 M barium
chloride and the other containing 0.3 M diammonium
hydrogen phosphate in 0.3 M HCl, deionized water, or 0.3
M NH3. Students are also provided with six 13 × 100 mm test
tubes, which they number 1−6and weigh. Using a combination
of volumetric pipets, 1, 2, 3, 4, 5, or 6 mL of the BaCl2 solution
is placed in the appropriately numbered test tube. Using a
different set of volumetric pipets, the phosphate solution is
placed in the test tubes in reverse order of volumes so that the
total volume in each test tube is 7 mL. A precipitate appears
immediately in each of the test tubes. The mixture is stirred
with a glass rod and the test tubes are placed in a hot water bath
for 15 min to digest the precipitates. The test tubes are then
centrifuged and the supernatant is discarded. The precipitate is
washed first with water and then with 95% ethanol, and
centrifuged each time; supernatants are discarded. The test
tubes are set in a beaker and placed in an oven to dry at 120 °C.
The dry, cool test tubes containing precipitate are weighed
again. The experimental procedure can be completed in a 3 h
laboratory period. Additional experimental details are described
in the Supporting Information.
A graph is constructed by plotting mass of precipitate versus
mole fraction of barium in the test tube, and by determining the
mole fraction at which the mass of precipitate would be a
maximum, the formula of the precipitate in the unknown is
deduced. The limiting reactant in each branch of the graph is
also determined by calculation.
■ HAZARDS
Barium chloride dihydrate and diammonium hydrogen
phosphate are hazardous in case of skin contact, eye contact,
ingestion, or inhalation. Care must be taken when handling
concentrated hydrochloric acid and ammonia in order to dilute
them. Hazards of 95% ethanol are due to ingestion, eye contact,
and inhalation of high vapor concentrations. Use gloves and
prepare all solutions in the fume hood. The phosphates of
barium are not classified as hazardous to human health.
However, clean up of the waste must be done scrupulously
because phosphate compounds are hazardous to aquatic
environments. A detailed procedure for efficient cleanup is
given in the Supporting Information.
■ RESULTS AND CONCLUSION
This experiment was tested over three years by a total of 31
second-year undergraduates in three quantitative analysis
chemistry classes, and the final version was carried out by
students in one general chemistry class, comprising 50 first-year
undergraduates. The analytical chemistry students carried out
the experiment individually. General chemistry students worked
in pairs. Of a total of 25 pairs of students, one pair was not able
to obtain the correct formula of the precipitate. The error was
Journal of Chemical Education Laboratory Experiment
dx.doi.org/10.1021/ed3004337 | J. Chem. Educ. XXXX, XXX, XXX−XXXB
found to be due to not allowing the precipitate to digest long
enough so that the discarded supernatant was cloudy and
contained some of the precipitate. Instructors should be aware
of this error that students are liable to make. All the students
obtained a narrow range of values for mole fraction of barium
of their precipitates, which, when rounded using appropriate
rules of significant figures, yielded the correct formula.
Representative student results taken from both, general
chemistry and analytical chemistry classes are given in Figure 1.
General chemistry and analytical chemistry students obtained
similar results, showing that this experiment can be carried out
successfully by students of varying caliber and experience.
The point of intersection of the two branches of each graph,
A, B, and C give the optimum mole fraction of barium in each
precipitate under acidic, neutral, and basic conditions,
respectively. The point of intersection is obtained by setting
equal to each other the equations of the best-fit lines of the two
branches of each graph. The points (0,0) and (1,0) are included
in the graphs although they are not experimentally determined.
These points aid in calculating the equations of the straight
lines so that the optimum mole fraction of barium in each
precipitate may be accurately determined.
In addition to the concept of stoichiometric relationships
between ions that form compounds, this lab experiment gave
students practice in calculations involving limiting reagents and
the use of a spreadsheet for the manipulation of data. The main
goal of this experiment was met when students were successful
in deducing the formula of an ionic compound by determining
the mole ratios of its constituent ions. The secondary objective
was also met when students successfully determined the
limiting reagent in each branch of the graph. It was found that
general chemistry students became more confident at carrying
out limiting reagent calculations because this laboratory
experiment reinforced the chemical principles that they were
learning in class during the same week. In the case of the
analytical chemistry students, who had not carried out this
experiment during the previous year, the calculations served to
reinforce previously learned concepts. Both groups needed help
with the steps involved in the spreadsheet manipulations.
To shorten the time in which the precipitates are brought to
constant mass, a modification was tried by filtering the
precipitate under vacuum, washing with water and ethanol,
and placing the filter papers and precipitates in an oven heated
to about 80 °C. However, it was found that the filter papers
themselves then lost moisture, which they gained while placed
on the pan of the analytical balance, necessitating a waiting
period before the mass stabilized. The error in the mass of the
filter paper led to lack of reproducibility in the final results.
Also, the logistics of keeping track of each set of six small filter
papers and their contents without contaminating them proved
to be formidable. Hence, the method described here of drying
the precipitates in the test tubes was adopted.
■ ASSOCIATED CONTENT
*S Supporting Information
Student handout; notes for the instructor including instructions
on the preparation of solutions and unknowns, and clean up
procedures. This material is available via the Internet at http://
pubs.acs.org.
■ AUTHOR INFORMATION
Corresponding Author
*E-mail: rindrali@stetson.edu.
Notes
The authors declare no competing financial interest.
■ ACKNOWLEDGMENTS
The authors are grateful to Eric W. Hoffman for invaluable help
with the graphics.
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Figure 1. Continuous variation plots of barium chloride and
diammonium hydrogen phosphate: (A) acidic conditions, (B) neutral
conditions, and (C) basic conditions.
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