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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 55
Using this in the expression above for µ and switching to molar quantities
µ = 1
Cp ,m
(T (∂Vm
∂T
)
p
− Vm) = 1
Cp ,m
(T [R
p
+ a
RT2
] − Vm)
�e above expression for Vm is used once more
µ = 1
Cp ,m
(T [R
p
+ a
RT2
] − {RT
p
+ b − a
RT
})
= 1
Cp ,m
(RT
p
+ a
RT
− RT
p
− b + a
RT
)
= 1
Cp ,m
( 2a
RT
− b)
which is the required result.
2E Adiabatic changes
Answers to discussion questions
D2E.2 In an adiabatic expansion the system does work but as no energy as heat is
permitted to enter the system, the internal energy of the system falls and so
consequently does the temperature. From the First Law, ∆U = w because q =
0. However, the change in internal energy is also related to the temperature
change and the heat capacity: ∆U = CV∆T . Equating these two expressions
for ∆U gives w = CV∆T , or dw = CVdT for an in�nitesimal change.
For a reversible expansion, the work is dw = −pdV . Equating these two ex-
pressions for the work gives −pdV = CVdT . �is equation is the point from
which the relationships between pressure, volume and temperature for a re-
versible adiabatic expansion are found: the heat capacity comes into the �nal
expressions via this route.
In words, the key thing here is that in an adiabatic process there is a change
in temperature, so it is not surprising that the properties of such a process are
related to the heat capacity because this quantity relates the energy and the
temperature rise.
Solutions to exercises
E2E.1(b) Carbon dioxide is a linear polyatomic molecule which has three degrees of
translational and two degrees of rotational freedom. From the equipartition
theorem
CV ,m = 1
2 × (νt + νr + 2νv) × R
where νt is the number of translational degrees of freedom, νr is the number
of rotational degrees of freedom and νv is the number of vibrational degrees of
freedom.