Physical Chemistry Third Edition

166 4 The Thermodynamics of Real Systems

Exercise 4.4

Calculate∆Sfor the expansion of 1.000 mol of argon from 10.00 atm to 1.000 atm at 298.15 K,

assuming the truncated pressure virial equation of state. Compare your result with that obtained

by assuming argon to be ideal.

PROBLEMS

Section 4.2: Fundamental Relations for Closed Simple
Systems

4.4Thefundamental equationorfundamental relation of

thermodynamicsfor a particular system is a formula giving

Sand a function ofU,V, andn, or givingUas a function of

S,V, andnfor that system. If this relation is known, all

thermodynamic information about the system can be

obtained from it. For an ideal monatomic gas with constant

heat capacity,^1

SnS 0 /n 0 +nRln(U/U 0 )^3 /^2 (V/V 0 )(n/n 0 )−^5 /^2

whereS 0 ,n 0 , andV 0 are constants.

a. Solve this equation forUU(S,V,n).

b. Use Eq. (4.2-5) to obtain an expression forT. Use this

expression to obtain an expression forUas a function of

Tandn.

c.Use Eq. (4.2-6) to obtain an expression forP. Use this

expression to obtain an expression forPas a function of

T,V, andn.

4.5A system obeys the fundamental thermodynamic relation

UU(S,V,n)Kn^5 /^3 (V−nb)−^2 /^3 e^2 S/^3 nR−

n^2 a

V

whereKis a constant. Find expressions forP,T, andμ.

Show that the system obeys the van der Waals equation

of state.

4.6Consider a gas obeying the truncated pressure virial

equation of state

PVmRT+A 2 P+A 3 P^2

where the pressure virial coefficientsA 2 andA 3 depend

onT.

(^1) H. B. Callen,Thermodynamics, Wiley, New York, 1960, pp. 26ff, 53ff.
a.Find an expression for (∂S/∂P)T,nfor this gas.
b.Write an expression forG(T,P 2 ,n)−G(T,P 1 ,n) for
this gas.
c.Find the value of∆Gfor pressurizing 2.500 mol of
argon from 1.000 atm to 25.00 atm at 298.15 K. Assume
thatA 3 ≈0.
4.7A gas is represented by the truncated virial equation of state
PVm/RT 1 +B 2 /Vm+B 3 /Vm^2
where the virial coefficients depend onT.
a.Find an expression for the molar entropy change for an
isothermal volume change of the gas.
b.Find the value of∆Sfor the expansion of 2.000 mol of
argon from 5.00 L to 30.00 L at a constant temperature of
298.15 K. Assume thatB 3 ≈0. Compare your answer
with the value assuming that argon is ideal.
4.8Consider a gas that obeys the van der Waals equation of
state.
a.Find an expression for (∂S/∂V)T,nfor this gas.
b.Find the value of∆Sfor the isothermal expansion of
2.000 mol of argon from a volume of 5.00 L to a volume
of 30.00 L at 298.15 K, assuming the van der Waals
equation of state. Compare with the value for the same
change in state assuming argon to be ideal.
4.9Consider a gas obeying the Redlich–Kwong equation of
state.
a.Find an expression for (∂S/∂V)T,nfor this gas.
b.Find the value for∆Sfor the isothermal expansion
of 2.000 mol of argon from a volume of 10.00 L to
a volume of 40.00 L at 298.15 K, assuming the
Redlich–Kwong equation of state. Compare with the
value for the same change in state assuming argon to
be ideal.