Physical Chemistry Third Edition

4.3 Additional Useful Thermodynamic Identities 173

PROBLEMS

Section 4.3: Additional Useful Thermodynamic Identities

4.10 Consider a gas obeying the van der Waals equation of state.

a.Find an expression for (∂U/∂V)T,n(the internal

pressure). Explain why this is an intensive

quantity.

b. Find the value of (∂U/∂V)T,nfor argon at 298.15 K and

a molar volume of 0.0244 m^3 mol−^1 , using the result of

part a. Explain why this is independent of temperature.

Compare the internal pressure with the pressure of the

gas at this temperature and molar volume.

c.Find the value of the internal pressure (∂U/∂V)T,nfor

argon at 298.15 K and a molar volume of

5. 0 × 10 −^5 m^3 mol−^1 (a liquid-like value). Compare

the internal pressure with the pressure of the gas at this

temperature and molar volume.

d.Find the value of (∂U/∂V)T,nfor argon at 298.15 K and

a molar volume of 0.0244 m^3 mol−^1 , using the

truncated virial equation of state as in Exercise 4.5.

Compare your answer with that of part b.

4.11 a.Show that

CPVT α(∂P/∂T)S,n

whereαis the coefficient of thermal expansion.

b.Show that

CP−CV(P+Pint)αV

wherePintis the internal pressure, (∂U/∂V)T,n.

4.12 It is shown in the theory of hydrodynamics^2 that the speed
of sound,vs, is given by

v^2 s

Vm

MκS

whereκSis the adiabatic compressibility,Vmis the molar

volume, andMis the molar mass.

a.Show that

v^2 s

VmCP

MκTCV

whereκTis the isothermal compressibility.

(^2) H. Lamb,Hydrodynamics, 6th ed., Cambridge University Press, New
York, 1932, p. 477ff.
b. Find the speed of sound in air at 298.15 K
and 1.000 atm, assuming a mean molar mass of
0.29 kg mol−^1 and assuming thatCV, m 5 R/2.
c.Find the speed of sound in helium at 298.15 K and
1.000 atm.
d.For both parts b and c, find the ratio of the speed of
sound to the mean speed of the gas molecules, given by
Eq. (9.4-6).
4.13 Using the formula in the previous problem, find the speed
of sound in liquid benzene at 20◦C. Assume that the value
ofCP, mat 25◦C can be used.
4.14 Derive the following equation:
(
∂U
∂V
)
T,n
−P+
(
∂H
∂P
)
T,n
−V

T,n
4.15 Consider a gas that obeys the truncated pressure virial
equation of state
PVmRT+A 2 P+A 3 P^2
where the pressure virial coefficients depend on
temperature.
a. Find an expression forμJT, the Joule–Thomson
coefficient, for this gas.
b. Show that the Joule–Thomson coefficient in part a does
not vanish in the limit of zero pressure, even though the
Joule–Thomson coefficient of an ideal gas vanishes.
c.Evaluate the Joule–Thomson coefficient for argon at
298.15 K in the limit of zero pressure, assuming that
CP, m 5 R/2. Use values in Example 2.15 and the fact
thatA 2 B 2.
4.16 a. Evaluate the Joule–Thomson coefficient for carbon
dioxide atT 298 .15 K andP 1 .000 bar, assuming
thatCP,m 5 R/2. Use the fact thatA 2 B 2 and
thatA 3 
1
RT
(
B 3 −B^22
)

. Obtain a value ofB 3 as in
Problem 1.46 from the van der Waals
parameters.
b. Repeat the calculation for 15.00 bar and the same
temperature.
4.17 a.Find the change in enthalpy if 1.000 mol of liquid water
is pressurized at 0.00◦C from 1.000 atm to 50.000 atm.