4.3 Additional Useful Thermodynamic Identities
173
P R O B L E M S
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 ,n for 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 ,n for
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 ,n for 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
C P VT α(∂P/∂T ) S,n
where α is the coefficient of thermal expansion.
b. Show that
C P − C V (P + P int ) αV
where P int is the internal pressure, (∂U/∂V ) T ,n .
4.12 It is shown in the theory of hydrodynamics 2 that the speed
of sound, v s , is given by
v
2
s
V m
Mκ S
where κ S is the adiabatic compressibility, V m is the molar
volume, and M is the molar mass.
a. Show that
v
2
s
V m C P
Mκ T C V
where κ T is 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 that C V,m 5R/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
of C P,m at 25 ◦ C can be used.
4.14 Derive the following equation:
∂U
∂V
T ,n
−P +
∂H
∂P
T ,n
− V
∂P
∂V
T ,n
4.15 Consider a gas that obeys the truncated pressure virial
equation of state
PV m RT + A 2 P + A 3 P
2
where the pressure virial coefficients depend on
temperature.
a. Find an expression for µ JT , 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
C P,m 5R/2. Use values in Example 2.15 and the fact
that A 2 B 2 .
4.16 a. Evaluate the Joule–Thomson coefficient for carbon
dioxide at T 298.15 K and P 1.000 bar, assuming
that C P,m 5R/2. Use the fact that A 2 B 2 and
that A 3
1
RT
B 3 − B 2
2
. Obtain a value of B 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.
173
P R O B L E M S
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 ,n for 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 ,n for
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 ,n for 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
C P VT α(∂P/∂T ) S,n
where α is the coefficient of thermal expansion.
b. Show that
C P − C V (P + P int ) αV
where P int is the internal pressure, (∂U/∂V ) T ,n .
4.12 It is shown in the theory of hydrodynamics 2 that the speed
of sound, v s , is given by
v
2
s
V m
Mκ S
where κ S is the adiabatic compressibility, V m is the molar
volume, and M is the molar mass.
a. Show that
v
2
s
V m C P
Mκ T C V
where κ T is 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 that C V,m 5R/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
of C P,m at 25 ◦ C can be used.
4.14 Derive the following equation:
∂U
∂V
T ,n
−P +
∂H
∂P
T ,n
− V
∂P
∂V
T ,n
4.15 Consider a gas that obeys the truncated pressure virial
equation of state
PV m RT + A 2 P + A 3 P
2
where the pressure virial coefficients depend on
temperature.
a. Find an expression for µ JT , 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
C P,m 5R/2. Use values in Example 2.15 and the fact
that A 2 B 2 .
4.16 a. Evaluate the Joule–Thomson coefficient for carbon
dioxide at T 298.15 K and P 1.000 bar, assuming
that C P,m 5R/2. Use the fact that A 2 B 2 and
that A 3
1
RT
B 3 − B 2
2
. Obtain a value of B 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.