166
4 The Thermodynamics of Real Systems
Exercise 4.4
Calculate ∆S for 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.
P R O B L E M S
Section 4.2: Fundamental Relations for Closed Simple
Systems
4.4 The fundamental equation or fundamental relation of
thermodynamics for a particular system is a formula giving
S and a function of U, V , and n, or giving U as a function of
S, V , and n for 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
S nS 0 /n 0 + nRln (U/U 0 )
3/2 (V /V 0 )(n/n 0 )
−5/2
where S 0 , n 0 , and V 0 are constants.
a. Solve this equation for U U(S, V , n).
b. Use Eq. (4.2-5) to obtain an expression for T . Use this
expression to obtain an expression for U as a function of
T and n.
c. Use Eq. (4.2-6) to obtain an expression for P. Use this
expression to obtain an expression for P as a function of
T , V , and n.
4.5 A system obeys the fundamental thermodynamic relation
U U(S, V , n) Kn
5/3 (V − nb)
−2/3 e
2S/3nR
−
n 2 a
V
where K is a constant. Find expressions for P, T , and µ.
Show that the system obeys the van der Waals equation
of state.
4.6 Consider a gas obeying the truncated pressure virial
equation of state
PV m RT + A 2 P + A 3 P
2
where the pressure virial coefficients A 2 and A 3 depend
on T .
1 H. B. Callen, Thermodynamics, Wiley, New York, 1960, pp. 26ff, 53ff.
a. Find an expression for (∂S/∂P) T ,n for this gas.
b. Write an expression for G(T , P 2 , n) − G(T , P 1 , n) for
this gas.
c. Find the value of ∆G for pressurizing 2.500 mol of
argon from 1.000 atm to 25.00 atm at 298.15 K. Assume
that A 3 ≈ 0.
4.7 A gas is represented by the truncated virial equation of state
PV m /RT 1 + B 2 /V m + B 3 /V
2
m
where the virial coefficients depend on T .
a. Find an expression for the molar entropy change for an
isothermal volume change of the gas.
b. Find the value of ∆S for 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 that B 3 ≈ 0. Compare your answer
with the value assuming that argon is ideal.
4.8 Consider a gas that obeys the van der Waals equation of
state.
a. Find an expression for (∂S/∂V ) T ,n for this gas.
b. Find the value of ∆S for 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.
4.9 Consider a gas obeying the Redlich–Kwong equation of
state.
a. Find an expression for (∂S/∂V ) T ,n for this gas.
b. Find the value for ∆S for 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.
4 The Thermodynamics of Real Systems
Exercise 4.4
Calculate ∆S for 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.
P R O B L E M S
Section 4.2: Fundamental Relations for Closed Simple
Systems
4.4 The fundamental equation or fundamental relation of
thermodynamics for a particular system is a formula giving
S and a function of U, V , and n, or giving U as a function of
S, V , and n for 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
S nS 0 /n 0 + nRln (U/U 0 )
3/2 (V /V 0 )(n/n 0 )
−5/2
where S 0 , n 0 , and V 0 are constants.
a. Solve this equation for U U(S, V , n).
b. Use Eq. (4.2-5) to obtain an expression for T . Use this
expression to obtain an expression for U as a function of
T and n.
c. Use Eq. (4.2-6) to obtain an expression for P. Use this
expression to obtain an expression for P as a function of
T , V , and n.
4.5 A system obeys the fundamental thermodynamic relation
U U(S, V , n) Kn
5/3 (V − nb)
−2/3 e
2S/3nR
−
n 2 a
V
where K is a constant. Find expressions for P, T , and µ.
Show that the system obeys the van der Waals equation
of state.
4.6 Consider a gas obeying the truncated pressure virial
equation of state
PV m RT + A 2 P + A 3 P
2
where the pressure virial coefficients A 2 and A 3 depend
on T .
1 H. B. Callen, Thermodynamics, Wiley, New York, 1960, pp. 26ff, 53ff.
a. Find an expression for (∂S/∂P) T ,n for this gas.
b. Write an expression for G(T , P 2 , n) − G(T , P 1 , n) for
this gas.
c. Find the value of ∆G for pressurizing 2.500 mol of
argon from 1.000 atm to 25.00 atm at 298.15 K. Assume
that A 3 ≈ 0.
4.7 A gas is represented by the truncated virial equation of state
PV m /RT 1 + B 2 /V m + B 3 /V
2
m
where the virial coefficients depend on T .
a. Find an expression for the molar entropy change for an
isothermal volume change of the gas.
b. Find the value of ∆S for 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 that B 3 ≈ 0. Compare your answer
with the value assuming that argon is ideal.
4.8 Consider a gas that obeys the van der Waals equation of
state.
a. Find an expression for (∂S/∂V ) T ,n for this gas.
b. Find the value of ∆S for 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.
4.9 Consider a gas obeying the Redlich–Kwong equation of
state.
a. Find an expression for (∂S/∂V ) T ,n for this gas.
b. Find the value for ∆S for 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.