2.9 Problems for This Chapter
123
parameters for Xe are a = 5.6850 dm
6 mol
−2 and b = 0.05900 dm
3 mol
−1 .
Compare your calculated values with an experimental value deduced from the
data provided by Michels et al. [A. Michels, T. Wassenaar, and P. Lowerse,
Physica 20, 99–106 (1954)].
14. The expressions obtained in Examples 2.8 and 2.9 for the molar internal energy
U(T , V ) and entropy S(T , V ) for a Van der Waals fluid may be written as
U(T , V ) = U ideal gas (T )−a/V and S(T , V ) = S ideal gas (T )+R ln(V −b). For
fixed temperature and pressure, the Gibbs energy typically exhibits two minima
as a function of molar volume, corresponding to values of the molar volumes
of the liquid and gaseous phases of the Van der Waals fluid, with the deeper
minimum corresponding to the thermodynamically stable phase. Evaluate G as
a function of ln V from V = 0.02 dm
3 to V = 400 dm
3 for a fixed pressure
P = 1 bar and for temperatures 354, 374, 394 K, for a Van der Waals fluid
having parameters a = 4.843 dm
6 bar mol
−2 , b = 0.016 dm
3 mol
−1 . Plot G vs.
ln V for each of these three temperatures and discuss the relative stabilities of
the liquid and gaseous phases at these temperatures.
15. By employing the criterion that G = G g − G = 0 for the process liquid
⇐⇒ gas at the normal boiling temperature T b , determine from the results of
Problem 14 the value of T b for the Van der Waals fluid, as well as the molar
volumes of the two co-existing phases at the boiling temperature. Develop
expressions for H vap (T b ) and S vap (T b ) for this Van der Waals fluid and
evaluate them.
16. Use the ideal gas, Van der Waals, and Redlich–Kwong equations of state to
compute the pressure of ethane gas at temperature T = 350 K and molar
density ρ = 14.241 mol dm
−3 , given that the Van der Waals parameters for
ethane are a = 5.56145 dm
6 mol
−2 bar, b = 0.06380 dm
3 mol
−1 , and that
the Redlich–Kwong parameters for ethane are A = 98.831 dm
6 mol
−2 bar K
1
2 ,
B = 0.045153 dm
3 mol
−1 . Compare your computed pressures with the
experimental pressure 506.6 bar.
17. The Benedict–Webb–Rubin equation of state, proposed in 1940, is an 8parameter equation:
P = RT ρ + (BRT − A − C 0 T
−2 )ρ
2
+ (bRT − a)ρ
3
+ aαρ
6
+ cT
−2 ρ
3 (1 + γ ρ
2 )e
−γ ρ 2 ,
with ρ ≡ 1/V , and A, B, C 0 , a, b, α, c, and γ the eight parameters.
For ethane, these parameters have the values A = 4.25062 dm 6 mol −2 bar,
B = 0.0627724 dm 3 mol −1 , C 0 = 1.81972×10 5 dm 3 mol −2 K 2 bar, a =
0.34973 dm 9 mol −3 bar, b = 0.011122 dm 6 mol −2 , c = 3.3201 ×
10 4 dm 9 mol −3 K 2 bar, α = 2.43389×10 −4 dm 9 mol −3 , γ = 0.01118 dm 6 mol −2 ,
while the universal gas constant has the value R = 0.083145 dm 3 mol −1 K −1 bar.
Calculate the pressure of an ethane sample with molar volume V =
0.07022 dm 3 mol −1 at temperature T = 350 K using the Benedict–Webb–
123
parameters for Xe are a = 5.6850 dm
6 mol
−2 and b = 0.05900 dm
3 mol
−1 .
Compare your calculated values with an experimental value deduced from the
data provided by Michels et al. [A. Michels, T. Wassenaar, and P. Lowerse,
Physica 20, 99–106 (1954)].
14. The expressions obtained in Examples 2.8 and 2.9 for the molar internal energy
U(T , V ) and entropy S(T , V ) for a Van der Waals fluid may be written as
U(T , V ) = U ideal gas (T )−a/V and S(T , V ) = S ideal gas (T )+R ln(V −b). For
fixed temperature and pressure, the Gibbs energy typically exhibits two minima
as a function of molar volume, corresponding to values of the molar volumes
of the liquid and gaseous phases of the Van der Waals fluid, with the deeper
minimum corresponding to the thermodynamically stable phase. Evaluate G as
a function of ln V from V = 0.02 dm
3 to V = 400 dm
3 for a fixed pressure
P = 1 bar and for temperatures 354, 374, 394 K, for a Van der Waals fluid
having parameters a = 4.843 dm
6 bar mol
−2 , b = 0.016 dm
3 mol
−1 . Plot G vs.
ln V for each of these three temperatures and discuss the relative stabilities of
the liquid and gaseous phases at these temperatures.
15. By employing the criterion that G = G g − G = 0 for the process liquid
⇐⇒ gas at the normal boiling temperature T b , determine from the results of
Problem 14 the value of T b for the Van der Waals fluid, as well as the molar
volumes of the two co-existing phases at the boiling temperature. Develop
expressions for H vap (T b ) and S vap (T b ) for this Van der Waals fluid and
evaluate them.
16. Use the ideal gas, Van der Waals, and Redlich–Kwong equations of state to
compute the pressure of ethane gas at temperature T = 350 K and molar
density ρ = 14.241 mol dm
−3 , given that the Van der Waals parameters for
ethane are a = 5.56145 dm
6 mol
−2 bar, b = 0.06380 dm
3 mol
−1 , and that
the Redlich–Kwong parameters for ethane are A = 98.831 dm
6 mol
−2 bar K
1
2 ,
B = 0.045153 dm
3 mol
−1 . Compare your computed pressures with the
experimental pressure 506.6 bar.
17. The Benedict–Webb–Rubin equation of state, proposed in 1940, is an 8parameter equation:
P = RT ρ + (BRT − A − C 0 T
−2 )ρ
2
+ (bRT − a)ρ
3
+ aαρ
6
+ cT
−2 ρ
3 (1 + γ ρ
2 )e
−γ ρ 2 ,
with ρ ≡ 1/V , and A, B, C 0 , a, b, α, c, and γ the eight parameters.
For ethane, these parameters have the values A = 4.25062 dm 6 mol −2 bar,
B = 0.0627724 dm 3 mol −1 , C 0 = 1.81972×10 5 dm 3 mol −2 K 2 bar, a =
0.34973 dm 9 mol −3 bar, b = 0.011122 dm 6 mol −2 , c = 3.3201 ×
10 4 dm 9 mol −3 K 2 bar, α = 2.43389×10 −4 dm 9 mol −3 , γ = 0.01118 dm 6 mol −2 ,
while the universal gas constant has the value R = 0.083145 dm 3 mol −1 K −1 bar.
Calculate the pressure of an ethane sample with molar volume V =
0.07022 dm 3 mol −1 at temperature T = 350 K using the Benedict–Webb–