2.8 Thermodynamics of Real Gases
107
μ α (T , P ) = μ
◦
α (T ) + RT ln
f α
f ◦
rather than employing Eq. (2.3.7c) to determine the chemical potential of chemical
species α. Recall also that f ◦ = P ◦ = 1 bar. The change in the Gibbs energy for
the chemical reaction (2.7.7) for real gases is then obtained as
((G) react (T , P , ξ) = ((G
◦ ) react (T ) + RT ln
(f C (ξ )/f ◦ ) ν C (f D (ξ )/f ◦ ) ν D
(f A (ξ )/f ◦ ) ν A (f B (ξ )/f ◦ ) ν B
,
rather than as given by Eq. (2.7.19). At equilibrium, ((G) react = 0, the fugacities
achieve their equilibrium values, f α = f α (ξ e ), and
((G
◦ ) react (T ) = −RT ln K f ,
with K f the relevant equilibrium constant. Upon employing the defining relation for
the activity coefficients γ α for the various chemical species in the reaction mixture,
we may relate K f to K P via
K f = K γ K P ,
with K γ ≡ γ
ν C
C γ
ν D
D /(γ
ν A
A γ
ν B
B ). A deviation of K γ from one thus provides a measure
of the nonideality of the gases participating in the chemical reaction.
2.8.4 The Law of Corresponding States
We shall see that the nature of the deviations from ideal gas behaviour is illustrated
well by the Van der Waals gas, for which the compressibility factor takes the form
Z VdW (P , T ) =
V
V − Nb
−
aN
k B T V
(2.8.21a)
=
1
1 −
Nb
V
−
aN
k B T V
.
(2.8.21b)
The form for the first term in Eq. (2.8.21b) facilitates its expansion as a power series
in Nb/V ≡ b/v for Nb V , which will be the case for a gas density not too far
removed from that of an ideal gas. The expanded form can be written as
Z VdW (P , T ) 1 +
b −
a
k B T
1
v
+
b 2
v 2 + · · ·
(2.8.22a)
107
μ α (T , P ) = μ
◦
α (T ) + RT ln
f α
f ◦
rather than employing Eq. (2.3.7c) to determine the chemical potential of chemical
species α. Recall also that f ◦ = P ◦ = 1 bar. The change in the Gibbs energy for
the chemical reaction (2.7.7) for real gases is then obtained as
((G) react (T , P , ξ) = ((G
◦ ) react (T ) + RT ln
(f C (ξ )/f ◦ ) ν C (f D (ξ )/f ◦ ) ν D
(f A (ξ )/f ◦ ) ν A (f B (ξ )/f ◦ ) ν B
,
rather than as given by Eq. (2.7.19). At equilibrium, ((G) react = 0, the fugacities
achieve their equilibrium values, f α = f α (ξ e ), and
((G
◦ ) react (T ) = −RT ln K f ,
with K f the relevant equilibrium constant. Upon employing the defining relation for
the activity coefficients γ α for the various chemical species in the reaction mixture,
we may relate K f to K P via
K f = K γ K P ,
with K γ ≡ γ
ν C
C γ
ν D
D /(γ
ν A
A γ
ν B
B ). A deviation of K γ from one thus provides a measure
of the nonideality of the gases participating in the chemical reaction.
2.8.4 The Law of Corresponding States
We shall see that the nature of the deviations from ideal gas behaviour is illustrated
well by the Van der Waals gas, for which the compressibility factor takes the form
Z VdW (P , T ) =
V
V − Nb
−
aN
k B T V
(2.8.21a)
=
1
1 −
Nb
V
−
aN
k B T V
.
(2.8.21b)
The form for the first term in Eq. (2.8.21b) facilitates its expansion as a power series
in Nb/V ≡ b/v for Nb V , which will be the case for a gas density not too far
removed from that of an ideal gas. The expanded form can be written as
Z VdW (P , T ) 1 +
b −
a
k B T
1
v
+
b 2
v 2 + · · ·
(2.8.22a)