2.8 Thermodynamics of Real Gases
103
temperature T B for which the second virial coefficient vanishes is known as the
Boyle temperature, T B .
To obtain an equation for the Gibbs energy for a fluid that is described by the
pressure virial equation (2.8.11a), we begin with Eq. (2.8.6) with V real fluid obtained
from the virial equation of state (2.8.11a), which becomes, with P i = P ideal ,
P
P ideal
dG T = Nk B T
P
P ideal
1
P
+ B
2 (T ) + B
3 (T )P + · · ·
dP .
The Gibbs energy is then given by the virial series
G(T , P ) = G(T , P ideal ) + Nk B T
ln
P
P ideal
+ B
2 (T )(P − P ideal )
+
1
2 B
3 (T )(P
2
− P
2
ideal ) + · · ·
.
The expression for G(T , P ) can be simplified by starting from G(T , P ideal ), then
choosing P ideal = 0 for the lower limit of the integral over pressure, thereby
obtaining
G(T , P ) = G
◦ (T ) + Nk B T
ln
P
P ◦
+ B
2 (T )P +
1
2 B
3 (T )P
2
+ · · ·
(2.8.13)
as the expression for the Gibbs energy of a nonideal gas described in terms of the
pressure virial equation of state.
Although an expression of this type is rather appealing, we encounter the same
problem that we have seen for the equation of state itself, i.e., the Gibbs free energy
is no longer universal, as it was for the ideal gas. We see this explicitly from
Eq. (2.8.13), as the virial coefficients differ from one nonideal gas to another. As will
be apparent in Chap. 9, for example, such an expression would not be particularly
convenient for the calculation of equilibrium constants for chemical reactions, so
how do we overcome this problem?
2.8.3 Fugacity
To resolve this inconvenience, we shall generalize the concept of pressure by
defining a new thermodynamic function, f (T , P ), called fugacity, via
G(T , P ) ≡ G
◦ (T ) + Nk B T ln
f
f ◦
,
(2.8.14a)
103
temperature T B for which the second virial coefficient vanishes is known as the
Boyle temperature, T B .
To obtain an equation for the Gibbs energy for a fluid that is described by the
pressure virial equation (2.8.11a), we begin with Eq. (2.8.6) with V real fluid obtained
from the virial equation of state (2.8.11a), which becomes, with P i = P ideal ,
P
P ideal
dG T = Nk B T
P
P ideal
1
P
+ B
2 (T ) + B
3 (T )P + · · ·
dP .
The Gibbs energy is then given by the virial series
G(T , P ) = G(T , P ideal ) + Nk B T
ln
P
P ideal
+ B
2 (T )(P − P ideal )
+
1
2 B
3 (T )(P
2
− P
2
ideal ) + · · ·
.
The expression for G(T , P ) can be simplified by starting from G(T , P ideal ), then
choosing P ideal = 0 for the lower limit of the integral over pressure, thereby
obtaining
G(T , P ) = G
◦ (T ) + Nk B T
ln
P
P ◦
+ B
2 (T )P +
1
2 B
3 (T )P
2
+ · · ·
(2.8.13)
as the expression for the Gibbs energy of a nonideal gas described in terms of the
pressure virial equation of state.
Although an expression of this type is rather appealing, we encounter the same
problem that we have seen for the equation of state itself, i.e., the Gibbs free energy
is no longer universal, as it was for the ideal gas. We see this explicitly from
Eq. (2.8.13), as the virial coefficients differ from one nonideal gas to another. As will
be apparent in Chap. 9, for example, such an expression would not be particularly
convenient for the calculation of equilibrium constants for chemical reactions, so
how do we overcome this problem?
2.8.3 Fugacity
To resolve this inconvenience, we shall generalize the concept of pressure by
defining a new thermodynamic function, f (T , P ), called fugacity, via
G(T , P ) ≡ G
◦ (T ) + Nk B T ln
f
f ◦
,
(2.8.14a)