2.8 Thermodynamics of Real Gases
101
while the Gibbs energy can be obtained similarly from A VdW (T , V ; N) via
Eq. (2.3.6b), for example. Note that it is not convenient to provide the enthalpy
and Gibbs energy explicitly in terms of their natural thermodynamic variables T
and P , as the allowed values of V for a Van der Waals fluid are given by the roots
of a cubic equation in V (see Eq. (2.8.9a)).
2.8.2 The Virial Equation of State
Accurate equation of state measurements (i.e., experimentally-determined P V T
behaviour) for real gases show systematic deviations from the ideal gas law. These
deviations may conveniently be represented by the generalization
P V = Nk B T Z(P , T )
(2.8.10a)
of the ideal gas law. The factor Z(P , T ), which we may express as the ratio
Z(P , T ) ≡
P V
(P V ) ideal
=
P V
Nk B T
,
(2.8.10b)
is called the compressibility factor, and has the value 1 for an ideal gas. Because
real gases approach ideal gas behaviour as the pressure tends to zero, we have the
limiting behaviour
lim
P →0
Z(P , T ) = 1 ,
so that Z(P , T ) > 1 implies that P real gas > P ideal gas , while Z(P , T ) < 1 implies
that P real gas < P ideal gas for the same molar volume V . This factor is commonly
represented graphically in terms of isotherms on a Z vs. P plot.
Based upon the observation that high-temperature isotherms for large molar volumes of real gases do not differ significantly from ideal gas isotherms, Kamerlingh–
Onnes proposed in 1917 a pressure power series representation of the equation of
state, with the leading term given by the ideal gas law. Specifically, he proposed that
the product P V be represented as a power series in the gas pressure P as
P V = RT (1 + B
2 P + B
3 P
2
+ · · · ) ,
(2.8.11a)
or, perhaps more conveniently, as a power series in the molar (number) density ρ
(the reciprocal of the molar volume V ), as
P V = RT (1 + B 2 ρ + B 3 ρ
2
+ · · · ) .
(2.8.11b)
101
while the Gibbs energy can be obtained similarly from A VdW (T , V ; N) via
Eq. (2.3.6b), for example. Note that it is not convenient to provide the enthalpy
and Gibbs energy explicitly in terms of their natural thermodynamic variables T
and P , as the allowed values of V for a Van der Waals fluid are given by the roots
of a cubic equation in V (see Eq. (2.8.9a)).
2.8.2 The Virial Equation of State
Accurate equation of state measurements (i.e., experimentally-determined P V T
behaviour) for real gases show systematic deviations from the ideal gas law. These
deviations may conveniently be represented by the generalization
P V = Nk B T Z(P , T )
(2.8.10a)
of the ideal gas law. The factor Z(P , T ), which we may express as the ratio
Z(P , T ) ≡
P V
(P V ) ideal
=
P V
Nk B T
,
(2.8.10b)
is called the compressibility factor, and has the value 1 for an ideal gas. Because
real gases approach ideal gas behaviour as the pressure tends to zero, we have the
limiting behaviour
lim
P →0
Z(P , T ) = 1 ,
so that Z(P , T ) > 1 implies that P real gas > P ideal gas , while Z(P , T ) < 1 implies
that P real gas < P ideal gas for the same molar volume V . This factor is commonly
represented graphically in terms of isotherms on a Z vs. P plot.
Based upon the observation that high-temperature isotherms for large molar volumes of real gases do not differ significantly from ideal gas isotherms, Kamerlingh–
Onnes proposed in 1917 a pressure power series representation of the equation of
state, with the leading term given by the ideal gas law. Specifically, he proposed that
the product P V be represented as a power series in the gas pressure P as
P V = RT (1 + B
2 P + B
3 P
2
+ · · · ) ,
(2.8.11a)
or, perhaps more conveniently, as a power series in the molar (number) density ρ
(the reciprocal of the molar volume V ), as
P V = RT (1 + B 2 ρ + B 3 ρ
2
+ · · · ) .
(2.8.11b)