2.8 Thermodynamics of Real Gases
109
B
HS
2 (T ) =
2πσ 3
3
= 4v HS ,
which is independent of temperature and equal to four times the volume v HS of an
individual hard sphere. For the Sutherland model, with n = 6, we obtain B 2 (T ) as
B
SM
2 (T ) = B
HS
2 − 2π
∞
σ
e
βC 6 /R 6 − 1
R
2 dR ,
with β equal to 1/(k B T ). We cannot in general evaluate this expression further
exactly, but for βC 6 /R 6 1, we may expand the integrand and retain only the
first non-vanishing term to give
B
SM
2 (T ) =
2πσ 3
3
−
2πC 6
3k B T σ 3 .
If we compare this result with B VdW
2
(T ) given in Eqs. (2.8.22), we see that the Van
der Waals parameters a and b may be obtained as
a =
2πC 6
3σ 3 ,
b=
2πσ 3
3
,
so that a is thus associated (through C 6 ) with the long-range attractive dispersion
interaction and b corresponds to four times the volume v HS associated with a
hard sphere of diameter σ (often referred to as the ‘forbidden’ or ‘excluded’
volume). Given this association of the parameters a and b with attraction and the
volume occupied by an individual atom/molecule, a positive initial slope for a
compressibility factor isotherm thus indicates that molecular size effects dominate,
while a negative initial slope indicates that attractive forces dominate.
Clearly, the Van der Waals equation of state gives a qualitative explanation of real
gas behaviour through Z(P , T ). Note, however, that as the Van der Waals equation
of state is a function of two parameters a and b that depend upon the nature of the
specific gas being considered, we have lost the universality that we had with the
ideal gas equation of state.
Another way of viewing the Van der Waals equation is in the form of a cubic
equation in the volume V , namely,
V
3
− N
b +
k B T
P
V
2
+
aN 2
P
V −
aN 3 b
P
= 0 .
(2.8.23)
Figure 2.9 illustrates the behaviour of the Van der Waals pressure isotherms for a
and b parameters characterizing CO 2 . We see from this figure that for temperatures
T > 304.25 K, the isotherms are monotone decreasing functions of molar volume,
while for temperatures below 304.25 K, the Van der Waals equation for CO 2 gives
strongly non-monotonic behaviour over a fairly extensive range of molar volume V .
109
B
HS
2 (T ) =
2πσ 3
3
= 4v HS ,
which is independent of temperature and equal to four times the volume v HS of an
individual hard sphere. For the Sutherland model, with n = 6, we obtain B 2 (T ) as
B
SM
2 (T ) = B
HS
2 − 2π
∞
σ
e
βC 6 /R 6 − 1
R
2 dR ,
with β equal to 1/(k B T ). We cannot in general evaluate this expression further
exactly, but for βC 6 /R 6 1, we may expand the integrand and retain only the
first non-vanishing term to give
B
SM
2 (T ) =
2πσ 3
3
−
2πC 6
3k B T σ 3 .
If we compare this result with B VdW
2
(T ) given in Eqs. (2.8.22), we see that the Van
der Waals parameters a and b may be obtained as
a =
2πC 6
3σ 3 ,
b=
2πσ 3
3
,
so that a is thus associated (through C 6 ) with the long-range attractive dispersion
interaction and b corresponds to four times the volume v HS associated with a
hard sphere of diameter σ (often referred to as the ‘forbidden’ or ‘excluded’
volume). Given this association of the parameters a and b with attraction and the
volume occupied by an individual atom/molecule, a positive initial slope for a
compressibility factor isotherm thus indicates that molecular size effects dominate,
while a negative initial slope indicates that attractive forces dominate.
Clearly, the Van der Waals equation of state gives a qualitative explanation of real
gas behaviour through Z(P , T ). Note, however, that as the Van der Waals equation
of state is a function of two parameters a and b that depend upon the nature of the
specific gas being considered, we have lost the universality that we had with the
ideal gas equation of state.
Another way of viewing the Van der Waals equation is in the form of a cubic
equation in the volume V , namely,
V
3
− N
b +
k B T
P
V
2
+
aN 2
P
V −
aN 3 b
P
= 0 .
(2.8.23)
Figure 2.9 illustrates the behaviour of the Van der Waals pressure isotherms for a
and b parameters characterizing CO 2 . We see from this figure that for temperatures
T > 304.25 K, the isotherms are monotone decreasing functions of molar volume,
while for temperatures below 304.25 K, the Van der Waals equation for CO 2 gives
strongly non-monotonic behaviour over a fairly extensive range of molar volume V .