1.3 What the Interaction Brings
25
The correspondence indicates that the virial expansion is a successive expansion in
terms of contributions of molecular clusters.
Even if the molecule is non-spherical. the intermolecular interaction can be averaged out spherically for its angular dependence upon discussing the properties of
gas, since the intermolecular distance is large, For example, the largest contribution
of the interaction can be estimated by
β 1 =
1
2
f 12 d r 2
,
(1.84)
where the · · · · means the orientational average.
1.3.2 Liquefaction
Along with the systematic expansion of the equation of state of non-ideal gas discussed in the previous section, there exists a famous equation. It was derived by
applying corrections for the presence of both attraction and repulsion from the equation of the ideal gas, pV = N k B T , by van der Waals [27]. The equation is written
as
p + a
N
V
2
(V − N b) = N k B T,
(1.85)
where b stands for a molecular volume and a for a constant characterizing a two-body
attraction.
8 Depending on temperature T , the relation between pressure and volume
changes, as shown in Fig. 1.6. At high temperatures, the dependence is monotonous
and resembles the equation of state of the ideal gas. On the other hand, at low enough
temperature, the dependence is transversally sigmoidal, giving three volumes at the
same value of pressure. Two outer states are thermodynamically healthy, while the
positive slope [(∂ p/∂V ) T > 0] of the middle point means an unstable state. It is
reasonable to assign the two outer states to the liquid (L) with a smaller volume
and the gas (G) with a larger volume, ignoring the middle state. The pressure of
the coexistence of two phases at a temperature is determined by the equality of
their chemical potentials (molar Gibbs energy), μ L = μ G . By (∂G/∂ p) T = V , the
condition is expressed as
G
L
V dp = 0.
(1.86)
The integration is along the equation of states between coexisting liquid and gas
specified by L and G, respectively. This condition is known as Maxwell’s rule.
Figure 1.6 indicates that the difference in volume decreases with increasing temperature and that there exists a unique temperature, above which the coexistence of
8 In terms of Sect. 1.3.1, a and b is related as B(T ) = (b − a/k B T ) and C(T ) = b 2 .
25
The correspondence indicates that the virial expansion is a successive expansion in
terms of contributions of molecular clusters.
Even if the molecule is non-spherical. the intermolecular interaction can be averaged out spherically for its angular dependence upon discussing the properties of
gas, since the intermolecular distance is large, For example, the largest contribution
of the interaction can be estimated by
β 1 =
1
2
f 12 d r 2
,
(1.84)
where the · · · · means the orientational average.
1.3.2 Liquefaction
Along with the systematic expansion of the equation of state of non-ideal gas discussed in the previous section, there exists a famous equation. It was derived by
applying corrections for the presence of both attraction and repulsion from the equation of the ideal gas, pV = N k B T , by van der Waals [27]. The equation is written
as
p + a
N
V
2
(V − N b) = N k B T,
(1.85)
where b stands for a molecular volume and a for a constant characterizing a two-body
attraction.
8 Depending on temperature T , the relation between pressure and volume
changes, as shown in Fig. 1.6. At high temperatures, the dependence is monotonous
and resembles the equation of state of the ideal gas. On the other hand, at low enough
temperature, the dependence is transversally sigmoidal, giving three volumes at the
same value of pressure. Two outer states are thermodynamically healthy, while the
positive slope [(∂ p/∂V ) T > 0] of the middle point means an unstable state. It is
reasonable to assign the two outer states to the liquid (L) with a smaller volume
and the gas (G) with a larger volume, ignoring the middle state. The pressure of
the coexistence of two phases at a temperature is determined by the equality of
their chemical potentials (molar Gibbs energy), μ L = μ G . By (∂G/∂ p) T = V , the
condition is expressed as
G
L
V dp = 0.
(1.86)
The integration is along the equation of states between coexisting liquid and gas
specified by L and G, respectively. This condition is known as Maxwell’s rule.
Figure 1.6 indicates that the difference in volume decreases with increasing temperature and that there exists a unique temperature, above which the coexistence of
8 In terms of Sect. 1.3.1, a and b is related as B(T ) = (b − a/k B T ) and C(T ) = b 2 .