374
7 Classical Statistical Mechanics
Had we carried out a quantum mechanical derivation of B 2 (T ) we would have
found that the classical expression (7.2.31) for B 2 (T ) gives by far the dominant
contribution to the second virial coefficient at most temperatures. For light atoms
(molecules), especially at low temperatures, there will be quantum corrections to
the classical expression. A more complete expression for B 2 (T ) for atoms is [3]
B 2 (T ) = −2π
∞
0
e
−βV (r 12 )
− 1
r
2
12 dr 12
+
h 2
24πm(k B T ) 3
∞
0
e
−βV (r 12 )
dV
dr 12
2
r
2
12 dr 12 + O
h
3
.
(7.2.32)
The first term in this expression is the classical contribution and the second term is
referred to as the first semiclassical (translational) quantum correction. The second
term contributes only a small fraction of the total value for B 2 (T ). Second and
higher order correction terms (representing a series expansion in powers of h 2 ) also
exist, but generally contribute significantly less than the two lowest first-order terms.
For 4 He, for example, the first translational quantum correction term contributes
4.1% to B 2 (T ) at 256 K and 17.0% at 83.5 K, while for H 2 , the corresponding
percentages are 23.0% at 183 K and 3.0% at 592 K. For a heavier atom, such as
Ne, we find 6% at 35 K, 9.7% at 95 K, and 0.6% at 392 K. Full quantum mechanical
calculations of the second virial coefficient should be carried out for the helium
isotopes, the hydrogen isotopologues and, to a lesser extent, the neon isotopes, as the
semi-classical expressions are asymptotic expressions, and can fail for sufficiently
low temperatures. 1 Figure 7.1 displays results [4] obtained from fully quantum
mechanical calculations of the second virial coefficient for 4 He. The quantummechanical ideal gas contribution for helium is also shown in the inset figure: as
may be seen, it has essentially vanished for temperatures in excess of 60 K. The
temperature dependence shown in this figure is typical of the behaviour of second
virial coefficients.
The extension of these concepts to mixtures of gases is fairly straightforward.
For a binary mixture of two chemical species designated by A and B, for example,
the second virial coefficient for the mixture can be written as
B
mix
2 (T ) = x
2
A B 2,AA + 2x A x B B 2,AB + x
2
B B 2,BB ,
(7.2.33)
with x A , x B the mole fractions of A and B in the binary mixture. The second virial
coefficients B 2,AA and B 2,BB are those for the pure gaseous A and B components,
respectively, each of which can be associated, through Eq. (7.2.31), with the corresponding intermolecular interaction. The second virial coefficient B 2,AB , is similar
to that for the pure species A and B and, as it depends solely upon the intermolecular
1 For H 2 and He at temperatures below 75 K, 45 K, respectively, an expansion in powers of h 2 no
longer converges, and it is actually necessary to carry out full quantum mechanical calculations of
B 2 (T ).
7 Classical Statistical Mechanics
Had we carried out a quantum mechanical derivation of B 2 (T ) we would have
found that the classical expression (7.2.31) for B 2 (T ) gives by far the dominant
contribution to the second virial coefficient at most temperatures. For light atoms
(molecules), especially at low temperatures, there will be quantum corrections to
the classical expression. A more complete expression for B 2 (T ) for atoms is [3]
B 2 (T ) = −2π
∞
0
e
−βV (r 12 )
− 1
r
2
12 dr 12
+
h 2
24πm(k B T ) 3
∞
0
e
−βV (r 12 )
dV
dr 12
2
r
2
12 dr 12 + O
h
3
.
(7.2.32)
The first term in this expression is the classical contribution and the second term is
referred to as the first semiclassical (translational) quantum correction. The second
term contributes only a small fraction of the total value for B 2 (T ). Second and
higher order correction terms (representing a series expansion in powers of h 2 ) also
exist, but generally contribute significantly less than the two lowest first-order terms.
For 4 He, for example, the first translational quantum correction term contributes
4.1% to B 2 (T ) at 256 K and 17.0% at 83.5 K, while for H 2 , the corresponding
percentages are 23.0% at 183 K and 3.0% at 592 K. For a heavier atom, such as
Ne, we find 6% at 35 K, 9.7% at 95 K, and 0.6% at 392 K. Full quantum mechanical
calculations of the second virial coefficient should be carried out for the helium
isotopes, the hydrogen isotopologues and, to a lesser extent, the neon isotopes, as the
semi-classical expressions are asymptotic expressions, and can fail for sufficiently
low temperatures. 1 Figure 7.1 displays results [4] obtained from fully quantum
mechanical calculations of the second virial coefficient for 4 He. The quantummechanical ideal gas contribution for helium is also shown in the inset figure: as
may be seen, it has essentially vanished for temperatures in excess of 60 K. The
temperature dependence shown in this figure is typical of the behaviour of second
virial coefficients.
The extension of these concepts to mixtures of gases is fairly straightforward.
For a binary mixture of two chemical species designated by A and B, for example,
the second virial coefficient for the mixture can be written as
B
mix
2 (T ) = x
2
A B 2,AA + 2x A x B B 2,AB + x
2
B B 2,BB ,
(7.2.33)
with x A , x B the mole fractions of A and B in the binary mixture. The second virial
coefficients B 2,AA and B 2,BB are those for the pure gaseous A and B components,
respectively, each of which can be associated, through Eq. (7.2.31), with the corresponding intermolecular interaction. The second virial coefficient B 2,AB , is similar
to that for the pure species A and B and, as it depends solely upon the intermolecular
1 For H 2 and He at temperatures below 75 K, 45 K, respectively, an expansion in powers of h 2 no
longer converges, and it is actually necessary to carry out full quantum mechanical calculations of
B 2 (T ).