372
7 Classical Statistical Mechanics
we see that the second and third virial coefficients B 2 (T ) and B 3 (T ) are given in
terms of configuration integrals as
B 2 (T ) = −
1
2V
Z 2C − Z
2
1C
(7.2.22)
and
B 3 (T ) = −
1
3!V
V
Z 3C − 3Z 2C Z 1C + 2Z
3
1C
− 3
Z 2C − Z
2
1C
2
,
(7.2.23)
respectively. A word of caution: should the binary intermolecular interaction energy
die off with increasing intermolecular separation r 12 more slowly than r
−3
12 , a virial
expansion does not even exist because, as we shall see shortly, the second virial
coefficient B 2 (T ) diverges for such interactions. This means, for example, that in
the case of a plasma in which the charged species interact via the Coulomb law (i.e.,
as r
−1
12 ) there can be no virial expansion representation for the pressure.
7.2.1 Second Virial Coefficient for Monatomic Gases
We return to Eq. (7.2.1) for the partition function Z N . For N = 1, we retrieve our
previous result for a monatomic gas, namely
Z ≡ Z 1 =
2πmk B T
h 2
3
2
V =
V
3 ,
(7.2.24)
with the thermal de Broglie wavelength. Note that we have utilized the result
Z 1C = V for the single-molecule case. Moreover, for N > 1 we may also write
Z N (V , T ) as
Z N (V , T ) =
1
N!
Z NC (V , T )
3N
.
(7.2.25)
If we also replace −3 in this result by Z 1 /V from Eq. (7.2.24), we obtain
Z N (V , T ) =
1
N!
Z 1
V
N
Z NC ,
(7.2.26)
which is nothing other than a rearranged version of Eq. (7.2.10).
We wish now to relate the second virial coefficient appearing in Eq. (7.2.8) to
the intermolecular interaction. We shall consider only a monatomic gas in order to
avoid complications due to angle-dependencies in the intermolecular interaction.
7 Classical Statistical Mechanics
we see that the second and third virial coefficients B 2 (T ) and B 3 (T ) are given in
terms of configuration integrals as
B 2 (T ) = −
1
2V
Z 2C − Z
2
1C
(7.2.22)
and
B 3 (T ) = −
1
3!V
V
Z 3C − 3Z 2C Z 1C + 2Z
3
1C
− 3
Z 2C − Z
2
1C
2
,
(7.2.23)
respectively. A word of caution: should the binary intermolecular interaction energy
die off with increasing intermolecular separation r 12 more slowly than r
−3
12 , a virial
expansion does not even exist because, as we shall see shortly, the second virial
coefficient B 2 (T ) diverges for such interactions. This means, for example, that in
the case of a plasma in which the charged species interact via the Coulomb law (i.e.,
as r
−1
12 ) there can be no virial expansion representation for the pressure.
7.2.1 Second Virial Coefficient for Monatomic Gases
We return to Eq. (7.2.1) for the partition function Z N . For N = 1, we retrieve our
previous result for a monatomic gas, namely
Z ≡ Z 1 =
2πmk B T
h 2
3
2
V =
V
3 ,
(7.2.24)
with the thermal de Broglie wavelength. Note that we have utilized the result
Z 1C = V for the single-molecule case. Moreover, for N > 1 we may also write
Z N (V , T ) as
Z N (V , T ) =
1
N!
Z NC (V , T )
3N
.
(7.2.25)
If we also replace −3 in this result by Z 1 /V from Eq. (7.2.24), we obtain
Z N (V , T ) =
1
N!
Z 1
V
N
Z NC ,
(7.2.26)
which is nothing other than a rearranged version of Eq. (7.2.10).
We wish now to relate the second virial coefficient appearing in Eq. (7.2.8) to
the intermolecular interaction. We shall consider only a monatomic gas in order to
avoid complications due to angle-dependencies in the intermolecular interaction.