Deviation of gases from the ideal state
We start from the case of a weakly non-ideal gas where the only pair interaction is important.
The simplest case is the monoatomic gas where the interaction potential depends only on the
distance between the atoms. Since
H (p,q) = ∑a
p
2
a
2m
+V
we can write the free energy as
F = Fid +Fi
,
Fi = −T log·
1
VN
Z
···Z
e
−βV(q) ∏a
dVa
¸
.
Here Fid is the free energy of the ideal gas. The factor V
−N in the argument of the logarithm is
just because for the ideal gas the integral over the volume is V
N. It is convenient to rewrite the
above expression in the form
F = Fid −T log·
1+
1
VN
Z
···Z ³
e
−βV(q) −1
´
dV1 ...dVN
¸
.
Now let us make use of the assumption that the gas is rarefied and only two atoms can collide at
the same time. Then the integrand is not small only if some two atoms are close together. Such
a pair can be extracted in N(N−1)/2 ways. Thus the integral can be expresses as
N(N−1)
2VN
Z
···Z ³
e
−βV12 −1
´
dV1 ...dVN ≈
N
2
2V2
Z ³
e
−βV12 −1
´
dV1 dV2 .
Here V12 is the pair interaction potential. Since log(1+x) ≈ x at x ¿ 1 and the gas density N/V
is assumed to be small,
Fi = −
T N2
2V2
Z ³
e
−βV12 −1
´
dV1 dV2 .
129
130 CHAPTER 10. NON-IDEAL GASES
Since the interaction depends only on the distance between the atoms we can integrate out the
center of mass and get V. In this way we are left with the expression
Fi =
T N2
V
B(T), B(T) = 1
2
Z ³
1−e
−βV12´
dV,
where dV stands for relative coordinates. Since P = −∂F /∂V we obtain the following correction
to the equation of state,
P =
NT
V
·
1+
N
V
B(T)
¸
.
The corrections to other thermodynamic potentials can be found using the principle of small
increments. For example, Gi = NBP.
A typical dependence of the interaction potential energy is shown in Fig. 10.1
0.8 1 1.2 1.4 1.6 1.8 2
V
3
2
1
0
-1 2r0
F