Chapter 8
Classical Gases and Liquids
c
c
2010 by Harvey Gould and Jan Tobochnik
8 June 2010
We discuss approximation techniques for interacting classical particle systems such as dense
gases and liquids.
8.1 Introduction
Because there are only a few problems in statistical mechanics that can be solved exactly, we need
to Fnd approximate solutions. We introduce several perturbation methods that are applicable
when there is a small expansion parameter. Our discussion of interacting classical particle systems
will involve some of the same considerations and di±culties that are encountered in quantum Feld
theory (no knowledge of the latter is assumed). ²or example, we will introduce diagrams that
are analogous to ²eynman diagrams and Fnd divergences analogous to those found in quantum
electrodynamics. We also discuss the spatial correlations between particles due to their interactions
and the use of hard spheres as a reference system for understanding the properties of dense ³uids.
8.2 Density Expansion
Consider a gas of
N
identical particles each of mass
m
at density
ρ
=
N/V
and temperature
T
. We
will assume that the total potential energy
U
is a sum of twobody interactions
u
ij
=
u
(

r
i
−
r
j

),
and write
U
as
U
=
N
s
i<j
u
ij
.
(8.1)
The exact form of
u
(
r
) for electrically neutral molecules and atoms must be constructed by a
Frst principles quantum mechanical calculation. Such a calculation is very di±cult, and for many
purposes it is su±cient to choose a simple phenomenological form for
u
(
r
). The most important
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388
u
LJ
(r)
r
2.0
σ
3.0
σ
1.0
σ
ε
Figure 8.1: Plot of the LennardJones potential
u
LJ
(
r
), where
r
is the distance between the parti
cles. The potential is characterized by a length
σ
and an energy
ǫ
.
features of
u
(
r
) are a strong repulsion for small
r
and a weak attraction at large
r
. A common
phenomenological form of
u
(
r
) is the
LennardJones
or 612 potential shown in Figure 8.1:
u
LJ
(
r
) = 4
ǫ
bp
σ
r
P
12
−
p
σ
r
P
6
B
.
(8.2)
The values of
σ
and
ǫ
for argon are
σ
= 3
.
4
×
10
−
10
m and
ǫ
= 1
.
65
×
10
−
21
J.
The attractive 1
/r
6
contribution to the LennardJones potential is due to the induced dipole
dipole interaction of two neutral atoms.
1
The resultant attractive interaction is called the
van
der Waals
potential. The rapidly increasing repulsive interaction as the separation between atoms
is decreased for small
r
is a consequence of the Pauli exclusion principle. The 1
/r
12
form of the
repulsive potential in (8.2) is chosen only for convenience.
The existence of many calculations and simulation results for the LennardJones potential
encourages us to use it even though there are more accurate forms of the interparticle potential
for modeling the interactions in real ±uids, proteins, and other complex molecules.
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 Summer '10
 hcverma
 Energy, Statistical Mechanics, FC, CLASSICAL GASES

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