The objective is the average gas pressure P.
Consider the canonical ensemble for a volume of ideal gas. Each element of the
ensemble contains N gas molecules (here taken to be points having
mass m that do not interact with each other)
within in a volume V. V could be an 
cube
defined by the six planes ( x = 0, L); (y = 0, L); (z = 0, L); shape considerations
reappear in the homework. T enters via the canonical statistical weight 
, because 
.
The pressure is the force which the gas exerts on the walls of its container.
In order for this force to exist, there must exist a gas-wall potential U(s),
s being the distance from the wall to a particular atom. The force exerted
on the wall
by a single atom is 
, 
being the unit vector normal to the wall. U(s) is assumed to be short range,
i. e., U(s) = 0 for 
, 
being a non-infinitesimal
distance with 
. For typical forces, 
,
while 
m in conventional apparatus, so is not
a stringent requirement. The atoms cannot pass beyond the walls, so 
as 
, and similarly for the other walls.
The pressure P depends on the exact position of the gas atoms, so P varies
from element to element of the canonical ensemble. Similarly, in a real system
P fluctuates in time, changing as the gas molecules move within their
container.
Figure 6-1 sketches representative elements of the ensemble. Small circles are atoms; the dashed line represents the range of the atom-wall potential. The atoms are assumed to exert no forces on each other. While the atoms are moving, a correct representation of a single element of the ensemble shows atoms at a single instant in time, atoms at each instant having given, specified values of position and momentum. The force on the wall arises from the atom-wall potential, which is short range, so in Figure 6-1a two atoms are exerting a force on the wall. On the other hand, in the element of the ensemble represented by 6-1b only one atom is contacting the wall.
As seen in Figure 6-1, P is not the same in every element of the ensemble.
The ensemble average gives an average behavior, values of P in different
elements of the ensemble being spread around the average value. The variation
around the average is a calculable quantity. Pressure fluctuations
in macroscopic containers are not large, with respect to measurements made
using apparatus of the first kind, as discussed in Lecture 5.
Figure 6.1: Representative elements of the canonical ensemble.
Representative
elements of the canonical ensemble for the N-atom gas. Dashed line
represents the outermost range of the atom-wall potential.
For simplicity, limit the calculation to the force which the gas exerts on the wall at x = 0, neglecting interactions between gas atoms and other walls of the container. (Problem 6-5 eliminates this simplification) The total energy of the system is
where 
is the potential energy of atom i, and 
is the
distance between atom i and the wall.
The pressure is 
, <F> being the magnitude of the ensemble-average
force that the gas exerts on the container, the minus sign appearing because an
outwards force corresponds to a positive pressure. For this ensemble,
To evaluate eq. 6.3, the outer sum on i may be taken outside
the integrals. All terms of the sum are identical except for label, permitting
the replacement 
(identical terms) 
(one term).
The 
factors into terms depending only on 
or
only on 
, so the 6N-dimensional integral can be factored into a
product of simpler integrals. The momentum integrals all have the form 
, as evaluated in Lecture 4, and all cancel
between numerator and denominator. The integrals over the position coordinates
of each particle include terms 
, which also cancel
between numerator and denominator, leaving
The integrand in the numerator of eq. 6.4 is an exact
differential having value 
. In the denominator
On the rhs of eq. 6.5, in the second term U(x)
is always zero, so the second term is simply 
. Since , the 
covers only a very small region. Neglecting the
first term, and approximating 
for the second term,
eqs. 6.4 and 6.5 show
being the volume of the box. Eq. 6.6 is the ideal gas
law.
The detailed form of the repulsive wall-atom potential does not appear in
final result, eq. 6.6. So long as U(x) is short range, and
diverges as , the numerator of eq. 6.4 is
independent of the detailed form of U(x). However, the analysis of eq.
6.5 did incorporate an implicit assumption as to the form of
U(x). The rhs of eq. 6.5 was taken to be dominated by its
second term, i. e., we approximated
If U(x) for 
is negative, and sufficiently large in magnitude,
the above inequality fails. Indeed, for sufficiently negative U(x) one can
have 
and
Figure 6.2: Forms for the wall-atom potential energy U(x).
Figure 6.2 contrasts a potential appropriate for eq. 6.8 with a simple repulsive potential. If eq. 6.8 were true, one would have
and 
.
Eqs. 6.7 - 6.9, while presented as a mathematical
exercise, correspond to an important physical phenomenon. Consider the
hackneyed 22.4 liter container, filled at 273 K with 
moles of benzene
( 
) vapor. A freshman chemistry exercise predicts for this system
P of 
atmosphere. Suppose, however, that the container is
constructed not of stainless steel but of activated charcoal! Activated
charcoal strongly adsorbs most aromatic compounds, including benzene.
Adsorption is well-described by the potential of eq. 6.8 and
Fig. 6.2. Close to the wall, the molecule finds itself in a very deep
potential well which, if the kinetic energy of the molecule is not too large,
serves to trap the molecule against the container surface. Because adsorption
takes place, at equilibrium the benzene pressure inside an activated charcoal
container will be much less than expected from the ideal gas equation.
