In the canonical ensemble, the average energy is given by
Therefore,
Since
Thus,
In order to compute the average energy, therefore, one needs to be able to
compute the average of the potential 
. In general, this
is a nontrivial task, however, let us work out the average for the
case of a pairwise-additive potential of the form
i.e., U is a sum of terms that depend only the distance between
two particles at a time. This form turns out to be an excellent approximation
in many cases. U therefore contains N(N-1) total terms, and
becomes
The second line follows from the fact that all terms in the first line are the exact same integral, just with the labels changed. Thus,
Once again, we change variables to 
and 
.
Thus, we find that
Therefore, the average energy becomes
Thus, we have an expression for E in terms of a simple integral over the pair potential form and the radial distribution function. It also makes explicit the deviation from ``ideal gas'' behavior, where E=3NkT/2.
By a similar procedure, we can develop an equation for the pressure P in terms of g(r). Recall that the pressure is given by
The volume dependence can be made explicit by changing variables
of integration in 
to
Using these variables, becomes
Carrying out the volume derivative gives
Thus,
Let us consider, once again, a pair potential. We showed in an earlier lecture that
where 
is the force on particle i due to particle j.
By interchaning the i and j summations in the above expression, we
obtain
However, by Newton's third law, the force on particle i due to particle j is equal and opposite to the force on particle j due to particle i:
Thus,
The ensemble average of this quantity is
As before, all integrals are exactly the same, so that
Then, for a pair potential, we have
where u'(r) = du/dr, and 
. Substituting this into the
ensemble average gives
As in the case of the average energy, we change variables at this point to
and . This gives
Therefore, the pressure becomes
which again gives a simple expression for the pressure in terms only of the
derivative of the pair potential form and the radial distribution function.
It also shows explicitly how the equation of state differs from the
that of the ideal gas 
.
From the definition of g(r) it can be seen that it depends on the
density 
and temperature T: 
. Note, however,
that the equation of state, derived above, has the general form
which looks like the first few terms in an expansion about
ideal gas behavior. This suggests that it may be possible to develop
a general expansion in all powers of the density about
ideal gas behavior. Consider representing 
as such a power series:
Substituting this into the equation of state derived above, we obtain
This is known as the virial equation of state, and the coefficients
are given by
are known as the virial coefficients. The coefficient 
is
of particular interest, as it gives the leading order deviation from
ideal gas behavior. It is known as the second virial coefficient.
In the low density limit, 
and
is directly related to the radial distribution function.