Statistical mechanical perturbation theory
The methods we have treated up to now do not give good results much below the critical point. That is, they are not applicable to dense fluids. One means of obtaining equations of state for real liquids is to use statistical mechanical perturbation theory developed by Zwanzig. The assumption is that we have good model such a hard sphere model for a fluid. We take this as our reference system and then improve upon it using a perturbation. The potential energy can be separated into two parts
UN = UN(0) + UN(1)
where UN(0) is the potential of the unperturbed or reference system and UN(1) is the perturbation.
The configuration integral for this potential is
We multiply and divide by
to obtain
.
The second factor is the average of exp(-
bUN(1)) over the unperturbed reference system. This can be rewritten as
If UN(1) is small enough then we can expand it
We can express the free energy as A = -kT lnQ where Q = ZN/N!
L3N.
Thus,
where A0 is the free energy of the reference system and A(1) is the perturbation free energy.
We can express A(1) as power series in
b.
where the
wn are to be determined in terms of the UN(1).We have that
and
The first few
wn are:
The free energy is
The first term in the expansion,
w1 = áUN(1)ñ0 simplifies when UN(1) can be written as a sum of pair potentials
.
Using techniques developed for the radial distribution function we can write
where the joint pair probability and the radial distribution function are those of the reference system. This derivation has ignored higher order terms. It is extendable to include those if need be.
The technique shown above can be used to derive the van der Waal's equation of state. In this derivation the reference state will be that of a hard-sphere fluid. We will assume that the potential is pair-wise additive and can be expressed as u(r) = uHS(r) + u(1)(r). From the above considerations we have
An approximate hard sphere radial distribution function is a step function
This form is correct for hard spheres only in the limit of low density. Nonetheless, taking this as our reference system we have
In terms of configuration integral for perturbation theory we now have
The final approximation of van der Waal's theory is to assume that the hard sphere configuration integral ZN(0) is of the form VeffN, where the effective volume is determined by assuming that the volume available to a molecule in the fluid is effectively reduced by the volume taken up by the other hard spheres. Each hard sphere takes up 4
ps3/3, however we must divide this quantity by two since it arises from a pair of molecules interacting. Therefore, Veff = V - N2ps3/3 and we can write
We have seen that the partition function
can be used to derive the van der Waal's equation of state. Here we have shown how this partition function can be derived using statistical mechanical perturbation theory. While the van der Waal's equation is qualitatively correct we have also seen that it is not a very good quantitative representation of the equation of state of a liquid. The value of the perturbation theory shown here is that it is obvious how we would improve upon the equation of state we have derived.