van der Waals equation 1694.7 Perturbation theory and the van der Waals equation
Up to this point, the example systems we have considered (ideal gas, harmonic bead-
spring model, etc.) have been simple enough to permit an analytical treatment but lack
the complexity needed for truly interesting behavior. The theory of distributions pre-
sented in Section 4.6 is useful for characterizing structural and thermodynamic prop-
erties of real gases and liquids, and as Figs. 4.2, 4.3, and 4.5 suggest, these properties
reflect the richness that arises from even mildly complex interparticle interactions. In
particular, complex systems can exist in different phases (e.g. solid,liquid, or gas) and
can undergophase transitionsbetween these different states. By contrast, the ideal
gas, in which the molecular constituents do not interact, cannot exist as anything but
a gas.
In this section, we will consider a model system sufficiently complex toexhibit a
gas–liquid phase transition but simple enough to permit an approximate analytical
treatment. We will see how the phase transition manifests itself in the equation of
state, and we will introduce some of the basic concepts of critical phenomena (to be
discussed in greater detail in Chapter 16).
Before introducing our real-gas model, we first need to develop some important
machinery, specifically, a statistical mechanical perturbation theory for calculating
partition functions. To this end, consider a system whose potential energy can be
written in the form
U(r 1 ,...,rN) =U 0 (r 1 ,...,rN) +U 1 (r 1 ,...,rN). (4.7.1)Here,U 1 (r 1 ,...,rN) is assumed to be a small perturbation to the potentialU 0 (r 1 ,...,rN).
We define the configurational partition function for the unperturbed system, described
byU 0 (r 1 ,...,rN), as
Z(0)(N,V,T) =
∫
dr 1 ···drNe−βU^0 (r^1 ,...,rN). (4.7.2)Then, the total configurational partition function
Z(N,V,T) =
∫
dr 1 ···drNe−βU(r^1 ,...,rN) (4.7.3)can be expressed as
Z(N,V,T) =
∫
dr 1 ···drNe−βU^0 (r^1 ,...,rN)e−βU^1 (r^1 ,...,rN)Z(N,V,T) =
Z(0)(N,V,T)
Z(0)(N,V,T)
∫
dr 1 ···drNe−βU^0 (r^1 ,...,rN)e−βU^1 (r^1 ,...,rN)=Z(0)(N,V,T)〈e−βU^1 〉 0 , (4.7.4)where an average over the unperturbed ensemble has been introduced. In general, an
unperturbed average〈a〉 0 is defined to be