1549380323-Statistical Mechanics Theory and Molecular Simulation

178 Canonical ensemble

For example, a simple molecular system that can exist as a solid, liquid, or gas has a
critical point on the gas–liquid coexistence curve. Similarly, a ferromagnetic material
has a critical point on the coexistence curve between its two ordered phases. As a
critical point is approached, certain thermodynamic properties are observed to diverge
according to power laws that are characterized by the critical exponents. These will
be explored in more detail in Chapter 16. What is particularly fascinating about
these exponents is that they are the same across large classes ofsystems that are
otherwise very different physically. These classes are known asuniversality classes, and
their existence suggests that the local detailed interactions among particles become
swamped by long-range cooperative effects that dominate the behavior of a system at
its critical point.
Other critical exponents are defined as follows: The heat capacityCVatV=Vc
is observed to diverge asTapproachesTc, according to

CV∼|T−Tc|−α. (4.7.44)

Near the critical point, the equation of state is observed to behave as

P−Pc∼|ρ−ρc|δsign(ρ−ρc). (4.7.45)

Finally, the shape of the gas–liquid coexistence curve (in theρ–Tplane) near the
critical point forT < Tcbehaves as

ρL−ρG∼(Tc−T)β, (4.7.46)

whereρLandρGare the liquid and gas densities, respectively. The four exponents,α,
β,γ,δcomprise the four principal critical exponents.
In order to calculateα, we first compute the energy according to

E=−

∂

∂β

lnQ(N,V,T) =

∂

∂β

[βA(N,V,T)]

=−

∂

∂β

{

ln

[

(V−Nb)N

N!λ^3 N

]

+

βaN^2

V

}

. (4.7.47)

Computing the derivative, we obtainE= 3NkT/ 2 −aN^2 /V, so that the heat capacity
CV= (∂E/∂T) is independent ofTor simplyCV ∼ |T−Tc|^0. From this, it follows
that the van der Waals theory predictsα= 0. The value ofδcan be easily deduced as
follows: In the van der Waals theory, the equation of state has theanalytical form in
eqn. (4.7.35). Thus, we may expandPin a power series inρabout the critical values
according to

P=Pc+

∂P

∂ρ

∣

∣

∣

∣

ρc,Tc

(ρ−ρc)+

1

2

∂^2 P

∂ρ^2

∣

∣

∣

∣

ρc,Tc

(ρ−ρc)^2 +

1

6

∂^3 P

∂ρ^3

∣

∣

∣

∣

ρc,Tc

(ρ−ρc)^3 +···.(4.7.48)

The density derivatives of the pressure can be computed from thevolume derivatives
as