1549380323-Statistical Mechanics Theory and Molecular Simulation

168 Canonical ensemble

P

kT

=ρ−

2 πρ^2

3 kT

∫∞

0

dr r^3 u′(r)g(r), (4.6.69)

which is a simple expression for the pressure in terms of the derivative of the pair
potential form and the radial distribution function.
Eqn. (4.6.69) is in the form of an equation of state and is exact for pair-wise
potentials. The dependence of the second term onρandTis more complicated than
it appears becauseg(r) depends on bothρandT:g(r) =g(r;ρ,T). At low density,
however, where the thermodynamic properties of a system shouldbe dominated by
those of an ideal gas, the second term, which has a leadingρ^2 dependence, should be
small. This fact suggests that the low density limit can be accurately approximated
by expanding theρdependence ofg(r) in a power series inρ:

g(r;ρ,T) =

∑∞

j=0

ρjgj(r;T). (4.6.70)

Substituting eqn. (4.6.70) into eqn. (4.6.69) gives the equation of state in the form

P

kT

=ρ+

∑∞

j=0

Bj+2(T)ρj+2. (4.6.71)

Eqn. (4.6.71) is known as thevirial equation of state. The coefficientsBj+2(T) are
given by

Bj+2(T) =−

2 π

3 kT

∫∞

0

dr r^3 u′(r)gj(r;T) (4.6.72)

and are known as thevirial coefficients. Eqn. (4.6.71) is still exact. However, in the
low density limit, the expansion can be truncated after the first fewterms. If we stop
after the second-order term, for example, then the equation ofstate reads

P

kT

≈ρ+B 2 (T)ρ^2 (4.6.73)

with

B 2 (T)≈−

2 π

3 kT

∫∞

0

dr r^3 u′(r)g(r) (4.6.74)

sinceg 0 (r;T)≈g(r). Thus, thesecond virial coefficientB 2 (T) gives the leading order
deviation from ideal gas behavior. In this limit, the radial distributionfunction, itself,
can be approximated by (see Problem 4.5)

g(r)≈e−βu(r), (4.6.75)

and the second virial coefficient is given approximately by

B 2 (T)≈− 2 π

∫∞

0

dr r^2

(

e−βu(r)− 1

)

. (4.6.76)

These concepts will be important for our development of perturbation theory and the
derivation of the van der Waals equation of state, to be treated in the next section.