Physical Chemistry Third Edition

1134 27 Equilibrium Statistical Mechanics. III. Ensembles

to the flow of a compressible fluid in ordinary space and is governed by an equation

called the Liouville equation,^1 which is valid both for an equilibrium system and a

nonequilibrium system.

We consider only the equilibrium case so that the distribution of these points’ phase

space is time-independent. In quantum statistical mechanics, we had a discrete list of

possible states. In classical statistical mechanics, we have coordinates and momentum

components that can range continuously. We denote the probability distribution (prob-

ability density) for the ensemble byfand define the probability that the phase point

of a randomly selected system of the ensemble will lie in the 6N-dimensional volume

elementd^3 r 1 d^3 p 1 d^3 r 2 d^3 p 2 ···d^3 rNd^3 pNto be

Probabilityf(r 1 ,p 1 ,r 2 ,p 2 ,...rN,pN)d^3 r 1 d^3 p 1 d^3 r 2 d^3 p 2 ···d^3 rNd^3 pN

f(q,p)dqdp (27.4-1)

whered^3 r 1 stands fordx 1 dy 1 dz 1 ,d^3 p 1 stands fordpx 1 dpy 1 dpz 1 , and so forth, where

qstands for all of the coordinates,pstands for all of the momentum components, and

wheredqdpstands for the entire volume elementd^3 r 1 d^3 p 1 d^3 r 2 d^3 p 2 ···d^3 rNd^3 pN.

Classical mechanical formulas must agree with those obtained by taking the limit

of quantum mechanical formulas as masses and energies become large (thecorre-

spondence limit). This limit does not affect the formula representing the equilibrium

canonical probability density, so it must therefore be the same function of the energy as

that of quantum statistical mechanics. For a one-component monatomic gas or liquid

ofNmolecules without electronic excitation but with intermolecular forces, the clas-

sical energy (classical Hamiltonian functionH) is expressed in terms of momentum

components and coordinates:

HH(p,q)K +V 

1

2 m

∑N

i 1

(p^2 xi+p^2 yi+p^2 zi)+V(q) (27.4-2)

whereK is the kinetic energy andVis the potential energy of the system.

By analogy with Eq. (27.1-34), the normalized canonical probability density is

ff(p,q)f(H)

1

Zcl

e−H(p,q)/kBT (27.4-3)

whereZclis called theclassical canonical partition functionor the classicalphase

integral:

Zcl

∫∫

···

∫

e−H(p,q)/kBTd^3 r 1 d^3 r 2 d^3 r 3 ...d^3 rNd^3 p 1 d^3 p 2 ...d^3 pN



∫

e−H(p,q)/kBTdqdp (27.4-4)

If the system is in a container, the coordinate integrations in this formula range over all

values of the coordinates inside the container, and the momentum integrations range

from negative infinity to positive infinity. Since our theory is nonrelativistic, there is

no limitation on the speeds of the particles.

(^1) D. A. McQuarrie,Statistical Mechanics, Harper & Row, New York, 1976, p. 119ff.