Physical Chemistry Third Edition

27.1 The Canonical Ensemble 1123

state 2, and so on. Since the systems are macroscopic in size, they are distinguishable

and there is no exclusion principle.

Let the fraction of systems in system microstatekbe denoted by

pk

nk

n

(27.1-1)

wherenis the total number of systems in the ensemble. Since by the second postulate

every state corresponding to the same energy is assumed to be equally represented in the

ensemble, we assume thatpkdepends only on the energy eigenvalueEkcorresponding

to the statek:

pkp(Ek) (27.1-2)

We also make the important assumption thatpis the same function ofEfor all different

kinds of systems. In order to determine what kind of a mathematical functionpis,

we consider a system that is made up of two parts, which we call subsystem I and

subsystem II. Each part is closed and at constant volume, but the two parts are in

thermal contact with each other (as well as with the rest of the ensemble) so that both

are at the same temperature. We assume that the two subsystems occupy mechanical

states independently of each other and that the system energy is a sum of the subsystem

energies:

EkEI,k+EII,k (27.1-3)

The subscript I,krepresents the state of subsystem I corresponding to system statek,

and II,krepresents the corresponding state of subsystem II.

A canonical ensemble that represents the system contains many replicas of the

system, each made up of two subsystems. The fraction of the replicas of subsystem I

in the ensemble that occupy state I,kis

pI,kp(EI,k) (27.1-4a)

and the fraction of the replicas of subsystem II in state II,kis

pII,kp(EII,k) (27.1-4b)

Furthermore, the probability that the combined system is in statekis

pkp(Ek) (27.1-4c)

where by our hypothesisprepresents the same mathematical function of its argument

for the combined system and for each subsystem.

Probability theory asserts that the joint probability of two independent occurrences

is equal to the product of the probabilities of the two occurrences. Therefore

p(Ek)p(EI,k+EII,k)p(EI,k)p(EII,k) (27.1-5)

The exponential function is the only function that has this property. We writepin the

form

p(E)Ae−βE (27.1-6)

whereAandβare parameters that do not depend onE. To prevent the probability from

increasing indefinitely for large energy, we must require thatβ>0. Equation (27.1-6)

is not restricted to dilute gases or any other particular kind of system, nor is it restricted

to a system made up of two separate subsystems.