Physical Chemistry Third Edition

1122 27 Equilibrium Statistical Mechanics. III. Ensembles

27.1 The Canonical Ensemble

In Chapters 25 and 26 we applied statistical mechanics to dilute gases, based on

the probability distribution for states of molecules in a dilute gas with fixed energy,

fixed volume, and fixed number of molecules. This chapter presents an alternative

approach to statistical mechanics, which is in principle applicable to any kind of a

system. It is based on the concept of anensemble, which is an imaginary collection

of very many replicas of the real system. Each replica occupies the same macrostate

as the real system, but different members of the ensemble occupy different system

microstates.

If the equilibrium macrostate of a closed system is specified by fixed values of the

thermodynamic energyU, the volumeV, and number of moleculesN,amicrocanonical

ensembleis used to represent it. The systems of this ensemble have the same values of

E(withEequal toU),N, andVbut occupy different microstates. Our discussion in

Chapters 25 and 26 is equivalent to using the microcanonical ensemble to represent a

dilute gas.

Thecanonical ensemblerepresents a closed system at equilibrium with a macrostate

specified by values ofT,V, andN. The systems of the canonical ensemble are placed

in thermal contact with each other so that each system of the ensemble is in a constant-

temperature bath consisting of the other systems of the ensemble, and all have the same

temperature. Figure 27.1 schematically depicts a portion of such an ensemble.

One system with given

values of N,V, and T

Figure 27.1 A Canonical Ensemble.

Thegrandcanonicalensemblerepresents an open system with a macrostate specified

by values ofT,V, andμ(the chemical potential) for each substance present. Each

system of the ensemble is open to the other systems of the ensemble and can exchange

matter and energy with the other systems. Discussions of the grand canonical ensemble

can be found in statistical mechanics textbooks, but we will not discuss it.

The Postulates of Statistical Mechanics

The two postulates of equilibrium statistical mechanics introduced in Chapter 25 can

now be restated:

Postulate 1. Observed values of thermodynamic variables in the actual system can be

represented by ensemble averages of the corresponding mechanical variables.

Postulate 2. In any ensemble, each system microstate corresponding to the same values

ofE,V, andNis occupied by an equal number of systems.

The Canonical Probability Distribution

Imagine that the Schrödinger equation of the real system has been solved, and that

we have a list of its very many system energy eigenfunctions and energy eigenvalues,

each corresponding to the same volume and same number of molecules. The fact that

we cannot actually carry out this solution is unimportant until we attempt to carry out

calculations. The energy eigenfunctions are numberedΨ 1 ,Ψ 2 ,Ψ 3 , and so on, with

energy eigenvaluesE 1 ,E 2 ,E 3 , and so on. Although there are infinitely many such

states, we imagine that the ensemble contains so many systems that each of these

system states is occupied by some members of the ensemble. Letn 1 be the number of

systems in the ensemble that are in system state 1,n 2 be the number of systems in system