Physical Chemistry Third Edition

27.3 The Dilute Gas in the Canonical Ensemble 1131

It is remarkable that the thermodynamic functions for a dilute gas are given by the

same formulas for a canonical ensemble, in which the replica systems have various

energies, and for a microcanonical ensemble, in which all of the replica systems have

the same energy. In the canonical ensemble the probability of a system state of energy

Eis proportional toe−E/kBT. At ordinary temperatures,kBTis a very small quantity

of energy compared withE, the energy of an entire macroscopic system. Therefore

except for the very lowest-energy states,e−E/kBTis a very small and very rapidly

decreasing function ofE. The probability of a system energy level is also proportional

to the degeneracy of the level:

pE∝Ω(E)e−E/kBT (27.3-3)

whereΩ(E) is the number of system microstates corresponding to energyE. In Section

26.2 we estimated a value ofΩand found it to be a very large number (see Exam-

ple 26.1). Furthermore,Ωincreases very rapidly as a function ofE. Figure 27.2a

schematically shows the probability distribution of states,e−E/kBT, and Figure 27.2b

schematically showsΩ(E). Unfortunately, the extremely small size ofe−E/kBTand the

extremely large size ofΩ(E) cannot be shown accurately. The large factor is rising so

(c) (d)

(a) (b)

E

E E

e

E/k

TB

V

(E

)

Probability

P

5 VE

(E

)e



E/k

TB

Figure 27.2 The Probability of System States and Levels.(a) Probability distribution

of system states in the canonical ensemble as a function of energy (schematic). (b) The

degeneracy of system energy levels as a function of energy (schematic). (c) The canonical

probability of system energy levels as a function of energy (schematic). (d) The micro-

canonical probability of system energy levels as a function of energy (schematic).