Physical Chemistry Third Edition

1150 27 Equilibrium Statistical Mechanics. III. Ensembles

A great deal of research has been carried out on the classical statistical mechanics

of liquids and dense gases, both for equilibrium and for nonequilibrium states. The

interested reader is invited to consult some of the statistical mechanics textbooks listed

at the end of this book, as well as such journals as theJournal of Chemical Physics,

thePhysical Review, and theJournal of Physical Chemistry.

PROBLEMS

Section 27.6: The Classical Statistical Mechanics
of Dense Gases and Liquids

27.20Using the parameters in Exercise 27.11 for the
square-well representation of the potential energy
function of argon, calculate the second virial coefficient
of argon at 50◦C and at 150◦C. Compare with the
experimental values,− 1. 12 × 10 −^5 m^3 mol−^1 at 50◦C
and 0. 11 × 10 −^5 m^3 mol−^1 at 150◦C.

27.21The effective hard-sphere diameter of helium atoms at

293 K is 2.17× 10 −^10 m. Use the formula of

Eq. (27.6-11) to calculate the second virial coefficient

of helium at 293 K. The experimental value of this

coefficient is nearly temperature-independent, equal

to 1. 17 × 10 −^5 m^3 mol−^1 at− 100 ◦C and equal to

1.10× 10 −^5 m^3 mol−^1 at 150◦C.

Summary of the Chapter

Statistical mechanics can be studied through use of an ensemble, which is an imaginary

collection of many replicas of the physical system. All of the systems in the ensemble

are in the same macroscopic state as the physical system, but members of the ensemble

occupy all possible microscopic states compatible with the macroscopic state. The

canonical ensemble represents a system with known values ofT,V, andn, and each

system of the ensemble is in a constant-temperature bath consisting of the other systems

of the ensemble. The probability distribution for the canonical ensemble is

pip(Ei)

1

Z

e−Ei/kBT

wherekBis Boltzmann’s constant,Tis the absolute temperature, andZis the canonical

partition function, which is a sum over system states:

Z

∑

i

e−Ei/kBT

In the case that the system is a dilute gas,

Z

zn

N!

wherezis the same molecular partition function as in Chapters 25 and 26. We obtained

formulas for all of the equilibrium thermodynamic functions of a general system, and

found that for a dilute gas, the formulas obtained were the same as those in Chapters

25 and 26.

Statistical mechanics can also be based on classical mechanics, and a brief introduc-

tion to this subject was included in the chapter, based on the canonical ensemble. Since

classical states are specified by values of coordinates and momentum components,

the probability distribution for classical statistical mechanics is a probability density