Physical Chemistry Third Edition

1082 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

26.1 The Statistical Thermodynamics of a

Dilute Gas

In the previous chapter we obtained the probability distribution for molecular states in a

dilute gas and obtained a formula for the thermodynamic energy of a dilute gas in terms

of the partition function. Statistical mechanics will not be very useful until we obtain

formulas for the other thermodynamic functions, which is the topic of this chapter.

The Entropy of a Dilute Gas

If a macroscopic system such as a gas is at thermodynamic equilibrium, its entropy has

a well-defined constant value. However, the molecules are moving and occupy many

different molecular states without changing the macroscopic state or the value of the

entropy. A single macroscopic state and a single value of the entropy must correspond

to many microscopic states of the system. We define thethermodynamic probability

Ω, to be the number of microscopic states of the system that might be occupied for a

given macroscopic state of the system.

Boltzmann defined astatistical entropycorresponding to a macroscopic state with

given values of the energy, volume, and amount of substance:

SstkBln(Ω)+S 0 (definition of statistical entropy) (26.1-1)

wherekBis Boltzmann’s constant, equal to 1. 38066 × 10 −^23 JK−^1. The thermody-

namic entropy can have an arbitrary constant added to its value, and the same is true

of the statistical entropy. The constantS 0 can be set equal to zero, which corresponds

to setting the statistical entropy equal to zero for a system that is known to be in a

single microstate. We will see later in this chapter that this definition of the statistical

entropy leads to values of the entropy of dilute gases that agree with values of the

thermodynamic entropy.

We have already encountered the thermodynamic probability in the discussion of our

model system of four harmonic oscillators. For the macroscopic state that we studied,

Ωwas equal to 35. For a typical macroscopic system,Ωwill be much larger.

Exercise 26.1

Calculate the statistical entropy of our model system of four harmonic oscillators for the

macrostate discussed in Section 25.1.

EXAMPLE26.1

Estimate the value ofΩfor 1.000 mol of helium at 298.15 K and 1.000 atm, using the value

of the third-law entropy, 126 J K−^1.

Solution

ΩeS/kBexp

[

126 J K−^1

1. 38 × 10 −^23 JK−^1

]

e^9.^13 ×^10

24

 103.^96 ×^10

24

Ten raised to an exponent larger than Avogadro’s constant is averylarge number.