Problems 71number of microscopic states in the phase space accessible within a given ensemble.
Each ensemble has a particular partition function that depends on the macroscopic
observables used to define the ensemble. We will show in Chapters 3 to 6 that the
thermodynamic properties of a system are calculated from the various partial deriva-
tives of the partition function. Other equilibrium observables are computed according
to
A=〈a(x)〉=1
Z
∫
dxa(x)F(H(x)). (2.6.6)Note that the conditionf(x 0 )dx 0 =f(xt)dxtimplied by eqn. (2.5.11) guarantees that
the equilibrium average over the systems in the ensemble can be performed at any
point in time.
Eqns. (2.6.5) and (2.6.6) constitute the essence of equilibrium statistical mechanics.
As Richard Feynman remarks in his bookStatistical Mechanics: A Set of Lectures,
eqns. (2.6.5) and (2.6.6) embody “the summit of statistical mechanics, and the entire
subject is either the slide-down from this summit, as the [principles are] applied to var-
ious cases, or the climb-up where the fundamental [laws are] derived and the concepts
of thermal equilibrium... [are] clarified” (Feynman, 1998).^2 We shall, of course,
embark on both, and we will explore the methods by which equilibrium ensemble
distributions are generated and observables are computed for realistic applications.
2.7 Problems
2.1. Considernmoles of an ideal gas in a volumeVat pressurePand temperature
T. The equation of state isPV =nRTas given in eqn. (2.2.2). If the gas
containsNmolecules, so thatn=N/N 0 , whereN 0 is Avogadro’s number,
then the total number of microscopic states available to the gas can be shown
(see Section 3.5) to be Ω ∝VN(kT)^3 N/^2 , wherek= R/N 0 is known as
Boltzmann’s constant. The entropy of the gas is defined via Boltzmann’s
relation (see Chapter 3) asS=kln Ω. Note the total energy of an ideal gas
isE= 3nRT/2.
a. Suppose the gas expands or contracts from a volumeV 1 to a volumeV 2
at constant temperature. Calculate the work done on the system.
b. For the process in part a, calculate the change of entropy usingBoltz-
mann’s relation and using eqn. (2.2.19). Show that these two approaches
yield the same entropy change.
c. Next, suppose the temperature of the gas is changed fromT 1 toT 2 under
conditions of constant volume. Calculate the entropy change usingthe
two approaches in part a and show that they yield the same entropy
change.(^2) This quote is actually made in the context of quantum statistical mechanics (see Chapter 10);
however, the sentiment applies equally well to classical statistical mechanics.