Physical Chemistry Third Edition

27.5 Thermodynamic Functions in the Classical Canonical Ensemble 1141

plane polar coordinates. The coordinate axis in the

phase space for this coordinate extends only from

0to2πin the coordinate direction, since it represents

the angleφ. Draw an accurate trajectory in this phase

space for a particle of mass 1.000 kg orbiting in a

plane at a fixed distance of 1.000 m from a fixed

center, such that it makes one revolution every 1.000

second and has a constant speed.

b.Draw an accurate trajectory in the 2-dimensional

phase space of the coordinateρ, representing the

distance from the fixed origin, for the same motion.

27.5 Thermodynamic Functions in the Classical

Canonical Ensemble

We now obtain equations for the thermodynamic functions of a system represented by

a classical canonical ensemble. We first write general equations and then specialize

them for dilute gases.

The Energy

The ensemble average energy of a general system is

U〈E〉

1

Zcl

∫

H(p,q)e−H(p,q)/kBTdpdq (27.5-1)

wherepandqare abbreviations for the coordinates and momentum components

of all particles in the system. We use a mathematical trick similar to the one used

in Eq. (27.1-20), writing the integrand as a derivative and exchanging the order of

integration and differentiating:

U

1

Zcl

kBT^2

∫(

∂

∂T

[e−H(p,q)/kBT]

)

dpdqkBT^2

(

∂ln(Zcl)

∂T

)

V

(27.5-2)

The volumeVis held constant in the differentiation, since the potential energy depends

on the volume of the system. Equation (27.5-1) is the same as Eq. (27.2-1) except for

replacement of the quantum-mechanical canonical partition function by the classical

phase integral. There is a problem with this equation. The classical canonical partition

is not dimensionless, which is required for the argument of a logarithm. We discuss

this problem later.

The Entropy

For a classical system, the definition of the statistical entropy in Eq. (26.1-1) cannot be

used because we cannot count states in the same way as with a quantum system. The

classical definition of the statistical entropy is analogous to that in Problem 27.6.

Sst−kB

∫

f(p,q)ln[f(p,q)]dpdq (27.5-3)

where the integration is over the values of all coordinates and momenta. Use of

Eq. (27.4-3) gives

S−kB

∫

f(p,q)

[

−

E

kBT

−ln(Zcl)

]

dpdq

U

T

+kBln(Zcl) (27.5-4)