Gas in Harmonic Well

A classical system of N distinguishable noninteracting particles of mass m is placed in a three-dimensional harmonic well:

a) Find the partition function and the Helmholtz free energy.

b) Regarding V as an external parameter, find the thermodynamic force  conjugate to this parameter, exerted by the system; find the equation of state and compare it to that of a gas in a container with rigid walls.

c) Find the entropy, internal energy, and total heat capacity at constant volume.

SOLUTION

a) The partition function is given by a standard integral (where the molecules are indistinguishable):

(1)

The Helmholtz free energy F follows directly from the partition function:

(2)

b) We may find the force from F:

(3)

The equation of state is therefore analogous to the gas in a container with rigid walls, where

c) The entropy, energy, and heat capacity all follow in quick succession from F:

(4)

(5)

(6)