Ideal Gas in One-Dimensional Potential

a) An ideal gas of particles, each of mass m, at temperature τ, is subjected to an external force whose potential energy has the form

with 0 ≤ x ≤ ∞, A > 0, and n > 0. Find the average potential energy per particle.

b) What is the average potential energy per particle in a gas in a uniform gravitational field?

SOLUTION

a) The coordinate- and momentum-dependent parts of the partition function can be separated. The coordinate-dependent part of the partition function

(1)

For the potential in this case we have

(2)

where we substituted

and

The free energy associated with the coordinate-dependent part of the partition function is

(3)

The average potential energy is given by

(4)

For n = 2 we have a harmonic oscillator, and in agreement with the equipartition theorem

b) For n = 1, U = mgx, and the average potential energy per particle

which also agrees with the generalized equipartition theorem.