PHY 4523 Spring 2000 – Homework 4
Due by 5:00 p.m. on Wednesday, February 23.
Answer both questions. To receive full credit, you should explain your reasoning and show
all working. Please write neatly and remember to include your name.
1. Ideal gas in a gravitational field.
Do Reif Problem 7.2.
2. Generalized equipartition theorem.
Consider a classical system of N non-identical particles in d spatial dimensions, held
at a constant temperature T . The system is described by a Hamiltonian of the form
H=
N X
j=1
Aj |pj |r + Bj |qj |s ,
where qj and pj are d-vectors describing the position and momentum of particle j,
respectively. The parameters Aj and Bj are temperature-independent, positive constants.
(a) Compute the average energy of this system.
Hint: Follow the derivation of the standard equipartition theorem presented in
class and in Reif Section 7.5. Just as in the standard case, it should not be
necessary to evaluate any of the complicated-looking integrals which arise. Your
final answer should be quite simple and should not contain any mention of the
Aj ’s and Bj ’s.
(b) Find the heat capacity of this system. (For sticklers, I mean the heat capacity at
constant Aj and Bj .)
(c) As a special case of the above, calculate the heat capacity of a three-dimensional
ultra-relativistic ideal gas, for which
H=
N
X
j=1
|pj |c,
where c is the velocity of light.
Hint: You’ll need to consider what modification must be made to your general
results above when some of the Aj ’s and/or Bj ’s are zero.