Statistical Mechanics - Physics at Oregon State University

234 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS.
the probability that a student who bought a copy of Jackson also bought a copy
of Liboff is four out of eight, or 50%. Similarly, the probability that a student
who bought a copy of Liboff also bought a copy of Jackson is four out of six, or
66.67%.
Problem 8.
For an ideal gas we have U = 3
2 NkBT , where N is the number of particles. Use
the relation between the entropy S(U, N) and the multiplicity function g(U, N)
to determine how g(U, N) depends on U.
From
we find
which gives
and with
we get
1
T =
∂S
∂U N
∂S
=
∂U N
3
2 NkB
1
U
S = 3
2 NkB log(U) + C(N)
g(U, N) = e S
k B
g(U, N) = C ′ (N)U 3
2 N
Does this make sense? The kinetic energy of one particle is 2 k 2
total kinetic energy is given by:
Ek = 2
2m
N
z
i=1 j=x
k 2 ij
2m
and hence the
A surface of constant energy U is a hypersphere in 3N dimensions, with radius
R given by U = 2R 2
. The number of states with a given energy is proportional
2m
to the surface area of that sphere, which is proportional to R3N−1 . For large N
we ignore the -1 and hence the area is proportional to U 3
2 N .
Problem 9.