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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2005
Problem Set 7
Due Thursday, April 14, in lecture
 Problem 1 [15 points]
N Fermions A of spin 1/2 are introduced into a large volume V at
temperature T. Two Fermions may combine to create a Boson with
spin 0 via the interaction,
A + A <> A_{2}
Creation of the molecule A_{2} costs energy _{o} > 0.
At equilibrium, the system will contain N_{F} Fermions and
N_{B} Bosons. Provide expressions from which the ratio
N_{B}/N_{F} can be calculated, and perform the
calculation explicitly for T=0. What would this (T=0) ratio be,
if the particles were classical (i.e. quantum statistics can be neglected).
Explain the difference.
 Problem 2 [15 points]
A linear molecule of N identical atoms has a vibrational spectrum given by the angular frequencies,
_{m}=_{o}sin(m/2N),
for m = 1,2, ..., N1 (i.e. these are the frequencies of the normal modes of vibration).
a) Show, by explicit calculation, that the vibrational contribution to the specific heat of this molecule at very high temperature, T> , is
C = (N1)k_{B}
b) Show that for lower temperatures, such that 1/N << k_{B}T/_{o} << 1,
C ~ T
c) How does C vary with temperature at very low temperatures, such that
k_{B}T/_{o} << 1/N?
 Problem 3 [25 points]
Consider a gas with hard core interparticle interactions, i.e.
u(r)  =  infinity_{ }
0  for r<r_{o} for r>r_{o} 
(a) Compute the 2nd and 3rd virial coefficients, B_{2} and B_{3} for d=1 dimension (hard rods).
(b) Do the same for d=2 dimensions (hard disks).
(c) Compute the isothermal compressibility, _{T}, in terms of the virial expansion to 2nd order, i.e. through B_{2} and B_{3}.
(d) Compute the specific heat at constant pressure, C_{p}, in terms of the virial expansion to 2nd order, i.e. through B_{2} and B_{3}. (Hint: to compute C_{p} you might wish to use the formula that relates it to C_{V}, _{T}, and the coefficient of thermal expansion .)
(e) Using your results in parts (a) and (b), and using Pathria Eqs. (9.3.15) and (9.3.20) to get B_{2} and B_{3} for hard spheres in d=3 dimensions, make a table showing how _{T} and C_{p}C_{V} vary with dimension d for hard core interactions. Comment on any trends you see.
