Physics 418: Statistical Mechanics I
Prof. S. Teitel firstname.lastname@example.org ----- Spring 2002
Problem Set 4
Due Tuesday, April 16, in lecture
- Problem 1 [5 points]
The Stefan-Boltzmann law states that a black body at
temperature T radiates power per unit surface area equal to
sigma*T4 where sigma is a universal constant
independent of the material properties of the body.
Assuming that the sun and the earth are black bodies,
and that the earth is in thermal equilibrium with the
sun [i.e. energy absorbed = energy emitted] calclate
the temperature of the earth in terms of the temperature
of the sun. Look up the parameters you need in order
to compute a number for this estimate of earth's temperature.
- Problem 2 [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 <--> A2
Creation of the molecule A2 costs energy eo > 0.
At equilibrium, the system will contain NF Fermions and
NB Bosons. Provide expressions from which the ratio
NB/NF 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 3 [15 points]
a) Find the chemical potential µ for an ideal (non-relativistic) Fermi
gas at low temperature, to second order in T, at fixed pressure p.
Note, this is a different expansion from that at fixed density
N/V. (Hint: what is E/V at fixed p?)
b) The Gibbs free energy is related to the chemical potential
by G(T,p,N) = Nµ(T,p). The entropy can be derived from the
Gibbs free energy, and hence from the chemical potential, by using,
S = - (dG/dT)p,N = - N(dµ/dT)p
Using your result from part (a), find the entropy of the Fermi gas
to lowest order in T. Does your result agree with that given in
Pathria §8.1 Eq.(41), as derived there from the Helmholtz free energy?
- Problem 4 [15 points]
Consider an ideal Bose gas composed of molecules with an internal
degree of freedom. Assume that this internal degree of freedom
can have only one of two energy values, the ground state
eo = 0, and an excited state, e1.
Determine the Bose Einstein condensation temperature of the gas
as a function of e1. Show in particular that for
e1/kBT >> 1,
Tc/Tco = 1- (2/3)(1/zeta(3/2))e
where Tco is the transition temperature when e1
is infinite, and zeta is the Riemann zeta function.