Statistical Mechanics - Physics at Oregon State University

3.8. PROBLEMS FOR CHAPTER 3 65
Problem 4.
The state of a many body system is characterized by two quantum numbers,
n and m. The possible values of the quantum number n are 0, 1, 2, · · · , ∞, while
the values of m are in the range 0, 1, · · · , n. The energy of the system in the state
(n, m) is nω and the number of particles is m. Evaluate the grand partition
function for this system.
Problem 5.
An ideal gas of atoms with mass m is contained in a cylinder that spins
around with angular frequency ω. The system is in equilibrium. The distance
to the axis of the cylinder is r. The radius of the cylinder is R. Calculate the
density of the gas as a function of r.
Problem 6.
Extremely relativistic particles obey the relation E( k) = c| k|. Assume we
have a gas of these identical particles at low density, or n ≪ nQ(T ).
(A) Calculate the partition function Z1(T, V ) for N=1.
(B) Calculate Z(T, V, N).
(C) Calculate p(T, V, N), S(T, V, N), and µ(T, V, N).
Problem 7.
The chemical potential of an ideal gas is given by 3.101. Suppose n ≪
nQ(T ). In this case we have µ < 0. A bottle contains an ideal gas at such an
extremely low density. We take one molecule out of the bottle. Since µ < 0 the
Helmholtz free energy will go up. The equilibrium of a system is reached when
the Helmholtz free energy is minimal. This seems to favor molecules entering the
bottle. Nevertheless, we all know that if we open the bottle in an environment
where the density outside is lower than inside the bottle, molecules will flow
out. What is wrong with the reasoning in this problem?
Problem 8.
The only frequency of the radiation in a certain cavity is ω. The average
number of photons in this cavity is equal to e ω −1 kB T − 1 when the temperature
is T. Calculate the chemical potential for this system.
Problem 9.