
Home
Contact Info
Course Info
Calendar
Homework
Lecture Notes




PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2018
Problem Set 3
Due Monday, March 5, in lecture
 Problem 1 [10 points total]
In lecture we derived the canonical partition function Q_{N}(V,N) for the idea gas using a factorization method. One can also directly compute it by taking the Laplace transform of the microcanonical partition function Ω(E,V,N). Using our result from lecture,
Ω(E,V,N) = [ 
V h^{3} 
(2πmE)^{3/2} 
]^{N} 
1
[(3N/2)  1]! N! 
(Δ/E) 
compute directly the Laplace transform,
Q_{N}(T,V) = 
∞ ∫ 0 
dE Δ 
Ω(E,V,N)e^{βE} 
and show that you get the same result as found in lecture using the factorization method.
 Problem 2 [10 points total]
Consider the same situation as in problem 3 of the last Problem Set, i.e. a system of N distinguishable noninteracting objects, each of which can be in one of two possible states, "up" and "down", with energies +ε and ε. Assume that N is large.
(a) Working in the canonical ensemble, find the Helmholtz free energy A(T, N) as a function of temperature T and number N.
[Note: Having found Ω(E,N) in problem 3 of HW 2, you could compute the canoncial Q_{N}(T) by taking the Laplace transform of Ω(E,N) with respect to E. Don't do it this way! Instead, compute Q_{N}(T) by directly summing the Boltzmann factor over all states in the phase space.
(b) Starting from A(T, N) of part (a), find the canonical entropy and express it as a function of the average energy E and number N. Show that your result agrees with your answer for the entropy in the microcanonical ensemble, as computed in problem 3, in the large N limit.
 Problem 3 [20 points total]
Consider a classical gas of N indistinguishable noninteracting particles with ultrarelativistic energies, i.e. their kinetic energy  momentum relation is given by ε = pc, with c the speed of light and p the magnitude of the particle's momentum. The gas is confined to a box of volume V.
(a) [5 pts] Compute the canonical partition function for this system.
(b) [5 pts] Show that this system obeys the usual ideal gas law, pV = Nk_{B}T.
(c) [5 pts] Show that the total average energy is, E = 3Nk_{B}T (and hence using (b) gives, E/V = 3p).
(d) [5 pts] Show that the ratio of specific heats is, C_{p}/C_{V} = 4/3.
 Problem 4 [20 points total]
Consider a three dimensional classical ideal gas of atoms of mass m, moving in a potential energy
V(x) =  { 
½ αx^{2} +∞  x≥0 x<0 
The infinite potential for x < 0 may be viewed as a rigid wall filling the yz plane at x=0. The atoms, therefore, are attracted to this wall, but they move freely in the y and z directions. Let T be the temperature and n = N/A be the total number of atoms per unit area of the wall.
a) [5 pts] Calculate the local atomic density n(x) (number of atoms per unit volume) as a function of the distance x from the wall. Note that
b) [5 pts] Show that the ideal gas law (relating pressure, temperature, and density) continues to hold locally everywhere.
c) [5 pts] Find the pressure that the atoms exert on the wall. How does this pressure vary with temperature?
d) [5 pts] Calculate the energy and specific heat per unit area of the wall.

