Problem Set 3

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
h³ (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 non-interacting 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 non-interacting 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_BT.
(c) [5 pts] Show that the total average energy is, E = 3Nk_BT (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 αx²
+∞
x≥0
x<0

The infinite potential for x < 0 may be viewed as a rigid wall filling the y-z 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

n= ∞
∫
0 n(x) dx

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.