Problem Set 3

Problem Set 3

Due Wednesday, February 20, in lecture

Problem 1 [10 points total]
Consider a box of volume V. The box is split exactly in half by a thermally conducting imovable wall. Equal quantities of the same type of ideal gas fill each half of the box, and the system is in equilibrium. The total energy of the gas is fixed at E_T.
a) Show that the pressure of the gas on each side is the same (remember, the wall is imovable, so you can't just appeal to mechanical equilibrium).
b) Using the formular derived in lecture for the number of states Ω(E) of an ideal gas at total energy E, find the most likely value for the energy of the gas on one side of the box.
c) Using Ω(E), if P(E) is the probability distribution for the gas on one side of the box to have energy E, show that the relative width (i.e. the width divided by the average) of P(E) is proportional to 1/N^1/2, for sufficiently large N, where N is the number of particles in the gas. This shows that the fluctuation of E away from its average becomes negligibly small in the thermodynamic limit of N going to infinity. (For "width" you may use half width at half height or any other reasonable definition.)
Problem 2 [10 points total]
In lecture we will derive 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 3 [20 points total]
Consider 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 microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N₊ and N₋ of up and down objects).
(b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0.
(c) What happens if such a system (1) with T₁<0 comes into thermal contact with another such system (2) with T₂>0? Does T₁ increase or decrease? Does T₂ increase or decrease? In which direction does the heat flow?