PHY 418: Statistical Mechanics I
Prof. S. Teitel: stte@pas.rochester.edu ---- Spring 2024
Due Monday, February 26, uploaded to Blackboard by 11pm
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 ET.
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/N1/2, 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.)
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 number of states Ω(E,N), and then the entropy of the system S(E, N), as a function of fixed total energy E and number N (Hint: it may be useful to consider the numbers N+ and N− of up and down objects). Sketch S(E,N) as a function of E for fixed N, and show that it is not a monotonic increasing function of E. What is the key feature of this system that causes the dependence of S on E to be so qualitatively different from that of an ideal gas? [10 points]
(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. Sketch T vs E for fixed N. [7 points]
(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? [8 points]
In Notes Eq. (2.4.14) we derived the number of states Ω(E,V,N) for the idea gas of N point particles, which can be written as given below,
Ω(E,V,N) = [ | V h3 |
(2πmE)3/2 | ]N | 1
[(3N/2) - 1]! N! |
(ΔE/E) |
Note, compared to Eq. (2.4.14) the above expression has the factor N! in the denominator, as needed for indistinguishable particles, as described in Notes 2-7.
By taking the Laplace transform of Ω(E,V,N), directly compute the canonical partition function, as in Notes Eq. (2.8.6),
QN(T,V) = | ∞ ∫ 0 |
dE ΔE |
Ω(E,V,N)e-βE |
Using the resulting canonical partition function, compute the Helmholtz free energy A(T,V,N) and compare it to what was found in Problem Set 1.