1
Problem Set #4 ; due Monday November 8
1. Adsorption of an ideal gas on a surface. Consider a box at temperature T and
volume L3 containing an ideal gas of N spinless atoms. The inside surface of the box
exerts an external potential on a small fraction
1. This is an isolated system (energy given) so we need to use microcanonical ensemble formalism.
(a) The number of empty sites Ne = Nd , for obvious reasons. Since the energy is E = Ne e + Nd d ,
it follows that
E
Nd =
.
e + d
(b) The multiplicity of the
Problem Set 3: Due: Monday Oct 25 at 10:00 am.
12
mv + Kx 2 n where n is a
2
positive integer. What is the mean energy at high temperatures?
1. Consider a particle in a 1D potential well with energy E =
Applying the generalized equipartition theorem:
E
=
Problem Set #3 - Solution
March 5, 2010
Problem 1
a) Directly using the degeneracy and the energy, Z = (2j + 1) exp(E0 j (j + 1).
j =0
b) Note E+ = Ej +1 Ej = 2E0 (j + 1) and E = Ej Ej 1 = 2E0 . If one averages these, one
1
1
has E = 2 (E+ E ) = E (2j + 1
Problem Set #2 - Solution
January 02, 2010
Problem 1
a) As the bath is a heat bath, its temperature T0 is constant such that
Sbath =
dQ
CM T
=
T0
T0
dS =
which is positive as heat ows from the water to the bath as the water cools down. The change of
entro
Problem Set #1 - Solution
January 20, 2010
Problem 1 (Sturge 2.6)
a) The rubber band is stretched very slowly so this is a reversible process and the tension in the
rubber band is the same as the external pressure applied. This is an isothermal process su
Problem Set #4 - Solution
Wednesday March 17th, 2010
Problem 1
a) Particle are indistinguishable and non-interacting so we factorize
1N
z
N!
Z (T, V, N ) =
where
1
2
1
z (T, V ) = 3
h0
d3 x
d3 p
e
p2
2Bs
2m
1
s= 2
and h3 is the momentum space volume per m
1
Problem Set #2 2010; due Friday October 8
1. Suppose a particle has a single (non degenerate) ground state and a two fold
degenerate excited state at an energy above the ground state energy.
(a) What is the partition function for N such particles assumi
1
P403 Problem Assignment #1: Due Monday Sept 27 10:00 am
1. Estimate the multiplicity ( N , n ) and probability P( N , n ) for a spin magnet where
x = n / N is a small number and n = N N . See page 3 of the notes on microstates.
N!
N!
=
N ! N ! ( N + n )
Problem Set #3 - Solution
March 5, 2010
Problem 1
a) The density of states is derived from the number of states per unit volume inside a d-dimensional
sphere of radius k :
1 Vd k d 1
G(k ) = 3 d d
2
Ld
d
L
1
V k d is the volume of the fraction
2d d
d
cont
Problem Set #5 - Solution
Wednesday March 31th, 2010
Problem 1
For an ideal gas, the chemical potential is
= kB T ln(
where
Z1 = BV T
Z1
)
N
CV
R
3
The constant B would simply be (2mkB /h2 ) 2 if it was a monoatomic gas and comes from integrating over th
Problem Set 5 : Due: Mon. Nov 22.
1. Ideal gas with an internal degree of freedom:
Consider an ideal gas of N identical atoms which have a doubly degenerate excited state
at energy above the ground state energy. For simplicity let the ground state energy