Homework 5 solution
Scrambled Potentially Useful Equations
F = U
exp( s / )
Z=
s
JU = B T 4
3
CV = N
2
3
U=
2
Z1 = nQ V
cs
(6 2 N/V )
kB
m 3/2
nQ = (
)
2 2
(Ideal gas) = ln(n/nQ )
D =
F = ln Z
Boltzmann Factor = exp( )
Gibbs Factor = exp(N ) )
dU = d pd
Homework 9 solution
1) a) We have F = N (ln(nQ (V N b)/N ) + 1) N 2 a/V and =
F
|V,N
therefore
= N (ln(nQ (V N b)/N ) + 5/2)
b) We have U = F + therefore
3
U = N N 2 a/V
2
c) We have (p + N 2 a/V 2 )(V N b) = N or pV = (N /(V N b) N 2 a/V 2 )V therefore
Homework 7 solution
1. Heat Pump. (10 pts) Solve Problem 1 of Chapter 8 in K&K
a) Reversible process thus we have Sl = Sh or Ql = (Tl /Th )Qh . We have W = Qh Ql therefore
W
Th Tl
=
Qh
Th
If not reversible then it means there is creation of entropy theref
Homework 6a solution
1. Density of States.
1.1. (6 pts) Solve Problem 1 of Chapter 7 in K&K.
Lets compute D(k )dk . We have in general
D(k )dk = g
Cd k d1 dk
( 2 )d
L
where d is the dimension of the space and g the degeneracy of the state. In 1 dimension
Homework 5 solution
1. Fermi and Bose Statistics. (15 pts) Solve Problem 1 and 2 of Chapter 6 in K&K. At room
temperature, at what value of ( ) is there a 1% dierence in occupation number for the BoseEinstein Distribution and the Fermi-Dirac Distribution.
Homework 4 solution
1. Centrifuge. (10 pts) Solve Problem 1 of Chapter 5 in K&K.
The particles are under a potential energy
1
U (r) = m 2 r2
2
Lets calculate the partition function:
R
Z=A
0
exp( m 2 r2 )2rdr
2
Where A is a constant. Using rdr = 1/2d(r2 )
Homework 3 solution
1. Fun with Black Body Emission.
1.1. Rayleigh-Jean Approximation. (5 pts) Find the expression for the surface brightness
from a thermal source in the limit
show that it is a power law in frequency. This is
the so-called Rayleigh-Jea
Homework 2 solution
1. Connecting the Microcanonical Ensemble to the Canonical Ensemble. (10 pts) The
microcanonical distribution describes the probability of microstates of an isolated system while
the canonical ensemble describes the probability of micr
Homework 1 solution
1. Width of the multiplicity function. (10 pts) Expand the entropy of a paramagnet (N
1
spins) as a function of the number of up spins n about the maximum at n = N/2. (Use the Stirling
approximation logN ! = N log N N .) Show that the