Final - Statistical Physics
Winter Quarter 2004
1. Two identical ideal gases at pressures P1 and P2 and temperatures T1 and T2 are in
vessels of volumes V1 and V2 respectively. The vessels are then connected and gases
mix. Find the change of entropy after
Quiz 1: Thermodynamics
1. A gas obeys the equation of state
P=
NT
N 2 (T )
+
V
V2
where (T ) is a function of the temperature T only. The gas is initially at temperature
T and volume V and is expanded isothermally and reversibly to volume 2V . For this
ex
Quiz 2: Gibbs Distribution
1. Consider two electrons in a two-level atom in equilibrium at temperature T . The
level separation is .
a) Evaluate the partition function of this system and its free energy.
b) Find the entropy and the energy of the system in
Quiz 3: Quantum Gases
1. Determine the entropy and the heat capacity of a degenerate, extreme relativistic
( = pc) electron gas.
Hint:
0
0
f () d
exp [( ) /T ] + 1
N () h () d
exp [( ) /T ] + 1
F
0
2 2
T f ( )
6
f () d +
0
N () h () d +
2 2
T N (F ) h (F
Quiz 5: Blackbody Radiation. Debyes Theory.
1. Apply Debyes theory to a 2D solid. Specically, derive the temperature dependence
of the heat capacity in the limits of high and low temperature. For simplicity, assume
a simple lattice, v = 1, with N unit cel
Quiz 4: Quantum Gases
1. For a concentration n = N/A of extreme relativistic ( = pc) bosons in 2D, nd the
temperature of Bose-Einstein condensation.
Hint:
0
zdz
2
=
exp z 1
6
Solution
N=
d =
n=
g
2h2 c2
0
g d
exp [( ) /T ] 1
A (2d)
A (2pdp)
=
2
(2h)
(2hc)