Description
Possibly useful results
Z ∞
−∞
dxe−ax2+bx =
r
π
a
e
b
2
4a (for a > 0)
Γ(n) = (n − 1)! =
Z ∞
0
dxxn−1
e
−x
ln N! ≈ N ln N − N (for N 1)
1. (a) (2 points) Prove the relationship
∂S
∂V
T,N
=
∂P
∂T
V,N
(b) (4 points) Using only the properties P = NT/V and CV = T
∂S
∂T
N,V
=
3
2N for an ideal gas, find the adiabatic sound speed squared
c
2
s = −
V
2
Nm
∂P
∂V
S,N
in terms of T, N, and V using partial derivative relations. Here, m is the
mass of the particles in the gas. The result of part (a) should be useful.
2. Consider a classical, non-interacting gas of N indistinguishable particles that
can only move in one-dimension and obey the single-particle dispersion relation e = |~p|c (i.e. the Hamiltonian is given by H = ∑
N
i=1
|~pi
|c) and are confined
to a length L.
(a) (8 points) Find the canonical partition function when N = 1.
(b) (2 points) Find the canonical partition function for arbitrary N.
(c) (2 points) Calculate the entropy of the N particle system.
3. (8 points) Consider a systems with an equation of state given by
P =
E
V
V
V0
λ
E
E0
λ
and a temperature given by
T = E
E
E0
λ
,
where E0, V0, and λ are constants. Given the entropy at E0 and V0 is S(E0, V0) =
0, find the entropy at arbitrary energy E and volume V.