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1. Show that
∂E
∂N
T,V
= µ − T
∂µ
∂T
N,V
.
2. Prove the relationship
CP = CV + TV
α
2
P
κT
.
Since the isothermal compressibility is always greater than zero for a thermodynamically stable gas, this implies the heat capacity at constant pressure is
always greater than the heat capacity at constant volume.
3. Consider N spin-1/2 particles on a lattice (so that the particles are distinguishable) in a state with N/2 + n up spins. The Hamiltonian for this system is
H = − ∑j σjB, where σj = ±1. This is a simple model for a paramagnetic
system.
(a) Show that the total number of such microstates is
Ω(n) = N!
(N/2 + n)!(N/2 − n)!
.
(I just want you to go through what we did in lecture here.)
(b) If the total energy of the system is unspecified, the probability of a a particular value of n (which is proportional to the magnetization of the system) is p(n) = Ω(n)/2N since there are 2N possible states of the system.
Show that for N n, we have
p(n) ≈
r
2
πN
e
−2n
2/N
(hint: use Sterling’s formula including factors of ln(2πN).)
(c) Verify that p(n) is normalized.
(d) Use p(n) to calculate hn
2
i and hn
4
i.
(e) Assume that there are two paramagnets, each with N spins, in contact
with a total energy of zero. What is the root-mean-square value of n for
one of the systems?
4. Consider two identical particles that cannot occupy the same single-particle
state (i.e. fermions), in a 3-level system with single-particle energies 0, e, and
2e.
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(a) Find the canonical partition function ZN.
(b) Calculate the average energy. Write down the T = 0 and T = ∞ limits of
the average energy.
(c) Calculate the entropy of the system. Write down the T = 0 and T = ∞
limits of the average entropy.
(d) Repeat parts (a)-(c), but now assuming that the particles are indistinguishable but can occupy the same state.
(e) Repeat parts (a)-(c), but assume the particles are distinguishable and can
occupy the same state.
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