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Thermodynamics and Statistical Mechanics II - Home Exercise 6

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Thermodynamics and Statistical Mechanics II - Home Exercise 6
Thermodynamics and Statistical Mechanics II - Home Exercise 6
1. Gas-solid equilibrium
Consider the gas-solid equilibrium under the extreme assumption that the entropy of the
solid may be neglected over the temperature range of interest. Let ε0 be the cohesive
energy of the solid, per atom. Treat the gas as ideal and monoatomic. Make the
approximation that the volume accessible to the gas is the volume V of the container,
independent of the much smaller volume taken by the solid.
(a) Show that the total Helmholtz free energy of the system is given by- F = FS +Fg =
−NS ε0 + Ng τ [ln(ng λ3T ) − 1], where the total number of atoms, N = Ng + NS is
constant.
(b) Find the minimum of the free energy with respect to Ng ; show that in the equilibrium condition Ng = nq V e−βε0 .
(c) Find the equilibrium vapor pressure.
2. Simplified model for the superconducting transition
The Bc (τ ) curves of most
have shapes close to simple parabolas. Sup superconductors
2 pose that Bc (τ ) = Bc0 1 − ττc
. Assume that CS vanishes faster than linearly as
τ → 0. Assume also that CN is linear in τ , as for Fermi gas. Draw on the results of
problem solved in class to calculate and plot the τ dependence of the two entropies,
the two heat capacities and the latent heat of the transition. Show that
CS (τc )
CN (τc )
= 3.
3. Alben model
The following mechanical model illustrates the symmetry breaking aspect of second
order phase transitions. An airtight piston of mass M is inside a tube of cross sectional
area a (see figure). The tube is bent into a semicircular shape of radius R. On each
side of the piston there is an ideal gas of N atoms at a temperature T .
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The volume to the right of the piston is aR(π/2 − φ) while to the left it is aR(π/2 + φ).
The free energy of the system has the form
aR(π/2 − φ)
aR(π/2 + φ)
F = M gR cos φ − N kB T ln
+ ln
+2 .
N λ3T
N λ3T
(a) Explain the terms in F . Interpret the minimum condition for F (φ) in terms of
the pressures in the two chambers.
(b) Expand F to 4-th order in φ (the relevant order parameter), show that there is a
symmetry breaking transition and find the critical temperature Tc.
(c) Describe what happens to the phase transition if the number of atoms on the left
and right of the piston is N (1 + δ) and N (1 − δ), respectively (It is sufficient to
consider |δ| 1 and include a term ∼ −φδ in the expansion (b)).
(d) At a certain temperature the left chamber (containing N (1 + δ) atoms) is found
to contain a droplet of liquid coexisting with its vapor. Which of the following
statements may be true at equilibrium?
i. The right chamber contains a liquid coexisting with its vapor.
ii. The right chamber contains only vapor.
iii. The right chamber contains only liquid.
Explain!
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