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TSP Problem Sheet 2
Thermal and Statistical Physics
University of Cambridge
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University of Cambridge Cavendish Laboratory
TSP-2020/21 — Thermal and Statistical Physics (Part II)
Problem sheet 2: Equilibrium thermodynamics, basic statistical physics
- Multiphase mixtures A substance A can exist as solid, liquid or gas. Its chemical potential μ in each of these phases depends on pressure p and temperature T. By considering the number of variables and the number of equations in the equilibrium conditions
(i) μg(p, T ) = μ(p, T ) for vapour-liquid coexistence, and (ii) μg(p, T ) = μ(p, T ) = μs(p, T ) for vapour-liquid-solid coexistence
show that in a p − T phase diagram, vapour-liquid coexistence occurs along a line, whereas three-phase coexistence happens at a single point. We add a second substance B. Now, the chemical potentials also depend on the concentra- tion of each component (A, B) within each phase s, `, g. Three-phase coexistence of both substances requires
μAg (p, T, cAg ) = μA (p, T, cA ) = μAg (p, T, cAs )
μBg (p, T, cBg ) = μB (p, T, cB ) = μBg (p, T, cBs )
Note that cAg + cBg = 1 etc. By counting variables and constraints, show that three-phase coexistence for the mixture occurs along a line in the p − T phase diagram. Generalise this argument to the equilibrium of P phases in a mixture of C components, and use it to determine the number of free thermodynamic variables (or thermodynamic degrees of freedom) that can be adjusted independently while preserving the coexistence of all the phases in all the components.
- Partition Function The partition function of a system is
Z = exp
[
aT 3 V
]
,
where a is a positive constant. Obtain expressions for the Helmholtz free energy, the equation of state, the internal energy, the heat capacity at constant volume, and the chemical potential. Write the pressure as a function of the internal energy per unit volume. Can you identify the physical system that corresponds to such a partition function?
- Vacancies A crystalline solid contains N identical atoms on N lattice sites, and N interstitial sites to which atoms may be transferred at the energy cost εc. If n atoms are on interstitial sites, show that the configurational entropy is 2kB ln( N !/n! (N − n)!). Assuming n/N is small, and that vacancies are very rare, show by minimising the total free energy that the equilibrium proportion of atoms on interstitial sites n/N is 〈 n
N
〉
=
1
1 + exp(εc/ 2 kB T )
.
TSP-2020/21 Problem sheet 2 1 Michaelmas Term
TSP Problem Sheet 2
Module: Thermal and Statistical Physics
University: University of Cambridge
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