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TSP Problem Sheet 1
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 1: Thermodynamics
- Van der Waals gas Show that, for a van der Waals gas, the specific heat at constant volume, CV , obeys ( ∂CV ∂V
)
T
= 0.
- Potentials and thermodynamic variables The Gibbs free energy of an imperfect gas containing N molecules is given, in terms of its natural variables T , p and N , by
G = N kB T ln
(
p p 0
)
− N A(T ) p,
where p 0 is a constant and A is a function of T only. Derive expressions in terms of T , p, V , and N for: (a) the equation of state of the gas; (b) the entropy, S; (c) the enthalpy, H; (d) the internal energy, U ; (e) the Helmholtz free energy, F. Can all equilibrium thermodynamic information about the gas be obtained from a knowl- edge of: (i) F (T, V, N ); (ii) the equation of state and U (T, p, N )?
- Entropy of the monatomic gas The entropy of a monatomic ideal gas is given by the Sackur-Tetrode equation which can be written in the form:
S(U, V, N ) = N kB ln
{
α V N
(
U
N
) 3 / 2 }
,
where α is a constant to be derived later in the course. Invert this expression to get U (S, V, N ). From this, obtain the equation of state expressing p as a function of V, N and T.
- Analytic thermodynamics Use a Maxwell relation and the chain rule to show that for any substance the rate of change of T with p in a reversible adiabatic compression is given by ( ∂T ∂p
)
S
=
(
T
Cp
) (
∂V
∂T
)
p
.
Find an equivalent expression for the adiabatic rate of change of T with V , and check that both results are valid for an ideal monatomic gas.
- Brief Notes Write brief notes on thermodynamic equilibrium in closed and open systems.
TSP-2020/21 Problem sheet 1 1 Michaelmas Term
TSP Problem Sheet 1
Module: Thermal and Statistical Physics
University: University of Cambridge
