PHYSICS 140A : STATISTICAL PHYSICS
HW ASSIGNMENT #3
(1) For an ideal gas, show explicitly that
∂(T, S, N )
=
∂(p, V, N )
∂T
∂p
V,N
∂S
∂V
−
p,N
∂T
∂V
p,N
∂S
∂p
=1.
V,N
(2) A thermodynamic process takes place at constant ϕ, where ϕ is a particular function
of some state variables. In each of the following cases, find the heat capacity at constant ϕ,
i.e. Cϕ . You may assume N is constant in all cases.
(a) ϕ(T, V ) = V T −2 .
(b) ϕ(p, T ) = T ep/p0 .
(c) ϕ(p, V ) = p3 V .
(3) The entropy of a thermodynamic system S(E, V, N ) is given by
S(E, V, N ) = a E α V β N γ ,
where a is a dimensionful constant.
(a) Extensivity of S imposes a condition on (α, β, γ). Find this constraint.
(b) Even with the extensivity condition satisfied, the system may violate one or more stability criteria. Find the general conditions on (α, β, γ) which are thermodynamically
permissible.
(4) Express V
∂T
∂V
in terms of cp , αp , and κT . See §1.10.1 of the notes if you have
forgotten the definitions of the latter two quantities.
H
(5) An ideal gas expands isothermally from volume Vi to volume Vf .
(a) Assuming N is constant, what is the change in the Helmholtz free energy F ?
(b) Assuming µ is constant, what is the change in the Landau free energy Ω ?
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