A Prove The Three Maxwell Relationsvp Pvnp

(a) Prove the three Maxwell relations

(∂V/∂τ)p = – (∂σ/∂P)τ,
(∂V/∂N)p = + (∂μ/∂P)N,
(∂μ/∂τ)N = – (∂σ/∂N)τ,

Strictly speaking, (45α) should be written

(∂μ/∂τ)p,N = – (∂σ/∂P)τ,N,

And two subscripts should appear similarly in (45b) and (45c). It is common to omit those subscripts that occur on both sides of these equalities.
(b) Show with the help of (45a) and the third law of thermodynamics that the volume coefficient of thermal expansion

α = 1/V (∂V/∂τ)p

Approaches zero as τ → 0.

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