a) Show that U (S, V) = − T^2 ∂/∂T(F (T, V)/T)where T ≡ T (S, V) is to be understood as a function of S and V. b) Show that H (S, p) = − T^2 ∂/∂T(G (T, p)/T)where T ≡ T (S, p) is to be understood as a function of S and p.

Question

a) Show that

[latex] U(S,V) = - T^2\frac{\partial }{\partial T}\Bigl(\frac{F(T,V)}{T}\Bigr) [/latex].

where T ≡ T (S, V) is to be understood as a function of S and V.

b) Show that

[latex] H(S,p) = - T^2\frac{\partial }{\partial T}\Bigl(\frac{G(T,p)}{T}\Bigr) [/latex].

where T ≡ T (S, p) is to be understood as a function of S and p.

Accepted Answer

a) The internal energy U is related to the free energy F and expressed in terms of the state variables S and V as,

[latex] U(S, V) = F \Bigl(T(S,V),V\Bigr) + T (S, V) S [/latex].

Using the definition (4.26) and the chain rule, it can be recast as,

[latex] S (T, V, \left\{ N_A\right\} ) = − \frac{\partial F(T,V,\left\{ N_A\right\} )}{\partial T} [/latex]. (4.26)

[latex] U(S, V) = F \Bigl(T(S,V),V\Bigr) - T (S, V) \frac{\partial F \Bigl(T(S,V),V\Bigr) }{\partial T } = -T (S,V)^2\frac{\partial }{\partial T} \Biggl(\frac{F \Bigl(T(S,V)V\Bigr) }{T} \Biggr) [/latex].

b) Likewise, the enthalpy H is related to the Gibbs free energy G and expressed in terms of the state variables S and p as,

[latex] H(S, p) = G \Bigl(T(S,p),p\Bigr) + T (S, p) S [/latex].

Using the definition (4.40) and the chain rule, it can be recast as,

[latex] S (T, p, \left\{ N_A\right\} ) = − \frac{\partial G(T,p,\left\{ N_A\right\} ) }{\partial T} [/latex]. (4.40)

[latex] H(S, p) = G \Bigl(T(S,p),p\Bigr) - T (S, p) \frac{\partial G \Bigl(T(S,p),p\Bigr) }{\partial T } = -T (S,p)^2\frac{\partial }{\partial T} \Biggl(\frac{G \Bigl(T(S,p)p\Bigr) }{T} \Biggr) [/latex].

Question 4.6: a) Show that U (S, V) = − T^2 ∂/∂T(F (T, V)/T)where T ≡ T (S...

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