1. Considere os potenciais termodinâmicos U, F, H e G e seus diferenciais
    d U=T d S-P d V+\mu d n
    d F=-S d T-P d V+\mu d n
    d H=T d S+V d P+\mu d n
    d G=-S d T+V d P+\mu d n
    Encontre todas as relações de Maxwell.

Passo 1

O enunciado apresenta quatro potenciais termodinâmicos cujos diferenciais são
dU = T\,dS - P\,dV + \mu\,dn,
dF = -S\,dT - P\,dV + \mu\,dn,
dH = T\,dS + V\,dP + \mu\,dn,
dG = -S\,dT + V\,dP + \mu\,dn.
O conteúdo aqui é a obtenção das relações de Maxwell a partir da igualdade das derivadas parciais mistas de funções de duas variáveis (S e V para U, T e V para F, S e P para H, T e P para G), mantido n constante.

Passo 2

Para um potencial \Phi(x,y,n) com diferencial
d\Phi = A(x,y,n)\,dx + B(x,y,n)\,dy + \mu\,dn,
temos
A = \Bigl(\frac{\partial \Phi}{\partial x}\Bigr)_{y,n}, \quad B = \Bigl(\frac{\partial \Phi}{\partial y}\Bigr)_{x,n}.
Pela igualdade das derivadas cruzadas (teorema de Schwarz),
\frac{\partial A}{\partial y}\Big|_{x,n} = \frac{\partial B}{\partial x}\Big|_{y,n}.
Dessa forma obtemos as quatro relações de Maxwell.

Passo 3

Para U(S,V,n):
\Bigl(\frac{\partial U}{\partial S}\Bigr)_{V,n} = T, \quad \Bigl(\frac{\partial U}{\partial V}\Bigr)_{S,n} = -P.
Aplicando Schwarz,
\frac{\partial T}{\partial V}\Big|_{S,n} = -\frac{\partial P}{\partial S}\Big|_{V,n}.

Passo 4

Para F(T,V,n):
\Bigl(\frac{\partial F}{\partial T}\Bigr)_{V,n} = -S, \quad \Bigl(\frac{\partial F}{\partial V}\Bigr)_{T,n} = -P.
Logo,
\frac{\partial(-S)}{\partial V}\Big|_{T,n} = \frac{\partial(-P)}{\partial T}\Big|_{V,n} \quad\Longrightarrow\quad \frac{\partial S}{\partial V}\Big|_{T,n} = \frac{\partial P}{\partial T}\Big|_{V,n}.

Passo 5

Para H(S,P,n):
\Bigl(\frac{\partial H}{\partial S}\Bigr)_{P,n} = T, \quad \Bigl(\frac{\partial H}{\partial P}\Bigr)_{S,n} = V.
Daí
\frac{\partial T}{\partial P}\Big|_{S,n} = \frac{\partial V}{\partial S}\Big|_{P,n}.

Passo 6

Para G(T,P,n):
\Bigl(\frac{\partial G}{\partial T}\Bigr)_{P,n} = -S, \quad \Bigl(\frac{\partial G}{\partial P}\Bigr)_{T,n} = V.
Portanto
\frac{\partial(-S)}{\partial P}\Big|_{T,n} = \frac{\partial V}{\partial T}\Big|_{P,n} \quad\Longrightarrow\quad \frac{\partial S}{\partial P}\Big|_{T,n} = -\frac{\partial V}{\partial T}\Big|_{P,n}.

Resposta

\frac{\partial T}{\partial V}\Big|_{S,n} = -\,\frac{\partial P}{\partial S}\Big|_{V,n}, \quad \frac{\partial S}{\partial V}\Big|_{T,n} = \frac{\partial P}{\partial T}\Big|_{V,n},
\frac{\partial T}{\partial P}\Big|_{S,n} = \frac{\partial V}{\partial S}\Big|_{P,n}, \quad \frac{\partial S}{\partial P}\Big|_{T,n} = -\,\frac{\partial V}{\partial T}\Big|_{P,n}.

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