### #36_常識集_Maxwell_Relations

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U = (TdS – pdV +µdN)dU = TdS – pdV +µdN
H = U + pVdH = TdS + Vdp + µdN
F = U – TSdF = – SdT – pdV +µdN
G = U + pV – TSdG = – SdT + Vdp + µdN

$\textup{d}U=\left&space;(\frac{\partial&space;U}{\partial&space;S}&space;\right&space;)_{V,&space;N}\textup{d}S+\left&space;(\frac{\partial&space;U}{\partial&space;V}&space;\right&space;)_{S,&space;N}\textup{d}V+\left&space;(\frac{\partial&space;U}{\partial&space;N}&space;\right&space;)_{S,V}\textup{d}N$

$\left&space;(\frac{\partial&space;U}{\partial&space;S}&space;\right&space;)_{V,&space;N}=T$$\left&space;(\frac{\partial&space;U}{\partial&space;V}&space;\right&space;)_{S,&space;N}=-p$$\left&space;(\frac{\partial&space;U}{\partial&space;N}&space;\right&space;)_{S,V}=\mu$

$\frac{\partial^2&space;U}{\partial&space;V\partial&space;S}=\left&space;(\frac{\partial&space;T}{\partial&space;V}&space;\right&space;)_{S}=\frac{\partial^2&space;U}{\partial&space;S\partial&space;V}=&space;-\left&space;(\frac{\partial&space;p}{\partial&space;S}&space;\right&space;)_{V}$

$\left&space;(\frac{\partial&space;T}{\partial&space;V}&space;\right&space;)_{S}=&space;-\left&space;(\frac{\partial&space;p}{\partial&space;S}&space;\right&space;)_{V}$

最後我們還要再介紹一個很重要的東西，叫做reciprocal relations & cyclical relations。假設一個函數滿足下面的關係：f(x, y, z)=0。因為實際上這個算式只有兩個變數是獨立變數，所以我可以寫出x = x(y, z), y=y(x, z)。對x & y分別做全微分可以得到：

$\textup{d}x&space;=&space;\left&space;(\frac{\partial&space;x}{\partial&space;y}&space;\right&space;)_{z}\left&space;(\frac{\partial&space;y}{\partial&space;x}&space;\right&space;)_{z}\textup{d}x&space;+&space;\left&space;(&space;\left&space;(\frac{\partial&space;x}{\partial&space;y}&space;\right&space;)_{z}\left&space;(\frac{\partial&space;y}{\partial&space;z}&space;\right&space;)_{x}+&space;\left&space;(\frac{\partial&space;x}{\partial&space;z}&space;\right&space;)_{y}\right&space;)\textup{d}z$