Maxwell's Equation of Thermodynamic Relations.

A pure substance existing in a single phase has only two independent variables. Of the eight quantities p, V, T, S , U, H , F (Helmholtz function), and G ( Gibbs Function ) any one may be expressed as a function of any two others.

For a pure substance undergoing an infinitesimal reversible process,

1.dU = TdS + pdV.

2.dH = dU + pdV + Vdp = TdS + Vdp.

3.dF = dU - TdS - SdT = -pdV + SdT.

4. dG = dH + TdS - SdT = Vdp - SdT.

Since U,H,F and G are thermodynamic properties and exact differentials of the type

dz = Mdx + Ndy, then

(d*M / d*y) x = ( d*N / d*x)y

Applying this to four equations

(d*T / d*V) s = - ( d*p / d*S) v

(d*T / d*p)s = (d*V / d* S)p

(d*p / d*T)v = (d*S / d* V)t

(d*V / d*T) p = - ( d*S / d*P)t

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