Thermodynamic Identities Proof - Gibbs and Helmholtz

[Vuldoraq]

okay,

G=U+PV-TS

G/T = U/T + PV/T - S

$$H = -T^2 \frac{\partial (U/T +PV/T - S)}{\partial T}$$

Okay let's go from here;

$$G/T=U/T+PV/T-S$$

$$d(G/T)/dT=\frac{\frac{dU}{dT}T-U}{T^{2}}+\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT})}{T^{2}}-\frac{dS}{dT}$$

Now let dU equal it's identity again let P be constant, simplify and rearrange. Should get you to the answer. All my d's should be partial d's.

 

[TFM]

So:

$$d(G/T}/dT=\frac{\frac{dU}{dT}T-U}{T^{2}}+\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT}}{T ^{2}}-\frac{dS}{dT}$$

dU = Tds + pdv

$$d(G/T}/dT=\frac{\frac{Tds + pdv}{dT}T-U}{T^{2}}+\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT}}{T ^{2}}-\frac{dS}{dT}$$

so:


$$d(G/T}/dT=\frac{T\frac{ds}{dT} + p\frac{dv}{dT}T-U}{T^{2}}+\frac{TV\frac{dP}{dT} + TP\frac{dV}{dT}}{T^{2}}-\frac{dS}{dT}$$

Pressure is constant, so can get rid of dP

$$d(G/T}/dT=\frac{T\frac{ds}{dT} + p\frac{dv}{dT}T-U}{T^{2}}+ TP\frac{dV}{dT}}{T^{2}}-\frac{dS}{dT}$$

Does this look okay?

$$d(G/T}/dT= T\frac{ds}{dT} + p\frac{dv}{dT}T-\frac{U}{T^{2}}+T(V\frac{dP}{dT}+P\frac{dV}{dT}}-\frac{dS}{dT}$$

 

[Vuldoraq]

Okay let's go from here;

$$G/T=U/T+PV/T-S$$

$$d(G/T)/dT=\frac{\frac{dU}{dT}T-U}{T^{2}}+\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT})}{T^{2}}-\frac{dS}{dT}$$

Now let dU equal it's identity again let P be constant, simplify and rearrange. Should get you to the answer. All my d's should be partial d's.

Sorry I made an error,

This,

$$\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT})}{T^{2}}$$

Should be this,

$$\frac{T(V\frac{dP}{dT}+P\frac{dV}{dT})-PV}{T^{2}}$$