Working with Maxwell's equations

[AndrewBworth]

Hello all - I've been trying to work out an example from a book, and I don't quite understand the math.

show that (δT/δV)s = - (δP/δS)v

solution (δ/δV (δU(S,V)/δS)v)s = (δ/δS(δU(S,V)/δV)s)v
(δ/δV (δ(TdS - PdV)/δS)v)s = (δ/δS(δ(TdS-PdV)/δV)s)v
(δT/δV)s = -(δP/δS)v

I don't understand the substitution or the last step

 

[Chestermiller]

I don't understand the notation. The way I learned it is as follows:

$dU=TdS-PdV$

We must also have that:

$$dU = \left(\frac{\partial U}{\partial S}\right)_VdS+\left(\frac{\partial U}{\partial T}\right)_SdV$$

Therefore, comparing both equations, we have:

$$T=\left(\frac{\partial U}{\partial S}\right)_V$$
$$-P=\left(\frac{\partial U}{\partial T}\right)_S$$

 

[AndrewBworth]

Right I don't understand it either they've inserted the differential expression into a partial fraction I don't know how to work with it.

 

[Chestermiller]

Right I don't understand it either they've inserted the differential expression into a partial fraction I don't know how to work with it.

So their notation in hinky. Can you get to the final result from my last two equations?

Chet

 

[AndrewBworth]

Sure no sweat.

 

[AndrewBworth]

Ah I figured it out