# I need help deriving this Maxwell relation

1. Dec 17, 2013

### Yekonaip

1. The problem statement, all variables and given/known data
By considering small changes in enthalpy,
and using the central equation, derive the Maxwell relation

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

2. Relevant equations
H=U+pV
dU=TdS-pdV

3. The attempt at a solution
So the way I attempted this was to get an expression for dH,
dH=dU+pdV+Vdp
dH=TdS+Vdp

Honestly I don't know the reason that pV splits up into pdV and Vdp but just know that it does that so I'd appreciate it if someone could input on that?

Then I used the fact that,
$$dH=\left ( \frac{\partial H}{\partial S} \right )_{p}dS+\left ( \frac{\partial H}{\partial p} \right )_{S}dp$$
and that
$$\frac{\partial^{2} H}{\partial S\partial p}=\frac{\partial^{2} H}{\partial p\partial S}$$

This basically yields me with the relation,

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

So I know this is a correct relation, however it isn't the relation that the question requires. I'm not really sure where to go from here, is there a way to rearrange the equation I ended up with? Or was this a Red Herring and I should have gone about it a different way?

Thank you for your time. I really appreciate it!

2. Dec 17, 2013

### vanhees71

Since, as independent variables in the partial derivatives you have $T$ and [/itex]V[/itex], you should use the corresponding thermodynamic potential, for which these variables are "natural". This is the free energy rather than the enthalpy, i.e.,
$$F(T,V)=U-TS.$$
Write down the first Law, i.e., evaluate $\mathrm{d} F=\ldots$ and use the commutativity of the 2nd derivatives as in your example with the enthalpy.

3. Dec 17, 2013

### Yekonaip

Sorry I may have been a little vague. I understand what you're saying but the question specifically asks to consider Enthalpy and use the equation,
$$H=U+pV$$

4. Dec 17, 2013

### Staff: Mentor

From the central equation and the definition of F, it follows that dF = -SdT-PdV. This should give you the required maxwell relation. The equation for the enthalpy is consistent with all this too. So, who knows what the question means.

Chet

5. Dec 18, 2013

### hjelmgart

That is just the chain rule from simple calculus. d(pV) = pdV+Vdp

For the rest of it, you did indeed prove one of the Maxwell relations, although I am not sure, how to do it for the one you had to find, using the enthalpy. An idea would be to try using:

U = -T^2 (∂(F/T)/∂T)V

in the equation for the enthalpy, or some other way of expressing U, that allows you to introduce F

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