(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to derive the following relation:

[tex]\alpha_{S}=\alpha_{P}\frac{1}{1-\frac{\kappa_{T}}{\kappa_{S}}}[/tex]

2. Relevant equations

Hopefully you can see that my notation |P means at constant pressure, I could not find a better way to do this, any ideas?

[tex]\alpha_{S}=\frac{1}{V}\frac{\partial V}{\partial T} |P[/tex]

[tex]\alpha_{P}=\frac{1}{V}\frac{\partial V}{\partial T} |S[/tex]

[tex]\kappa_{T}=\frac{1}{V}\frac{\partial V}{\partial P} |T[/tex]

[tex]\kappa_{S}=\frac{1}{V}\frac{\partial V}{\partial P} |S[/tex]

3. The attempt at a solution

I have no intuition about this problem, so i have just been trying everything I can think of and nothing works. I think the first step is pretty obvious:

[tex]\alpha_{s}=\frac{\alpha_{P}}{\frac{\partial V}{\partial T}|P}\frac{\partial V}{\partial T}|S[/tex]

after this it just seems like a guessing game, trying to apply the proper identity to lead me to the answer. Note that I used one of the Maxwell relations somewhere in here. Here is my last ditch effort. I'm pretty sure the identity I made up is not true, but this seems to have gotten me closest to the answer, and it would take me days to type up all my false leads.

[tex]dV = \frac{\partial V}{\partial T}|P dT + \frac{\partial V}{\partial P}|T dP[/tex]

From here, I made the almost certainly false conclusion that

[tex]\frac{\partial V}{\partial T}|S = \frac{\partial V}{\partial T}|P + \frac{\partial V}{\partial P}|T\frac{\partial P}{\partial T}|S[/tex]

Plugging this back into my first step gives:

[tex]\alpha_{s}= \frac{\alpha_{P}}{1 + \frac{V\kappa_{T}\frac{\partial P}{\partial T}|S}{\frac{\partial V}{\partial T}|S}} [/tex]

Which seems very close to me, but I still can't finish it off and I'm pretty sure I cheated to get there anyway. Any help would be so greatly appreciated, this has got to be a pretty simple problem and it is driving me insane!

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