How to derive this thermodynamic math identity

[Hiero]

Homework Statement

See image below

Relevant Equations

$dS = \beta(dE - \mu dn +PdV)$

$\frac{\partial x}{\partial y}\Big \rvert _z = - \frac{\partial x}{\partial z}\Big \rvert _y \frac{\partial z}{\partial y}\Big \rvert _x$
(Maybe relevant maybe not.)

So the Legendre transforms are straightforward; define $S_1=S-\beta E$ and $S_2= S-\beta E + \beta \mu n$ then we get:

$dS_1 = -Ed\beta - \beta \mu dn + \beta PdV$
$dS_2 = -Ed\beta + nd(\beta \mu) + \beta PdV$

And so by applying the equality of mixed partials of $S_1$ and $S_2$ we can conclude (similar to the Maxwell relations)

$\frac{\partial E}{\partial n }\Big \rvert _{\beta , V}=\frac{\partial (\beta \mu)}{\partial \beta }\Big \rvert _{n , V}$

$\frac{\partial E}{\partial (\beta \mu)}\Big \rvert _{\beta , V}=-\frac{\partial n}{\partial \beta }\Big \rvert _{\beta \mu , V}$

I feel like these must be relevant because why else would he mention those Legendre transforms in the same problem. However I cannot figure out how to derive the given formula. I tried a lot of things (no point in typing them) that didn’t lead to it. I mention the “maybe relevant” equation because it was used in the same chapter.

Thought about it for more than a day with no new ideas. Thanks in advance.

 

[Charles Link]

This is a long one, with all of the partials, and would take all day to write out in Latex, but I think I verified it to be correct. Let me summarize: Take $E_1(\beta,V, \beta \mu )=E_2(\beta,V, n)$. The problem is to take $\frac{\partial{E_1}}{\partial{\beta}}_{V,\beta \mu}$ and you do it on $E_2$. You write $n=n(\beta,V, \beta \mu)$. It helps to write differentials $dE=dE1=dE_2$ and write out all of the terms. You also write a differential for $dn$, and it gets a term with $d (\beta \mu )$. You then write $\beta \mu=\beta \mu (\beta, V,n)$, and write out the differential for $d (\beta \mu)$. You have two degrees of freedom, so let $dV=dn=0$. This gives $(\frac{\partial{n}}{\partial{\beta}})_{V, \beta \mu}=-(\frac{\partial{n}}{\partial{\beta \mu}})_{\beta,V} ( \frac{\partial{\beta \mu}}{\partial{\beta}})_{V,n}$.

 

[Charles Link]

To add to the above $dE=(\frac{\partial{E_1}}{\partial{\beta}}) d \beta +(...) dV+(...) d (\beta \mu)=(\frac{\partial{E_2}}{\partial{\beta}}) d \beta + (...)dV+(\frac{\partial{E_2}}{\partial{n}})_{\beta,V} dn$, where $dn=(...)d \beta+(...)dV+(...)d (\beta \mu)$. for this part, you let $dV=d(\beta \mu)=0$, and take the partial on $\beta$.

 

[Charles Link]

Note that post 3 refers to the first step. Once you have that, you need what I think is a Maxwell type relation with the partials of $dn$ (described in post 2 at the bottom) to get the final result.
Edit:The equation that you posted with the x,y, and z is this same expression. (The $V$ as a 4th variable is irrelevant).

 

[Hiero]

@Charles Link

You said a couple things I didn’t follow, like how you mentioned to take $\beta \mu$ to be a function of $(\beta, \mu, n)$, and how you said something was equivalent to my x,y,z formula.

But what I mainly gathered is that you took $E_1(\beta, V, \beta \mu) = E_2(\beta, V, n(\beta, V, \beta \mu))$ and then differentiated w.r.t beta (effectively using the chain rule) to conclude:

$\frac{\partial E_1}{\partial \beta }\Big \rvert _{\beta \mu , V} = \frac{\partial E_2}{\partial \beta }\Big \rvert _{n, V}+\frac{\partial E_2}{\partial n }\Big \rvert _{\beta, V}\frac{\partial n}{\partial \beta }\Big \rvert _{\beta \mu, V}$

That’s a great starting point, but the main thing I didn’t understand is how did you conclude the following:

This gives $(\frac{\partial{n}}{\partial{\beta}})_{V, \beta \mu}=-(\frac{\partial{n}}{\partial{\beta \mu}})_{\beta,V} ( \frac{\partial{\beta \mu}}{\partial{\beta}})_{V,n}$.

In my OP I found a different Maxwell type relation for that same term, if we move the negative to the other side:

$\frac{\partial E}{\partial (\beta \mu)}\Big \rvert _{\beta , V}=-\frac{\partial n}{\partial \beta }\Big \rvert _{\beta \mu , V}$

Your relation does yield the answer (if we drop the subscripts from E) but I didn’t follow how you found it.
[Edit: nevermind; see next post]

Its strange; all the other problems in this chapter were pretty trivial.

And I completely understand about all the Latex haha, I’m on mobile so copy+paste is a life saver

Thanks for taking the time to tackle this problem with me!

 

[Hiero]

Edit:The equation that you posted with the x,y, and z is this same expression. (The $V$ as a 4th variable is irrelevant).

Oh wow I just realized what you meant! It’s not a Maxwell relation it’s the x,y,z relation! Wow. So it was relevant after all hahah.

Well then I guess that solves it! Thanks so much!