Which should I use to prove this?

Outrageous

(∂s/∂P) at constant volume =1/T ×(∂u/∂T)at constant volume × (∂T/∂P) at constant volume

s(P,v)
Tds= du +pdv
h= u + pv
h(P,v)
u(P,v)
or other formula?
How am I going to choose ? Please guide

Thank

haruspex

I'm not expert on gas theory, so I'll just treat this as an algebraic question.
Of those 'formulae', only two are equations. One of those two involves h, not mentioned anywhere else. So that leaves Tds= du +pdv as the only candidate.
What would the constant volume version of that equation look like?

Outrageous

The others also can become equation like s is function of v and p
So ∂s=(∂s/∂p)dp + (∂s/∂v)dv
Then h=u + pv , can be dh= du + Pdv + vdp
This is all just dealing with mathematics , please teach me how to choose
Do you mean why did I put constant volume there? It means by keeping volume constant then only differentiate.
Thank

Bipolarity

Have you considered using the Maxwell relations?

BiP

haruspex

No, I mean take this equation: Tds= du +pdv
and turn it into an equation involving partial derivatives, v being constant.

Outrageous

I am not really understand what is Maxwell ,
I only can get (∂s/∂P) at constant volume = (1/T)(∂h/∂P)constant volume -(v/T)

haruspex

Please post your working to get that.

Outrageous

Trial

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Outrageous

This first , only that second deriavative

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Chestermiller

You need to follow BiP's advice and familurize yourself with the derivation and application of the Maxwell equations.