What is the relationship between entropy and pressure in thermodynamics?

[happyparticle]

Homework Statement

Derive $(\frac{\partial H}{\partial P})_t$ in term of $P,V,T, \beta, \kappa, c_p$
Let H = H(T,P)

Relevant Equations

$ds = (\frac{\partial S}{\partial H})_p dH + (\frac{\partial S}{ \partial P})_H dP$

$dH = (\frac{\partial H}{\partial T})_p dT + (\frac{\partial H}{ \partial P})_T dP$

Hi,
Starting from dS in term of H and P, I'm trying to find $(\frac{\partial H}{\partial P})_t$ in term of $P,V,T, \beta, \kappa, c_p$.
Here what I did so far.

$ds = (\frac{\partial S}{\partial H})_p dH + (\frac{\partial S}{ \partial P})_H dP$

$ds = (\frac{\partial S}{\partial H})_p [ (\frac{\partial H}{\partial T})_p dT + (\frac{\partial H}{ \partial P})_T dP] + (\frac{\partial S}{ \partial P})_H dP$

$\frac{\partial S}{\partial P} = (\frac{\partial S}{\partial H})_p (\frac{\partial H}{\partial T}) (\frac{\partial T}{\partial P}) + (\frac{\partial H}{\partial P})_T (\frac{\partial S}{\partial H})_P + (\frac{\partial S}{\partial P})_H$

$(\frac{\partial S^2}{\partial T \partial P}) = (\frac{\partial S}{\partial H})_p (\frac{\partial H}{\partial T}) (\frac{\partial T}{\partial P}) + (\frac{\partial H}{\partial P})_T (\frac{\partial S}{\partial H})_P + \frac{d}{dP}(\frac{\partial S}{\partial P})_H$

$(\frac{\partial H}{\partial P})_T = -(\frac{\partial H}{\partial S})_P (\frac{\partial S}{\partial H})_P (\frac{\partial H}{\partial T})(\frac{\partial T}{\partial P})$

From there, I'm stuck. I don't see how I can get $P,V,T, \beta, \kappa, c_p$

Thank you

 

[Chestermiller]

Start with the exact differential dG=-SdT+VdP, which means that $$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$$

 

[happyparticle]

I have to start with ds in term of dh and dp.
I just saw in the original post, those P should be in index.

I finally found something using $Tds = dh - vdp$