- #1
happyparticle
- 438
- 20
- 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
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