Question

6. Prove the following relation:

Answer 1

We have, H = U + PV

or dH = dU+PdV+VdP = TdS+Vdp

partial derivatives , (with V at constant T )

($\partial$H / $\partial$V)_T = T ($\partial$S/$\partial$V) _T + V ($\partial$P/$\partial$V) _T

or ($\partial$H / $\partial$V)_T = T ($\partial$P/$\partial$T) _V + V ($\partial$P/$\partial$V) _T   (maxwell relation ($\partial$S/$\partial$V) _T = ($\partial$P/$\partial$T) _V )\

Euler theorem ;  ($\partial$P/$\partial$T) _V ($\partial$T/$\partial$V) _P ($\partial$V/$\partial$P) _T  = -1

Using this,   ($\partial$H / $\partial$V)_T = T ($\partial$P/$\partial$T) _V - V  ($\partial$P/$\partial$T) _P ($\partial$T/$\partial$V) _V

_or   ($\partial$H / $\partial$V)_T = ($\partial$P/$\partial$T_{) V} ( T- V  ( ($\partial$T/$\partial$V) _P ) .............................i

We solve, L.H.S.

on multiplying by, V²/V²

we get    V² ($\partial$P/$\partial$T_{) V} ( T/V² - (1/V)* ( ($\partial$T/$\partial$V) _P )) ..................................ii

we have ,   ($\partial$(T/V) /$\partial$V) _P =  (1/V)* ( ($\partial$T/$\partial$V) _P ) -  T/V² ............................iii

From i, ii, and iii

we get ,   ($\partial$H / $\partial$V)_T = - V² ($\partial$P/$\partial$T_{) V}   ($\partial$(T/V) /$\partial$V) _P