Heat Capacity relations for 1st order phase transition

[Carbon123]

Homework Statement

Prove the following relation for which clausius equation holds :
Cs=Cp-αV(ΔH/ΔV)
Where Cs=∂q/∂T at constant S and is the heat capacity in the coexistence line of 2 phases

Homework Equations

dq=dU+dW
dP/dT=ΔH/(ΔV*T)

The Attempt at a Solution

I do not fully understand why q could change even though S is constant .Is the problem correct ? I thought dqrev=Tds .Isn't Cp in first order transition in coexistence line near infinite ?
Anyway trying to differentiate by dT to dq=dU+dW where dw =pdv gives ∂q/∂T=∂U/∂T+p∂V/∂T I know that U =H-pV so ∂U/∂T =∂H/∂T-∂(pV)/∂T.
I know that ∂H/∂T = Cp but cannot prove the next relation.Can anyone give me a guidance for the next step ? I do not know how to prove the -αVΔH/ΔV though αV=∂V/∂T at constant P. Thanks in advance !

 

[rude man]

What is α?