How to Calculate Latent Heat at a New Point on the Coexistence Curve?

[fluidistic]

Homework Statement

I'm stuck in the following problem: In a particular solid-liquid phase transition the point $T_0$, $P_0$ lies on the coexistence curve. The latent heat of vaporization at this point is $l_0$. A nearby point on the existence curve has pressure $P_0+p$ and temperature $T_0+t$; the local slope of the coexistence curve in the P-T plane is p/t. Assuming $v$, $c_p$, $\alpha$ and $\kappa _T$ to be known in each phase in the vicinty of the states of interest, find the latent heat at the point $P_0+p$, $T_0+t$.

Homework Equations

$l=T_{\text{transition}}(s_{\text{liq.}}-s_{\text{sol.}})$. (1)
$c_p=\left ( \frac{\partial s}{\partial T} \right ) _P$
$\alpha = \frac{1}{v} \left ( \frac{\partial v}{\partial T} \right ) _P$
$\kappa _T =-\frac{1}{v} \left ( \frac{\partial v}{\partial P} \right )$

The Attempt at a Solution

So I want to use equation (1). My idea is to expression the entropies of the liquid and solid phase as functions of alpha, kappa, etc.
So I've thought of s as a function of P and T to start with. I then took the total differential of s to reach $ds=-\alpha v dP+c_p dT$. I've tried to make appear $\kappa _T$ without any success.
That's basically where I've been stuck for the last days, almost a week now. My friend told me to make a Taylor's expansion for the latent heat, so I've written down $l(T) \approx l(T_0)+ l'(T_0)(T-T_0)$ though I don't think that's correct since the latent heat depends on both the temperature and entropy; not the temperature alone.
So I'm looking for getting the $\Delta s$. I've ran out of ideas, any tip is welcome.

 

[Chestermiller]

Check out the derivation of the Clapeyron equation for vapor-liquid equilibrium. The derivation for going from solid to liquid should parallel this.

 

[fluidistic]

I've solved the problem via a Taylor's expansion, the derivation up to the solution is rather lengthy in latex. This differs quite a lot from the 2 derivations given in wikipedia of the Clausius-Clapeyron's relation.
So I'll write final answer that I got:

Spoiler

l(T_0+t)\approx l_0 + \frac{tl_0}{T_0}+t(c_P^{\text{liq}}-c_P^{\text{sol.}})-T_0p(\alpha ^{\text{liq.}}v ^{\text{liq.}}-\alpha ^{\text{sol.}}v ^{\text{sol.}})

. I've found no dependence on $\kappa _T$.

 

[Chestermiller]

I've solved the problem via a Taylor's expansion, the derivation up to the solution is rather lengthy in latex. This differs quite a lot from the 2 derivations given in wikipedia of the Clausius-Clapeyron's relation.
So I'll write final answer that I got:

Spoiler

l(T_0+t)\approx l_0 + \frac{tl_0}{T_0}+t(c_P^{\text{liq}}-c_P^{\text{sol.}})-T_0p(\alpha ^{\text{liq.}}v ^{\text{liq.}}-\alpha ^{\text{sol.}}v ^{\text{sol.}})

. I've found no dependence on $\kappa _T$.

You're right. The Clapeyron equation is not the way to go. I've solved this problem starting with general equation for the differential change in enthalpy dH=C_pdT+V(1-Tα)dP in conjunction with the Clapeyron equation, and have confirmed your result. Very nice job.