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.

 

[paranormal]

I can suggest the following approach to solve this problem:

1. Start by understanding the given information and variables: We know that at the point $T_0$, $P_0$ lies on the coexistence curve and the latent heat of vaporization is $l_0$. We also know the values of pressure and temperature at a nearby point on the coexistence curve, $P_0+p$ and $T_0+t$ respectively. Additionally, we have the values of $v$, $c_p$, $\alpha$ and $\kappa _T$ for each phase in the vicinity of the states of interest.

2. Use the given equations to express the entropies of the liquid and solid phases as functions of pressure and temperature: From the given equations, we can express the entropies of the liquid and solid phases as: $s_{liq.}=s_{liq.}(P,T)$ and $s_{sol.}=s_{sol.}(P,T)$. We can then take the difference between these two entropies to get $\Delta s$, which is the change in entropy during the phase transition.

3. Use the given values to calculate the local slope of the coexistence curve: We are given the values of pressure and temperature at a nearby point on the coexistence curve, $P_0+p$ and $T_0+t$ respectively. We can use these values to calculate the slope of the coexistence curve in the P-T plane, which is given by $p/t$.

4. Use the calculated values to find the latent heat at the point $P_0+p$, $T_0+t$: Now, we have all the necessary information to calculate the latent heat at the point $P_0+p$, $T_0+t$. Using equation (1), we can express the latent heat as: $l(T_0+t)=T_{\text{transition}}\Delta s$. We already have the value of $\Delta s$ from step 2, and we can calculate $T_{\text{transition}}$ using the local slope of the coexistence curve from step 3.

5. Check your answer: Finally, it is always a good practice to check your answer and make sure it is reasonable and makes sense. You can do this by comparing your calculated value of latent