# Specific heat in the curve of equilibrium

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1. Jul 16, 2017

### Dario SLC

1. The problem statement, all variables and given/known data
Consider a system formed by two phases of a substance that consists of a single class of molecules. Determine the specific heat $c$ of a vapor pressure and temperature $p$ $T$ on the curve of liquid-vapor equilibrium. Consider the steam as an ideal gas.
Data: $c_p$, $c_v$, $l$ heat of the phase transition of liquid-vapor.

2. Relevant equations
$$c_X=T\frac{\partial S}{\partial T}_X$$
Clausius-Clapeyron equation:
$$\frac{dp}{dT}=\frac{l_{l-v}}{T(v_l-v_v)}$$
when $v_l$ is the specific volume of the liquid and $v_v$ is the specific volume of the vapor

3. The attempt at a solution
Well I think that like $S=S(T,p(T))$ then:
$$c_CC=T\frac{\partial S}{\partial T}_p+T\frac{\partial S}{\partial p}_T\frac{dp}{dT}$$
for the rule of the chain, and $c_{CC}$ it is the specific heat on the curve Clausius-Clapeyron.
For the Maxwell relation:
$$\frac{\partial S}{\partial p}_T=-\frac{\partial V}{\partial T}_p$$
then gathering all
$$c_{CC}=c_p-\cancel{T}\frac{\partial V}{\partial T}_p\frac{l_{l-v}}{\cancel{T}(v_l-v_v)}$$
when $v_l\ll v_v$ and using that it is a ideal gas for your equation of state:
$$c_{CC}=c_p+\frac{R}{p}\frac{l_{l-v}}{v_v}=c_p+\frac{c_p-c_v}{p}\frac{l_{l-v}}{RT/p}$$
like the presure $p$ it is equal for all curve the expression for specific heat it will (I hope):
$$c_{CC}=c_p+\frac{l_{l-v}}{T}$$

2. Jul 21, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Jul 24, 2017

### Dario SLC

Hi! the problem it has resolved by the same arguments that I showed. The only difference it is that the denominator is $v_v-v_l$, when $v_v$ is the volume of vapor, therefore
$$C_{CC}=c_P-\frac{l_{v-l}}{T}$$
and the reasoning was fine!

Thanks a lot and this it's not anymore problem!