# Specific heat in the curve of equilibrium

• Dario SLC

## Homework Statement

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.

## Homework 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

## 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):
$$c_{CC}=c_p+\frac{l_{l-v}}{T}$$