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

 

[anathema]

Thank you for your interesting question regarding the specific heat of a substance at a given vapor pressure and temperature on the liquid-vapor equilibrium curve. I would like to provide my input and attempt to solve this problem.

Firstly, let's define some variables for better understanding. Let $c$ be the specific heat of the substance, $p$ be the vapor pressure, $T$ be the temperature, $l$ be the heat of phase transition from liquid to vapor, and $v_l$ and $v_v$ be the specific volumes of the liquid and vapor, respectively.

Using the definition of specific heat, we can write:
$$c = T\frac{\partial S}{\partial T}$$
where $S$ is the entropy of the substance.

Next, we can use the Clausius-Clapeyron equation to relate the change in vapor pressure $p$ with respect to temperature $T$ to the heat of phase transition $l$ and the specific volumes of the liquid and vapor, as follows:
$$\frac{dp}{dT} = \frac{l}{T(v_l - v_v)}$$

Now, using the chain rule, we can write the specific heat at a given vapor pressure and temperature as:
$$c_{p} = T\frac{\partial S}{\partial T}\bigg\rvert_{p} = T\frac{\partial S}{\partial T}\bigg\rvert_{T}\frac{dT}{dp}\bigg\rvert_{p}$$

Substituting the expression for $\frac{dp}{dT}$ from the Clausius-Clapeyron equation, we get:
$$c_{p} = T\frac{\partial S}{\partial T}\bigg\rvert_{T}\frac{l}{T(v_l - v_v)}$$

Next, we can use the Maxwell relation to relate the change in entropy with respect to pressure to the change in volume with respect to temperature, as follows:
$$\frac{\partial S}{\partial p}\bigg\rvert_{T} = -\frac{\partial V}{\partial T}\bigg\rvert_{p}$$

Substituting this in the previous equation, we get:
$$c_{p} = T\left(-\frac{\partial V}{\partial T}\bigg\rvert_{p}\right)\frac{l}{T