We will utilize the Carnot cycle to derive an important relationship, known as the Clausius-Clapeyron Equation or the first latent heat equation. This equation describes how saturated vapor pressure above a liquid changes with temperature and also how the melting point of a solid changes with pressure.
Let the working substance in the cylinder of a Carnot ideal heat engine be a liquid in equilibrium with its saturated vapor and let the initial state of the substance be T− and es,
Leg1-2 Let the cylinder be placed on a source of heat at temperature T and let the substance expand isothermally until a unit mass of the liquid evaporates. In this transformation the pressure *** constant at es, and the substance passes from state 1 to 2. If the specific volumes of liquid and vapor at temperature T are αl and αv, respectively, the increase in the volume of the system in passing from 1 to 2 is (αv − αl). Also the heat absorbed from the source is Lv where Lv is the latent heat of vaporization.
Leg2-3 The cylinder is now placed on a nonconducting stand and a small adiabatic expansion is carried out from 2 to 3 in which the temperature falls from T to T − dT and the pressure from es − des.
Leg3-4 The cylinder is placed on the heat sink at temperature T − dT and an isothermal and isobaric compression is carried out from state 3 to 4 during which vapor is condensed.
Leg4-1 We finalize by an adiabatic compression from es − des and T − dT to es and T.
All the transformations are reversible, so We can define the efficiency as in the Carnot Cycle
And in this specific case of an infinitesimal cycle We can define the efficiency as in the Carnot Cycle
The work done in the cycle is equal to the area enclosed on a p − V diagram.
Therefore
Which is the Clausius-Clapeyron Equation
