Intuitive Derivation of the Clausius-Clapeyron Equation

The equilibrium vapor pressure above a pure liquid in a closed container results from equal rates of condensation (R_C) and evaporation (R_E). Expressions for these rates are in equations E-1 and E-2. Equating the rates and solving for the vapor pressure, P_vap, gives E-3.

Here k₁ and k₂ are the rate constants for evaporation and condensation; A is the exposed surface area of liquid in the container, and T is the Kelvin temperature. The temperature dependence of the rate of evaporation, explicitly acknowledged in 1, arises from the necessity for a molecule to escape the potential well of the liquid phase (evaporate). The number of molecules with sufficient thermal energy to escape is governed by the Maxwell-Boltzmann distribution of kinetic energies. It follows that k₁ and R_E are also governed by this distribution and should show a similar temperature dependence. The rate constant for condensation, k₂, is essentially independent of temperature, as no minimum thermal energy is required for a molecule to fall into a potential well. Therefore, from E-3, P_vap should show the same temperature dependence as the evaporative rate constant. It remains only to show the mathematical form of this dependence.

To escape the intermolecular forces in the liquid, a molecule must have kinetic energy which exceeds a critical value KE_o, which will have magnitude about equal to the depth of the liquid potential well. According to the M-B distribution, the fraction of molecules with kinetic energies equal to or exceeding this value is proportional to the exponential in E-4.

Since the evaporative rate constant, k₁, is proportional to the number of molecules with KE > KE_o, E-5 and E-6 follow.

C is a constant of proportionality. Conversion to natural logarithmic form gives E-7.

KE_o is the minimum kinetic energy for escape of a single molecule. Conversion to kinetic energy per mole gives E-8

Replacing the molar minimum KE of escape with the potential well depth per mole, roughly equal to the enthalpy of vaporization of the liquid, gives essentially the Clausius-Clapeyron equation:

The origins of the temperature dependence of vapor pressure in the statistical distribution of molecular kinetic energies is clear.

In Chapter 10, Raoult's Law and a brief explanation of its form were presented. This appendix presents a somewhat more detailed version of the argument used in Chapter 10. Please refer to Figure 10-16 for a visualization of the solvent and solution surfaces. In pure solvent, all surface molecules are solvent. The rate of evaporation is therefore given by equation E-10, where A is the total surface area of liquid:

In the solution, a certain fraction of surface molecules are those of solute, which are non-volatile (cannot escape to the vapor phase). These reduce the surface area available for escape of solvent molecules to a lower value, A', so the solvent evaporation rate is lower than in pure solvent:

Here A' is the surface area occupied by solvent molecules. The fraction of surface area occupied by solvent is directly proportional to the mole fraction of solvent, assuming that solute and solvent molecules are of similar size. Thus

In contrast, molecules returning from the vapor to the liquid phase (condensing) can enter the liquid at any point on the surface, regardless of whether solvent or solute molecules are there, so the rate of condensation is unaffected by the solute molecules at the surface:

P^o_vap and P_vap for solvent and solution, respectively, may be calculated by equating R_C and R_E:

For pure solvent: k₁A = k₂AP^o_vap

For the solution: k₁A' = k₂AP_vap

From equation E-12, A' = X₁A, so

The reduced value of R_E for the solution means that R_C becomes equal to R_E at a lower value of the vapor pressure. From equations E-15 and E-16, we conclude that P_vap = X₁P^o_vap. This is Raoult's Law.