Clausius - Clapeyron Equation

Calculating Enthalpy
A plot of ln (K) as a function of 1/T for the Sabatier reaction between 400 and 600 ºC. The slope of the line is positive. Multiplying the slope by R gives a H of -185 kJ for the Sabatier reaction.

If you plot the natural logarithm of an equilibrium constant for gases, K_p, as a function of the inverse Kelvin temperature, you get a straight line. This line has a slope of -H/R where:

H = the standard enthalpy change for the reaction
R = the ideal gas constant, 8.314 J/mol·K

It is possible to show that

which is known as the Clausius - Clapeyron equation. If you know the value of K_p at one temperature (K₁ at T₁) this equation allows you to calculate the value of K_p at a second temperature (K₂ at T₂). This equation assumes that H does not change with temperature, which is only partially correct.

Deriving this equation is beyond the scope of this material. However, it is derived from the variation of reaction rates with temperature, which depends on the forward and reverse reaction's activation energies. Notice also that K_p expressed in terms of gas pressures must be used.