Clausius–Clapeyron Vapor Pressure

Worked Example

Consider an idealized substance with reference temperature T₀ = 300 K, reference pressure P₀ = 100,000 Pa, and constant molar latent heat H = 40,000 J/mol.

We want to calculate the vapor pressure at T = 350 K. The universal gas constant is R = 8.314462618 J/(mol K).

The Clausius-Clapeyron equation in integral form is:
ln(P / P₀) = (H / R) * (1 / T₀ - 1 / T)

1. Substitute the values into the log ratio expression: ln(P / 100000) = (40000 / 8.314462618) * (1 / 300 - 1 / 350)
2. Calculate the intermediate log ratio: ln(P / 100000) = 2.290902000856053
3. Calculate final pressure: P = 100000 * exp(2.290902000856053) = 988384.8901775192 Pa

The log-pressure slope with respect to temperature is given by d(ln P)/dT = H / (R T²).
For T = 350 K, this is 40000 / (8.314462618 * 350²) = 0.03927260572896089 K^-1.

Interactive Experiment

You can explore this model interactively in the Clausius–Clapeyron Vapor Pressure module.

1. Vary the temperature and observe the calculated pressure and log ratio changes.
2. Vary the molar latent heat and observe how the log-pressure slope changes, and how the straight-line plot of ln(P) versus 1/T shifts and changes slope according to the latent heat. When H=0, the line is perfectly horizontal and pressure remains constant.

Limitations

The model relies on several idealizations. It assumes that the vapor behaves as an ideal gas and that the molar volume of the liquid phase is negligible compared to the vapor phase.

Furthermore, it assumes the latent heat of vaporization H is constant over the temperature range considered, which is an approximation. Real substances have latent heats that vary with temperature, and this illustrative set of parameters does not describe real water. The mathematical boundary H=0 is treated as a continuous constant-pressure state, though it is not a physical first-order phase transition.

References

• OpenStax Chemistry 2e, Phase Transitions.

For more details, see https://openstax.org/books/chemistry-2e/pages/10-3-phase-transitions. The referenced materials elaborate on the approximations needed to derive the simplified linear relationship.