Clausius-Clapeyron Equation
The Clausius-Clapeyron equation relates vapor pressure changes to temperature and heat of vaporization.
Learn with examples.
The Formula
The Clausius-Clapeyron equation describes how the vapor pressure of a substance changes with temperature. It allows you to calculate the vapor pressure at any temperature if you know it at another temperature and the heat of vaporization.
Rudolf Clausius and Benoit Paul Emile Clapeyron developed this relationship in the 19th century. It is derived from thermodynamic principles applied to the liquid-gas phase boundary. The equation assumes the enthalpy of vaporization remains constant over the temperature range.
This equation explains why liquids boil at lower temperatures at high altitudes where atmospheric pressure is lower. It also predicts how pressure cookers work by showing that higher pressure raises the boiling point. The equation applies to any phase transition, including sublimation (solid to gas).
Variables
| Symbol | Meaning |
|---|---|
| P₁, P₂ | Vapor pressures at temperatures T₁ and T₂ |
| ΔHvap | Enthalpy of vaporization (J/mol) |
| R | Gas constant (8.314 J/(mol·K)) |
| T₁, T₂ | Temperatures (kelvin, K) |
Example 1
Water boils at 100°C at 1 atm. Its heat of vaporization is 40,700 J/mol. At what temperature does it boil at 0.5 atm?
ln(0.5/1.0) = −(40700/8.314)(1/T₂ − 1/373)
−0.6931 = −4893(1/T₂ − 0.002681)
1/T₂ = 0.002681 + 0.0001416 = 0.002823
T₂ = 354 K = 81°C (water boils about 19°C cooler at half atmospheric pressure)
Example 2
Ethanol has vapor pressure 44 mmHg at 20°C and 135 mmHg at 50°C. What is its heat of vaporization?
ln(135/44) = −(ΔH/8.314)(1/323 − 1/293)
1.121 = −(ΔH/8.314)(−3.17 × 10⁻⁴)
ΔH = 1.121 × 8.314 / 3.17 × 10⁻⁴ ≈ 29,400 J/mol = 29.4 kJ/mol
When to Use It
Use the Clausius-Clapeyron equation to calculate how vapor pressure changes with temperature.
- Predicting boiling points at different altitudes
- Designing distillation and evaporation processes
- Determining heats of vaporization from pressure-temperature data
- Understanding weather phenomena like cloud formation