Clausius–Clapeyron Equation

\[\ln\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)\]

Variables

P2vapor pressure at temperature T2

P1vapor pressure at temperature T1

dHvap[ΔHvap]enthalpy of vaporization (J/mol)

Rgas constant (8.314 J·mol⁻¹·K⁻¹)

T2final absolute temperature (K)

T1initial absolute temperature (K)

Description

What is this formula?

The Clausius–Clapeyron equation relates vapor pressure to temperature and allows prediction of vapor pressure changes from thermodynamic data.

It is one of the most important equations in phase equilibrium and thermodynamics.

When to use it

Use this equation to estimate vapor pressure at a new temperature, determine enthalpy of vaporization, or analyze liquid-vapor equilibrium behavior.

Example

Given:

P1 = 101.3 kPa

T1 = 373.15 K

T2 = 393.15 K

ΔHvap = 40650 J/mol

R = 8.314 J·mol^-1·K^-1

Formula:

P2 = P1·e^((-ΔHvap/R)(1/T2−1/T1))

Substitution:

P2 = 101.3 × e^((-40650/8.314)(1/393.15−1/373.15))

Result:

P2 ≈ 193 kPa

The vapor pressure increases significantly as temperature rises.

Applications

Phase equilibrium

Distillation design

Chemical engineering

Thermodynamics

Atmospheric science

Materials processing

Note

This integrated form assumes that the enthalpy of vaporization remains approximately constant over the temperature range considered and that the vapor behaves ideally.

For large temperature intervals or highly non-ideal systems, more advanced vapor-pressure correlations may provide greater accuracy.

The Clausius–Clapeyron equation is widely used for estimating boiling points, vapor pressures, and phase-transition properties.