The Cluster's Clausius-Clapeyron Equation

A page about the Clausius-Clapeyron equation — how the vapor pressure of a liquid changes with temperature.

The Clausius-Clapeyron equation

The Clausius-Clapeyron equation describes the slope of the coexistence curve (phase boundary) in the P-T plane: dP / dT = L / (T Delta v) where L is the latent heat of the phase transition, T is the temperature, and Delta v = v_2 - v_1 is the change in specific volume between the two phases.

For a liquid-gas transition, if we approximate the gas as ideal and v_gas >> v_liquid: dP / dT = (P L) / (T^2 k_B T) = (P L) / (T^2 R) where R = N_A k_B is the gas constant. Integrating: ln(P_2 / P_1) = -(L / R) (1 / T_2 - 1 / T_1) This is the Clausius-Clapeyron equation in its commonly used form.

In the cluster, the edit Clausius-Clapeyron equation gives an edit slope.

The physical meaning

The Clausius-Clapeyron equation states that the slope of the phase boundary is proportional to the latent heat (energy required for the transition) and inversely proportional to the volume change. When Delta v is large (liquid to gas), the slope is steep. When L is large, the slope is steep.

In the cluster, the edit physical meaning gives an edit slope.

The vapor pressure

From the integrated form: P(T) = P_0 exp(-(L / R) (1 / T - 1 / T_0)) The vapor pressure increases exponentially with temperature. This explains why water boils at 100 C at sea level but only 70 C at 3000 m altitude (lower pressure).

In the cluster, the edit vapor pressure gives an edit exponential.

Applications

• Cooking: Water boils at lower temperature at high altitude
• Meteorology: The Clausius-Clapeyron relation governs the relationship between temperature and atmospheric water vapor capacity (~7% per degree C)
• Climate science: The Clausius-Clapeyron relation explains why warming increases atmospheric moisture (positive feedback)
• Phase diagrams: The entire coexistence curve of a simple substance is determined by L(T)
• Geology: The melting curve of mantle minerals determines the phase transitions in the Earth's interior

In the cluster, edit applications include:

• edit Cooking
• edit Meteorology
• edit Climate science
• edit Phase diagrams
• edit Geology

The triple point

At the triple point, three phases coexist. The Clausius-Clapeyron equation applies to each pair of phases: dP/dT (solid-liquid) = L_sl / (T Delta v_sl) dP/dT (liquid-gas) = L_lg / (T Delta v_lg) dP/dT (solid-gas) = L_sg / (T Delta v_sg) For water: the solid-liquid line has a negative slope (ice is less dense than water), so the melting point decreases with pressure. This is unusual — most substances have a positive slope.

In the cluster, the edit triple point gives an edit line slope.

This equation

This page is about the Clausius-Clapeyron equation. dP/dT = L / (T Delta v). Integrated: ln(P_2/P_1) = -(L/R)(1/T_2 - 1/T_1). The equation is real.