Clausius-Clapeyron Equation

Why it matters in thermodynamics and process design

The Clausius-Clapeyron equation links vapor pressure, temperature, and enthalpy of vaporization. It is widely used to estimate pressure-temperature behavior in evaporation, distillation, and safety analysis.

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

As temperature rises, saturation vapor pressure can increase strongly. The equation provides a practical approximation for process planning, equipment operation, and condition screening in laboratory and industrial workflows.

  • Estimate vapor pressure at a new temperature
  • Determine ΔHvap from measured data points
  • Find target temperature for a required vapor pressure
Crosslink: For equilibrium-constant temperature dependence, see the Van’t Hoff calculator.
Formulas (MathJax)
\[\ln\!\left(\frac{P_2}{P_1}\right)= -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)\]
\[\Delta H_{vap}= -R\,\frac{\ln(P_2/P_1)}{(1/T_2-1/T_1)}\]
\[\frac{1}{T_2}=\frac{1}{T_1}-\frac{R}{\Delta H_{vap}}\ln\!\left(\frac{P_2}{P_1}\right)\]

R = 8.314 J·mol⁻¹·K⁻¹; ΔHvap in kJ/mol is internally converted to J/mol.

Formula symbol legend
  • P₁, P₂ = vapor pressure at T₁ and T₂
  • T₁, T₂ = absolute temperature in Kelvin
  • ΔHvap = molar enthalpy of vaporization
  • R = universal gas constant
  • ln = natural logarithm


Detailed examples
Example 1: Vapor pressure rise for water
Given P₁ = 1.013 bar at T₁ = 373.15 K, ΔHvap = 40.66 kJ/mol, and T₂ = 393.15 K.
Result: P₂ increases substantially above 1 bar, showing strong temperature sensitivity.
Example 2: Data-based ΔHvap estimation
Using two measured pressure-temperature pairs, ΔHvap can be back-calculated.
This is useful for fluid characterization in lab practice and teaching.
Example 3: Target temperature planning
Given P₁, T₁, ΔHvap, and required P₂, solve for T₂ directly.
Helpful for process setpoint planning in evaporation and distillation.
Practical notes
• Always use Kelvin.
• Use consistent pressure units for P₁ and P₂.
• Approximation quality is best over moderate temperature ranges with near-constant ΔHvap.
Deeper context
Model assumptions

The integrated Clausius-Clapeyron form assumes approximately constant ΔHvap over the selected range and sufficiently ideal vapor behavior. For broader ranges, higher-fidelity property models may be required.

Engineering relevance

From vacuum drying to distillation and pressure safety checks, pressure-temperature dependence governs operation windows. Clausius-Clapeyron remains a core first-pass model for rapid engineering estimates.

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