Clausius-Clapeyron Calculator – Vapor Pressure, Boiling Points and ΔH_vap
The Clausius-Clapeyron equation links the vapor pressure of a pure liquid (or solid) to temperature through its enthalpy of vaporization. This calculator uses it three ways: to find the vapor pressure at a new temperature or the boiling point at a new pressure, to get ΔH_vap from two points or a least-squares fit of your own data, and to follow the melting line of ice with the full Clapeyron equation. British texts write vapour pressure; the physics is the same.
Where the equation comes from
Émile Clapeyron showed in 1834 that along any phase boundary dP/dT = ΔH/(T·ΔV). In 1850 Rudolf Clausius simplified it for evaporation: the vapor's volume dwarfs the liquid's and behaves almost ideally, so ΔV ≈ RT/P. That gives d ln P/dT = ΔH_vap/(RT²), and with ΔH_vap constant, ln(P₂/P₁) = −(ΔH_vap/R)·(1/T₂ − 1/T₁).
Boiling is a pressure match
A liquid boils when its vapor pressure equals the pressure around it. Climb a mountain and the air pressure falls, so water boils colder: about 94.5 °C in Denver and below 70 °C on Everest, roughly 1 °C lower for every 300 m near sea level. A pressure cooker does the opposite: at about 15 psi gauge, water boils near 121 °C. Vacuum distillation and rotary evaporators lower the pressure until a solvent boils at a gentle bath temperature; acetone boils at 40 °C under about 590 mbar.
Reading a ln P vs 1/T plot
Plot ln P against 1/T and you get a straight line whose slope is −ΔH_vap/R. Example 2B in the Fit data mode uses seven vapor pressures between 20 and 50 °C. The fit gives ΔH_vap = 30.51 ± 0.33 kJ/mol, and extending the line to 1 atm predicts a normal boiling point of 68.72 °C, with a 95 % interval of 67.61–69.82 °C. The boiling point lies outside the measured range, and the interval widens the further you extrapolate, because any uncertainty in the slope grows with distance from the centre of the data. n-Hexane (68.73 °C) falls inside it, which makes it a candidate, not an identification.
Trouton's rule
For many liquids, ΔH_vap divided by the normal boiling point is close to 85–88 J/(mol·K), a pattern called Trouton's rule. Water's ratio is about 109 J/(mol·K). Hydrogen bonds hold the liquid in an ordered network, so escaping into the vapor gains more entropy than usual. Alcohols are high for the same reason, which is why a Trouton estimate undershoots their ΔH_vap.
When the equation breaks
The simple form assumes ΔH_vap is constant, the vapor is ideal and the liquid's volume is negligible. The first assumption fails over wide ranges: start from water's boiling point and predict its vapor pressure at 25 °C, and you overshoot the measured 3170 Pa by about 18.5 %, because ΔH_vap grows as the temperature falls. Adding the heat-capacity change ΔCp (about −42 J/(mol·K) for water) cuts the error to a few percent. Near the critical point the vapor stops behaving ideally. Data tables therefore use empirical fits such as the Antoine or Wagner equations.
Melting under pressure
For melting, neither phase is a gas, so the full Clapeyron equation is needed. Ice is less dense than water, so ΔV_fus is negative and pressure lowers the melting point, but only by about 0.0074 °C per bar. A skater's blade lowers it by under 0.2 °C, nowhere near enough to explain skating at −10 °C; most of the slipperiness is credited to a thin premelted surface layer and frictional heating. The classic regelation demonstration, a weighted wire that slowly cuts through a block of ice which refreezes behind it, works only because the ice is held right at 0 °C, where a small shift is enough.
Same maths, different names
The Van 't Hoff Equation Calculator does the same thing for equilibrium constants, and the Arrhenius Equation Calculator for rate constants: each is ln of something against 1/T. For mixtures, use the Raoult's Law Calculator; for heat amounts, the Latent Heat Calculator.