The Clausius–Clapeyron Equation and Vapour Pressure

The general Clapeyron equation

At a phase boundary the two phases are in equilibrium, so the chemical potential of the substance is equal in both. Requiring that equality to persist as temperature and pressure change gives

This applies to any phase transition — solid to liquid, liquid to vapour, solid to solid. It gives the slope of the boundary on a phase diagram directly, and it is worth knowing in this general form before the simplified version.

The simplification for vaporisation

For a liquid–vapour transition two approximations are made:

• The volume of the liquid is negligible compared with the vapour, so ΔV is approximately the vapour volume.
• The vapour behaves ideally, so its volume is RT/P.

Substituting and rearranging gives

and integrating, assuming the enthalpy of vaporisation is constant over the range,

The three question types

The plotting version is the most reliable when several data points are available, since it uses all of them and reveals whether the assumption of constant enthalpy holds — curvature would indicate it does not.

The sign trap is the same as in the Arrhenius equation: the slope is negative and the enthalpy is positive, so write ΔH = −slope × R explicitly rather than trusting memory.

Where the assumptions fail

• Near the critical point, the liquid volume is no longer negligible compared with the vapour.
• At high pressure, the vapour is not ideal.
• Over a wide temperature range, the enthalpy of vaporisation is not constant — it decreases with temperature and vanishes at the critical point.

Being asked which assumption fails in a stated situation is a good discriminating question, and it requires knowing what was assumed rather than only the final formula.

Trouton's rule

For many liquids, the entropy of vaporisation at the normal boiling point is roughly constant at about 85 J K−¹ mol−¹. The physical reason is that the disorder gained on vaporising is similar for most liquids.

Liquids that deviate strongly are informative: those with substantial hydrogen bonding have higher values, because the liquid is more ordered than usual so more entropy is gained on vaporising. Identifying a hydrogen-bonded liquid from an anomalous Trouton value is a neat applied question.

Frequently asked questions

Why can the liquid volume be neglected?

Because a vapour occupies roughly a thousand times the volume of the same amount of liquid at ordinary conditions, so the liquid contributes negligibly to ΔV.

Why does the water phase boundary slope backwards?

Because ice is less dense than liquid water, so ΔV for melting is negative. With ΔH positive, the slope dP/dT is negative.

What does curvature in a ln P against 1/T plot indicate?

That the enthalpy of vaporisation is not constant over the range, which is expected over a wide temperature interval.

What does a high Trouton constant tell you?

That the liquid is unusually ordered, typically through hydrogen bonding, so more entropy is gained on vaporisation than for a normal liquid.