---
title: "Phase Diagrams, the Clapeyron Equation and the Phase Rule"
description: "A phase boundary is a line where two chemical potentials are equal. Setting them equal gives an equation for the slope of the line, and counting the constraints gives the number of variables that can "
canonical: https://lightmysky.com/learn/science/phase-diagrams-the-clapeyron-equation-and-the-phase-rule-mt_U-zj6r4WFM
source: https://lightmysky.com/learn/science/phase-diagrams-the-clapeyron-equation-and-the-phase-rule-mt_U-zj6r4WFM.md
retrieved: 2026-11-12
---

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# Phase Diagrams, the Clapeyron Equation and the Phase Rule

A phase boundary is a line where two chemical potentials are equal. Setting them equal gives an equation for the slope of the line, and counting the constraints gives the number of variables that can still be changed freely.

Subject: Science · Area: Chemistry · Ages 20 to 22
Page: https://lightmysky.com/learn/science/phase-diagrams-the-clapeyron-equation-and-the-phase-rule-mt_U-zj6r4WFM

## Ready when they can

- Reads a one-component phase diagram and names the triple point and critical point
- Derives the slope of a boundary from equal chemical potentials and explains the unusual slope for water
- Applies the Clausius-Clapeyron form to find an enthalpy of vaporisation from vapour pressure data
- Uses the phase rule to say how many variables are free in a stated system

## Lesson: Reading the map of phases

A one-component phase diagram maps which phase is stable at each pressure and temperature. Lines mark where two phases coexist. The triple point is the single spot where solid, liquid, and gas all meet. The liquid-gas line ends at the critical point, beyond which the two merge into one supercritical fluid.

**Example.** Each boundary is a line of equal chemical potentials, and that equality sets its slope through the Clapeyron equation. Water leans the odd way: its melting line slopes back because ice is less dense than liquid water. So squeezing ice at fixed temperature melts it.

For vaporisation the equation simplifies, because the gas takes far more room than the liquid. Plot the natural log of vapour pressure against one over temperature, and the slope is minus the enthalpy of vaporisation divided by R. The derivation neglects the liquid volume and treats the vapour as an ideal gas with Z near 1; the ideal gas step is what brings in the log. The plot stays straight only while the enthalpy is roughly constant over the range.

The phase rule counts your freedom: add the components to two, then subtract the phases. Pure water at its triple point has one component and three phases, so freedom is zero. Zero freedom means all three phases meet at exactly one temperature and pressure.

**Recap.** Read regions for stable phases, slopes for equal potentials, and count freedom with components plus two minus phases.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Real Gases, Fugacity and Activity](https://lightmysky.com/learn/science/real-gases-fugacity-and-activity-mt_HzRqV-KXfN)
- [Chemical Potential and Gibbs Energy That Depends on Composition](https://lightmysky.com/learn/science/chemical-potential-and-gibbs-energy-that-depends-on-composition-mt_Tdl6K-gFO3)

## Optional connections

- [The Laws of Logarithms](https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY)

## Opens up

- [Ideal and Real Solutions: Raoult, Henry and Excess Functions](https://lightmysky.com/learn/science/ideal-and-real-solutions-raoult-henry-and-excess-functions-mt_DRHoU0yg3Z)
- [Glass Transition, Crystallinity and Entanglement in Polymers](https://lightmysky.com/learn/science/glass-transition-crystallinity-and-entanglement-in-polymers-mt_vGEYO5ILqt)