Gibbs Phase Rule Calculator (degrees of freedom F = C − P + 2)

Understanding the Gibbs phase rule calculator — concept and principle

This calculator applies Gibbs' phase rule to find the number of degrees of freedom of a system in equilibrium: how many intensive variables, such as temperature, pressure and composition, you can change independently without a phase appearing or disappearing. It is used in physical chemistry, materials science and geology to read phase diagrams.

With liquid and vapour in equilibrium you can choose temperature or pressure, not both: the boiling curve.

How it is calculated

F = C − P + 2 (temperature and pressure both free)

F = C − P + 1 (pressure or temperature held fixed)

C = N − r − s

N is the number of chemical species, r the number of independent reactions, s other fixed constraints, P the number of phases.
Most phases that can coexist = C + 2 (or C + 1 when one variable is fixed).
Total intensive variables = (C − 1) × P + 2.

The principle

Each phase has a temperature, a pressure and C − 1 independent composition fractions. At equilibrium, temperature and pressure are the same in all phases, and the chemical potential of each component must be equal in every phase. Each equality removes one free variable. Counting variables minus constraints gives F = C − P + 2. Reactions and fixed ratios reduce the number of truly independent components.

Worked example

Default inputs: C = 1, P = 2, r = 0, s = 0, temperature and pressure free.

Independent components = 1 − 0 − 0 = 1.
F = 1 − 2 + 2 = 1, so the equilibrium is univariant (a line).
Most phases that can coexist = 1 + 2 = 3.
Total intensive variables = (1 − 1) × 2 + 2 = 2.

How to read the result

F = 0 is an invariant point: for pure water, ice, liquid and vapour coexist only at the triple point. F = 1 is a line: liquid water and steam coexist along the boiling curve, so fixing the pressure fixes the boiling temperature. F = 2 is an area, where you can vary temperature and pressure freely within a single phase.

Common mistakes

Counting species instead of independent components when reactions link them.
Using +2 for a diagram drawn at fixed pressure, such as most alloy diagrams.
Counting two solid forms of the same substance as one phase. Each distinct crystal structure is a separate phase.

Keep in mind: The phase rule counts degrees of freedom at equilibrium only. It says nothing about which phases form or how fast.

Frequently asked questions

Why are only three phases possible for a pure substance?

With C = 1, F = 3 − P cannot be negative, so P is at most 3.

Why do alloy diagrams use F = C − P + 1?

They are usually drawn at constant atmospheric pressure, which removes one variable.

Is a mixture of gases one phase?

Yes. Gases mix completely, so they form a single phase.

Sources: P. Atkins and J. de Paula, Physical Chemistry (Oxford University Press), chapter on phase diagrams

Checked 9 Nov 2026. J. W. Gibbs’ phase rule F = C − P + 2 and the counting of independent components (C = N − r − s): P. Atkins & J. de Paula, Physical Chemistry — standard result.

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