What this calculator does
Gibbs' phase rule connects the number of independently variable conditions of a system, its degrees of freedom, to how many chemically independent components it contains and how many phases are coexisting at equilibrium. It is one of the foundational relationships in physical chemistry and materials science, underlying every phase diagram.
Degrees of freedom is how many intensive variables (such as temperature, pressure, or composition) can be changed independently without a phase appearing or disappearing. A single-component system with one phase, such as liquid water alone, has two degrees of freedom (temperature and pressure can both vary); at the triple point, where three phases coexist, there are zero degrees of freedom left, and any change forces a phase to vanish.
The formula
Subtract the number of coexisting phases from the number of components, then add 2 for a standard system where both temperature and pressure can vary, or add 1 for a condensed system where pressure is held fixed and only temperature and composition matter.
| Term | Meaning |
|---|---|
| F | Degrees of freedom: the number of intensive variables that can be changed independently. |
| C | The number of chemically independent components in the system. |
| P | The number of phases (solid, liquid, gas, or distinct solid forms) coexisting at equilibrium. |
The inputs explained
| Field | What to enter |
|---|---|
| Number of components (C) | The number of chemically independent components, such as elements or compounds that cannot be produced from the others present. |
| Number of phases (P) | The number of distinct phases currently coexisting at equilibrium in the system. |
| System type | Choose the condensed option for systems where pressure is fixed (common in metallurgy and solid-state phase diagrams) and only temperature and composition vary. |
When to use it
Reading a phase diagram
At any point on a standard pressure-temperature phase diagram, the phase rule tells you how many of those two axes you are free to move along without a phase boundary being crossed.
Identifying an invariant point
A triple point, where three phases of a single component coexist, has zero degrees of freedom by the phase rule, which is exactly why it appears as a single fixed point on a phase diagram rather than a line or region.
Working with alloy or condensed systems
Metallurgical phase diagrams are usually drawn at fixed (atmospheric) pressure, so the condensed form of the rule, F = C − P + 1, applies instead of the standard version.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How degrees of freedom change as more phases coexist
A fixed 3-component system, as the number of coexisting phases increases.
Questions
What does it mean if F comes out negative?
A negative result signals that the proposed combination of components and phases cannot coexist at equilibrium under the assumptions of the rule; more phases have been specified than the system, at that number of components, can actually support simultaneously.
What counts as a "component"?
A component is a chemically independent constituent, meaning its amount cannot be expressed in terms of the others present. A pure substance is one component regardless of how many phases it forms; a solution of two substances that do not react is two components.
When should I use the condensed form of the rule (+1 instead of +2)?
Use it when pressure has been fixed as an experimental condition, typically atmospheric pressure, and does not count as a variable degree of freedom. This is standard for most solid and liquid alloy phase diagrams, where pressure effects are negligible.
Why does a pure liquid have 2 degrees of freedom?
With one component and one phase, F = 1 − 1 + 2 = 2: both temperature and pressure can be changed independently within the liquid region of the phase diagram without a new phase appearing, as long as neither crosses a phase boundary.
This calculator addresses degrees of freedom only; it does not attempt to identify or balance the phases themselves. For concentration-based equilibrium calculations elsewhere in a system, see the Kp / Kc equilibrium constant calculator.