What are degrees of freedom in the phase rule, and what do invariant and univariant systems mean?

In one line

Degrees of freedom (F) are the minimum number of independent variables, such as temperature, pressure and concentration, that must be fixed to define the system completely at equilibrium.

Explained simply

Think of degrees of freedom as how many settings you can change without destroying the balance. If F = 2, you can change temperature and pressure independently and the same phases stay. If F = 1, change one and the other is automatically decided. If F = 0, you cannot change anything; even a tiny change makes a phase vanish.

How it works

1. Identify C and P.
2. F = C − P + 2 (or F′ = C − P + 1 for condensed systems at constant pressure).
3. Name the system by F:
   • F = 0: invariant / non-variant
   • F = 1: univariant / monovariant
   • F = 2: bivariant
   • F = 3: trivariant

Key formula

\( F = C - P + 2 \)

For a one-component system, the maximum F is 2 (when P = 1), so its phase diagram can be drawn on a T–P graph. For two-component systems, F can be 3 (T, P, composition), so pressure is usually fixed to draw the diagram on a T–composition graph.

Quick example

• Gas in a cylinder (C = 1, P = 1): F = 2. You must state both T and P to define it, e.g., the volume of a gas depends on both.
• Water ⇌ vapour (C = 1, P = 2): F = 1. At 100 °C the vapour pressure must be 760 mm Hg; you cannot choose both.
• Ice ⇌ water ⇌ vapour (C = 1, P = 3): F = 0. Triple point: one fixed T and P.
• CaCO₃(s) ⇌ CaO(s) + CO₂(g) (C = 2, P = 3): F = 1. At a given temperature the CO₂ pressure (dissociation pressure) is fixed.
• NaCl solution (C = 2, P = 1, unsaturated): F = 3 (T, P and concentration).

Common mistakes

• Getting a negative F and not noticing a counting error. F can never be negative.
• Confusing degrees of freedom with the number of phases.
• Forgetting concentration as a variable in multi-component systems.

How MU asks it

Usually as part of "Explain phase, component and degree of freedom with examples" (5–6 marks), or as short numericals: "Calculate the degrees of freedom for..." (1 mark each). For full marks: definition, the meaning of 0/1/2, and at least two examples showing the calculation and its physical meaning.