Chemistry · Lesson

Phase equilibria, mixtures and phase diagrams

Read a phase diagram as a map of equilibrium constraints and conserved composition. Compare a homogeneous mixture with allowed phase combinations before claiming stability.

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By the end, you can

Connect common-tangent phase coexistence to component chemical potentials.
Apply a lever rule using the same amount basis as the composition axis.
Distinguish intensive phase-rule freedom and local coexistence slopes.

Compare all allowed phase combinations

At fixed T,p, a binary homogeneous phase has molar Gibbs energy g(x), where x is the mole fraction of B. A mixture of two phases at compositions xα and xβ has an amount-weighted average of their molar energies. This is a possible competing state at an intermediate overall composition.

For a differentiable g(x), μA = g − xg′ and μB = g + (1 − x)g′. A common tangent between two composition points has the same slope and intercept at both ends, matching both component potentials across phases. Stable phase combinations form the lower convex envelope, after comparison with all permitted competing phases. A tangent above a lower competing state would not establish stable coexistence.

An invented regular-solution curve separates local and global tests

Use the specified symmetric teaching model g_mix/(RT) = x ln x + (1 − x) ln(1 − x) + 3x(1 − x). Linear pure-component reference terms are omitted; adding them would not change its coexistence compositions. The endpoints x = 0 and 1 use the continuous limit of x ln x. This is a model, not a measured diagram of a named material.

The symmetric coexistence endpoints are approximately 0.07072 and 0.92928, where the common tangent is horizontal. At overall x = 0.500, a half-and-half combination lies at g_mix/(RT) ≈ −0.05834, below the homogeneous value +0.05685. The homogeneous midpoint also has negative curvature. The spinodal endpoints, where the curvature changes sign, are approximately 0.21132 and 0.78868; they differ from the coexistence endpoints.

A two-phase combination can lie below the homogeneous curve

g_mix / (RT)

Mole fraction x of B

Homogeneous midpoint and spinodal limits
Coexisting-phase combination
— Homogeneous regular-solution g_mix/(RT)

Underlying data

Mole fraction x of B	g_mix / (RT)
0.21132486540518708	-0.015706736463554416 · Homogeneous midpoint and spinodal limits
0.5	0.056852819440054714 · Homogeneous midpoint and spinodal limits
0.7886751345948129	-0.01570673646355436 · Homogeneous midpoint and spinodal limits
0.07072018167994484	-0.058341349441440093 · Coexisting-phase combination
0.3	-0.058341349441440093 · Coexisting-phase combination
0.5	-0.058341349441440093 · Coexisting-phase combination
0.7	-0.058341349441440093 · Coexisting-phase combination
0.9292798183200551	-0.058341349441440093 · Coexisting-phase combination

Original fixed-T,p binary regular-solution model, χ=3, with linear pure-component references omitted. The dashed curve shows homogeneous g_mix/(RT). Lower markers between the coexistence endpoints show allowed coexisting-phase combinations at the common-tangent height; no separate tangent line is drawn. Spinodal markers identify local-curvature limits, not coexistence endpoints. Surface and kinetic costs are excluded.

Conserve component amount on the chosen axis

For two supplied coexisting compositions xα < z < xβ, the fraction of total chemical amount in phase β is fβ = (z − xα)/(xβ − xα), and fα = 1 − fβ. The overall mole fraction obeys z = fαxα + fβxβ. This is the lever rule on an amount basis.

A separate invented tie line has xα = 0.200, xβ = 0.800 and overall z = 0.400. The β amount fraction is 1/3, not the distance fraction toward α by mistake. Phase amounts are not automatically phase masses or volumes. A mass-fraction axis instead requires a consistent mass balance; volumes additionally need density information.

Count intensive freedom and differentiate the coexistence condition

The conventional bulk equilibrium phase rule is F = C − P + 2, with C independent components and P phases. It counts independent intensive variables under the model’s constraints. Independent reactions or charge constraints affect the component count; externally fixing pressure removes one available freedom. Phase amount fractions are not counted by this F.

For pure-substance two-phase coexistence, the Clapeyron slope is dp/dT = ΔSₘ/ΔVₘ = ΔHₘ/(TΔVₘ), with changes taken in the same direction. Its sign depends on the volume change as well as the latent enthalpy. The ideal-vapor Clausius–Clapeyron approximation requires additional assumptions and is unsuitable for a generic solid–liquid slope or near-critical region.

Worked examples

Recover phase amounts from a mole-fraction tie line

Use the separate invented equilibrium tie line xα = 0.200, xβ = 0.800 and overall z = 0.400 for a total amount of 10.0 mol. Find the two phase amounts.

fβ = (0.400 − 0.200)/(0.800 − 0.200) = 1/3.
nβ = 10.0/3 ≈ 3.33 mol and nα = 20.0/3 ≈ 6.67 mol.
Check B amount: 0.200(20/3) + 0.800(10/3) = 4.00 mol.
Divide by 10.0 mol to recover overall z = 0.400.

Approximately 6.67 mol in phase α and 3.33 mol in β. These amount fractions do not determine volume fractions.

Keep a coexistence slope local

At one invented pure-substance coexistence point, T = 3.00 × 10² K, ΔHₘ = 6.00 × 10³ J mol⁻¹ and ΔVₘ = 1.00 × 10⁻⁶ m³ mol⁻¹ in the same transition direction. Find dp/dT.

Use ΔHₘ/(TΔVₘ), retaining the common molar basis.
The denominator is 300 × 10⁻⁶ K m³ mol⁻¹.
The result is 2.00 × 10⁷ Pa K⁻¹, or 20.0 MPa K⁻¹.
If the supplied volume change instead had the opposite sign, the slope would reverse sign.

The supplied local slope is +20.0 MPa K⁻¹. This single point does not establish a finite-temperature straight-line phase boundary.

Try it yourself

For xα = 0.200, xβ = 0.800 and z = 0.500, what is the β amount fraction?
Show the answer and reasoning
0.500.
The overall composition is halfway between the supplied endpoints.
For C = 2 and P = 2, what is F before and after externally fixing pressure?
Show the answer and reasoning
F = 2 before; one intensive freedom remains after fixing pressure.
Use C − P + 2, then apply the independent imposed constraint.
Does positive local curvature alone exclude a lower two-phase combination?
Show the answer and reasoning
No.
Local stability and global comparison are different tests.
Does a mole-fraction lever rule automatically produce phase volume fractions?
Show the answer and reasoning
No.
It gives chemical-amount fractions; volume conversion needs additional data.

What to carry forward

A common tangent matches both component potentials.
Compare local curvature with global competing phase states.
Use the same conserved basis as the composition axis.
Phase-rule freedom and coexistence slopes are intensive statements.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.