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Qwestrum Engineering360 · Chemical Engineering · Chemical Engineering Thermodynamics

Phase Equilibria

At equilibrium the fugacity of each species is equal in every phase; for vapour–liquid systems this reduces to modified Raoult’s law, y_iP = x_iγ_iP_i^sat, from which bubble points, dew points and K-values follow.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

$Raoult: \gamma_{i} = 1 for ideal solution$
$Bubble point: Σ x_{i} P_{i}^sat = P; dew point: Σ \frac{y_{i}}{P_{i}}^sat = 1$
Gibbs phase rule: F = C − P + 2

Topic details

Introduction

Phase equilibria underpins distillation, absorption and flash design. The topic asks you to compute bubble and dew points, construct T–x–y and P–x–y diagrams, and perform isothermal flash calculations. The governing statement is equality of fugacities; Raoult’s law is the ideal limit and the activity coefficient γ captures liquid-phase non-ideality that produces azeotropes.

Key relations & formulas

f_{i}^L = f_{i}^V

(phase equilibrium, same T and P)

y_{i} P = x_{i} \gamma_{i} P_{i}^sat

(modified Raoult, activity coefficient γ)

K_{i} = \frac{y_{i}}{x_{i}} = \gamma_{i} P_{i}^\frac{sat}{P}

(K-value)

Notation and sign conventions

Relation 1 —

f_{i}^L = f_{i}^V

f_{i}^L = f_{i}^V

(phase equilibrium, same T and P)

Write this relation with symbols exactly as in Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

y_{i} P = x_{i} \gamma_{i} P_{i}^sat

y_{i} P = x_{i} \gamma_{i} P_{i}^sat

(modified Raoult, activity coefficient γ)

Relation 3 —

K_{i} = \frac{y_{i}}{x_{i}} = \gamma_{i} P_{i}^\frac{sat}{P}

K_{i} = \frac{y_{i}}{x_{i}} = \gamma_{i} P_{i}^\frac{sat}{P}

(K-value)

Concept in depth

Equilibrium is a criterion of equal chemical potential, expressed practically as equal fugacity. In the vapour, fugacity is approximated by partial pressure; in the liquid it is the product of activity coefficient, mole fraction and pure-component vapour pressure. The bubble-point condition sums the vapour mole fractions to one, the dew-point condition sums liquid mole fractions to one, and both are simply the two ways of forcing the K-values to be consistent. The Gibbs phase rule tells you how many of T, P and composition you are free to set before the system is fully determined.

Assumptions and validity limits

State assumptions explicitly before using any relation for phase equilibria — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Chemical Engineering Thermodynamics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Chemical Engineering Thermodynamics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to phase equilibria.
4. Use equation 1:

f_{i}^L = f_{i}^V

.
5. Use equation 2:

y_{i} P = x_{i} \gamma_{i} P_{i}^sat

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Phase Equilibria appears in separation and reaction design. In Indian chemical curricula this topic is tested because it connects theory to phase equilibria and property models.

GATE and semester exams often combine phase equilibria with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use phase equilibria?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Errors include assuming ideal solution (γ = 1) when the mixture clearly forms an azeotrope, swapping the bubble- and dew-point summation conditions, and using vapour pressure at the wrong temperature. Forgetting that ΣK-weighted fractions must equal one when flashing is another.

Quick revision checklist

Before attempting phase equilibria problems, confirm you can:

Raoult: \gamma_{i} = 1 for ideal solution

Bubble point: Σ x_{i} P_{i}^sat = P; dew point: Σ \frac{y_{i}}{P_{i}}^sat = 1

3. Gibbs phase rule: F = C − P + 2

Revise the solved examples in Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Bubble point of an ideal binary

Problem

A liquid is 40 mol% A, 60 mol% B. At the temperature of interest P_A^sat = 120 kPa, P_B^sat = 60 kPa. Find the bubble pressure.

Solution

Assuming ideal (γ = 1): P = x_A P_A^sat + x_B P_B^sat = 0.4×120 + 0.6×60 = 48 + 36 = 84 kPa. Vapour composition y_A = 48/84 = 0.571.

Conceptual check — Phase Equilibria

Problem

In a Chemical Engineering Thermodynamics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of phase equilibria." What should a complete answer include?

Exams & GATE

VLE flash: Rachford-Rice Σ z_i(K_i−1)/(1+β(K_i−1)) = 0.

📖 Standard books (India)

Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott
Read: Syllabus unit
Phase equilibria and chemical thermo

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📚Key formulas & points