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Thermodynamics of Solutions

Solution thermodynamics describes mixtures through partial molar properties and an excess Gibbs energy G_ex, whose composition derivative gives activity coefficients; an ideal solution has G_ex = 0, ΔH_mix = 0 and only entropic mixing.
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.

  • Idealsolution:γi=1,ΔHmix=0,ΔSmix=−RΣxilnxiIdeal solution: \gamma_{i} = 1, \Delta H_{mix} = 0, \Delta S_{mix} = -R Σ x_{i} ln x_{i}
  • Positive deviation: γ > 1; negative: γ < 1 (azeotrope possible)
  • Partialmolarproperties:Gˉi=(∂G/∂ni)T,P,njPartial molar properties: Ḡ_i = (∂G/∂n_{i})_T,P,n_{j}

Topic details

Introduction

This topic explains where activity coefficients come from. You express any extensive property of a mixture as the mole-weighted sum of partial molar properties, define the excess property as the real-minus-ideal difference, and then differentiate an excess-Gibbs-energy model to obtain γ. These γ values close the loop back to vapour–liquid equilibrium.

Key relations & formulas

ΔGmix=RTΣxiln(γixi)\Delta G_{mix} = R T Σ x_{i} ln(\gamma_{i} x_{i})
(excess + ideal)
lnγi=(∂(GexRT)/∂ni)T,P,njln \gamma_{i} = (∂(\frac{G_{ex}}{RT})/∂n_{i})_T,P,n_{j}
(from excess Gibbs energy model)

Formulas (Indian textbook notation)

  • ΔHmix=−RT2Σxi(∂lnγi/∂T)P,x\Delta H_{mix} = -R T^{2} Σ x_{i} (∂ ln \gamma_{i} / ∂T)_P,x

Notation and sign conventions

Relation 1 —
ΔGmix=RTΣxiln\Delta G_{mix} = R T Σ x_{i} ln
ΔGmix=RTΣxiln(γixi)\Delta G_{mix} = R T Σ x_{i} ln(\gamma_{i} x_{i})
(excess + ideal)
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 —
lnγi=ln \gamma_{i} =
lnγi=(∂(GexRT)/∂ni)T,P,njln \gamma_{i} = (∂(\frac{G_{ex}}{RT})/∂n_{i})_T,P,n_{j}
(from excess Gibbs energy model)
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 3 —
ΔHmix=−RT2Σxi\Delta H_{mix} = -R T^{2} Σ x_{i}

Formulas (Indian textbook notation)

  • ΔHmix=−RT2Σxi(∂lnγi/∂T)P,x\Delta H_{mix} = -R T^{2} Σ x_{i} (∂ ln \gamma_{i} / ∂T)_P,x
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.

Concept in depth

A partial molar property is the contribution one added mole makes to a mixture property at fixed T, P and other amounts — it is what actually enters balances, not the pure-component value. Mixing produces two effects: an unavoidable entropy increase (molecules disperse) and an energetic effect captured by the excess Gibbs energy. When unlike interactions are weaker than like interactions, G_ex is positive, γ exceeds one, and a minimum-boiling azeotrope can form; the opposite sign gives a maximum-boiling azeotrope. The Gibbs–Duhem equation links the activity coefficients so they cannot be assigned independently.

Assumptions and validity limits

State assumptions explicitly before using any relation for thermodynamics of solutions — 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 thermodynamics of solutions.
4. Use equation 1:
ΔGmix=RTΣxiln\Delta G_{mix} = R T Σ x_{i} ln
.
5. Use equation 2:
lnγi=ln \gamma_{i} =
.
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

Thermodynamics of Solutions 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 thermodynamics of solutions with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use thermodynamics of solutions?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Common errors are treating pure-component properties as partial molar properties, forgetting the ideal entropy-of-mixing term, and assuming ΔH_mix = 0 for a non-ideal solution. Violating the Gibbs–Duhem consistency between the two γ values is a subtler mistake.

Quick revision checklist

Before attempting thermodynamics of solutions problems, confirm you can:
1.
Idealsolution:γi=1,ΔHmix=0,ΔSmix=−RΣxilnxiIdeal solution: \gamma_{i} = 1, \Delta H_{mix} = 0, \Delta S_{mix} = -R Σ x_{i} ln x_{i}

2. Positive deviation: γ > 1; negative: γ < 1 (azeotrope possible)
3.
Partialmolarproperties:Gˉi=(∂G/∂ni)T,P,njPartial molar properties: Ḡ_i = (∂G/∂n_{i})_T,P,n_{j}
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.

Ideal entropy of mixing

Problem

Compute the molar entropy of mixing for an equimolar ideal binary solution (R = 8.314 J/mol·K).

Solution

ΔS_mix = −R Σ x_i ln x_i = −8.314(0.5 ln0.5 + 0.5 ln0.5) = −8.314 × (−0.693) = 5.76 J/mol·K, positive as mixing always is.

Conceptual check — Thermodynamics of Solutions

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 thermodynamics of solutions." What should a complete answer include?

Exams & GATE

Link γ to VLE — common GATE topic on activity models.

📖 Standard books (India)

  • Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott

    Read: Syllabus unit

    Phase equilibria and chemical thermo

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