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Qwestrum Engineering360 · Chemical Engineering · Mass Transfer

Molecular Diffusion

Molecular diffusion is described by Fick’s law, flux proportional to concentration gradient; the key exam decision is whether the situation is equimolar counter-diffusion or diffusion of A through stagnant B, which adds the (1 − y_A) drift-flux correction.
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.

  • Units of D: m²/s; flux N: kmol/m²s
  • Diffusion in solids much slower than liquids/gases
  • Steady-state diffusion through a film controls interphase transfer

Topic details

Introduction

Treybal presents molecular diffusion as the microscopic basis of every separation. You must recognise the two canonical cases — equimolar counter-diffusion (as in distillation) and diffusion through a stagnant film (as in absorption or evaporation) — because the stagnant case carries a logarithmic mean concentration and a bulk-flow correction that the equimolar case lacks.

Key relations & formulas

NA=−DABdCAdxN_{A} = -D_{AB} \frac{dC_{A}}{dx}
(Fick first law, equimolar counter-diffusion)
NA=−(DAB(1−yA))CdyAdxN_{A} = -(\frac{D_{AB}}{(1 - y_{A})}) C \frac{dy_{A}}{dx}
(Stefan diffusion through stagnant B)
DAB∝T1.5PD_{AB} ∝ T^1.\frac{5}{P}
(Chapman-Enskog, gas diffusivity trend)

Notation and sign conventions

Relation 1 —
NA=−DABdCAdxN_{A} = -D_{AB} \frac{dC_{A}}{dx}
NA=−DABdCAdxN_{A} = -D_{AB} \frac{dC_{A}}{dx}
(Fick first law, equimolar counter-diffusion)
Write this relation with symbols exactly as in Mass Transfer Operations — Treybal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
NA=−N_{A} = -
NA=−(DAB(1−yA))CdyAdxN_{A} = -(\frac{D_{AB}}{(1 - y_{A})}) C \frac{dy_{A}}{dx}
(Stefan diffusion through stagnant B)
Write this relation with symbols exactly as in Mass Transfer Operations — Treybal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
DAB∝T1.5PD_{AB} ∝ T^1.\frac{5}{P}
DAB∝T1.5PD_{AB} ∝ T^1.\frac{5}{P}
(Chapman-Enskog, gas diffusivity trend)
Write this relation with symbols exactly as in Mass Transfer Operations — Treybal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Concept in depth

Fick’s first law says a species moves down its concentration gradient at a rate set by the diffusivity D. In equimolar counter-diffusion, A and B move in opposite directions at equal molar rates, so there is no net bulk flow and Fick’s law applies directly. When B is stagnant (it cannot cross a phase boundary), the diffusing A drags the mixture along, creating a convective drift that steepens the profile — this is Stefan diffusion, and the flux is amplified by the factor 1/(1 − y_A) or, integrated, by the log-mean of the inert concentration. Gas diffusivities rise with temperature and fall with pressure; liquid and solid diffusivities are orders of magnitude smaller.

Assumptions and validity limits

State assumptions explicitly before using any relation for molecular diffusion — 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 Mass Transfer 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 Mass Transfer 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 molecular diffusion.
4. Use equation 1:
NA=−DABdCAdxN_{A} = -D_{AB} \frac{dC_{A}}{dx}
.
5. Use equation 2:
NA=−N_{A} = -
.
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

Molecular Diffusion appears in distillation, absorption, and drying. In Indian chemical curricula this topic is tested because it connects theory to diffusion and interphase transfer.
GATE and semester exams often combine molecular diffusion with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use molecular diffusion?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

The signature error is using the equimolar formula for a stagnant-film problem, thereby dropping the drift-flux correction and the log-mean term. Others include unit slips between concentration and mole fraction and using gas-phase diffusivity trends for liquids.

Quick revision checklist

Before attempting molecular diffusion problems, confirm you can:
1. Units of D: m²/s; flux N: kmol/m²s
2. Diffusion in solids much slower than liquids/gases
3. Steady-state diffusion through a film controls interphase transfer
Revise the solved examples in Mass Transfer Operations — Treybal 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.

Equimolar counter-diffusion flux

Problem

For equimolar counter-diffusion, D_AB = 2×10⁻⁵ m²/s, film thickness 2 mm, ΔC_A = 1.5 kmol/m³. Find N_A.

Solution

N_A = D_AB·ΔC_A/δ = (2×10⁻⁵ × 1.5)/0.002 = 1.5×10⁻² kmol/m²·s. No drift correction is needed because bulk flow is zero.

Conceptual check — Molecular Diffusion

Problem

In a Mass Transfer semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of molecular diffusion." What should a complete answer include?

Exams & GATE

Treybal Ch. 2 — identify stagnant vs equimolar conditions.

📖 Standard books (India)

  • Mass Transfer Operations — Treybal

    Read: Syllabus unit

    Absorption, distillation, and extraction

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