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Topic GuidePart of Mass Transfer

Diffusion & Mass Transfer
Coefficients Guide

Fick's law, stagnant-film diffusion, equimolar counterdiffusion, and Sherwood/Schmidt numbers for the GATE Chemical Engineering paper.

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Overview

Before any equipment-specific calculation in distillation, absorption, or drying can be set up, there's a more basic question: how does a species actually move through a stationary or flowing medium at the molecular level? Diffusion & Mass Transfer Coefficients answers that, it's the microscopic foundation underneath every applied Mass Transfer subtopic in the GATE CH syllabus, playing the same conceptual role that conduction and convection fundamentals play for Heat Transfer.

The syllabus starts with Fick's first law of molecular diffusion, then works through the two classic 1-D diffusion problems that show up constantly on GATE: diffusion through a stagnant film (where one species is diffusing but the other is not moving at all, giving a slightly-more-complex "log-mean" concentration form) and equimolar counterdiffusion (where two species diffuse in opposite directions at equal molar rates, giving a simpler linear concentration profile). From there, dimensional analysis extends heat-transfer-style correlations (Nusselt, Prandtl) to their mass-transfer analogs (Sherwood, Schmidt), and film theory / two-film theory (introduced in the parent Mass Transfer guide) formalizes how these molecular-diffusion ideas scale up to real gas-liquid or gas-solid interfaces.

The heat-and-mass-transfer analogy is the single most useful mental shortcut in this subtopic: Fick's law mirrors Fourier's law, the mass transfer coefficient k mirrors the heat transfer coefficient h, and Sherwood/Schmidt numbers mirror Nusselt/Prandtl, recognizing this parallel structure means half of what you already know from the Heat Transfer guide transfers over directly.

Real GATE CH PYQ Frequency (2024–2026)

No questions in our 2024–2026 archive were tagged to this specific subtopic on its own, they're grouped under the broader Mass Transfer category. See the full topic weightage table for the real, computed numbers.

GATE Weightage

Diffusion & Mass Transfer Coefficients typically contribute 2–3 questions (about 3–5 marks) within the Mass Transfer section of the GATE CH paper, concentrated in stagnant-film and equimolar counterdiffusion numericals.

Sub-areaApprox. Marks
Fick's law & diffusion through a stagnant film~1–2
Equimolar counterdiffusion~1
Dimensionless numbers & mass transfer coefficients~1

Sub-area split is a directional estimate (our archive doesn't tag marks at this granularity), for the real, computed topic-level total, see "Real GATE CH PYQ Frequency" above.

Key Subtopics

1

Fick's first law

The fundamental diffusion flux equation, relating molar flux to the concentration gradient via the diffusivity DAB, analogous to Fourier's law in heat transfer.

2

Diffusivity estimation

Correlations (e.g., Chapman-Enskog for gases) for estimating the diffusion coefficient DAB when it isn't given directly, based on molecular properties and system temperature/pressure.

3

Diffusion through a stagnant film

One species (A) diffuses while the second (B) has zero net flux, bulk flow induced by A's own diffusion enhances the flux beyond simple Fick's law, captured by the log-mean concentration of inert B.

4

Equimolar counterdiffusion

Two species diffuse in exactly opposite directions at equal molar rates (net bulk flow is zero), gives a simpler, linear concentration profile without the log-mean correction.

5

Mass transfer coefficients

Local (kc, ky, kx) and overall (Kx, Ky) mass transfer coefficients that lump the complexity of the concentration boundary layer into a single proportionality constant between flux and a bulk concentration driving force.

6

Dimensionless numbers for mass transfer

Sherwood number (Sh, analogous to Nusselt), Schmidt number (Sc, analogous to Prandtl), and correlations like Sh = 0.023 Re^0.8 Sc^(1/3) that mirror heat-transfer correlations exactly.

7

Heat and mass transfer analogy

The Chilton-Colburn analogy relating mass transfer coefficients to friction factor and heat transfer coefficients, letting one measurement stand in for another in some engineering estimates.

Essential Formulas

Full formula reference →

NA=−DABdCAdz+xA(NA+NB)N_A = -D_{AB}\dfrac{dC_A}{dz} + x_A(N_A+N_B)

Fick's first law with the bulk-flow (convective) term, reduces to simple diffusion only when NA + NB = 0

NA=DABPRTz(1−yA2)lm(yA1−yA2)N_A = \dfrac{D_{AB}P}{RTz(1-y_{A2})_{lm}}\left(y_{A1}-y_{A2}\right)

Diffusion of A through stagnant B, flux enhanced by the log-mean mole fraction of inert B, (1−yA)lm

(1−yA)lm=(1−yA2)−(1−yA1)ln⁡[1−yA21−yA1](1-y_A)_{lm} = \dfrac{(1-y_{A2})-(1-y_{A1})}{\ln\left[\frac{1-y_{A2}}{1-y_{A1}}\right]}

Log-mean mole fraction of stagnant component B, used in the stagnant-film flux equation

NA=−NB=DABRTz(pA1−pA2)N_A = -N_B = \dfrac{D_{AB}}{RTz}(p_{A1}-p_{A2})

Equimolar counterdiffusion flux, a simpler linear form, no log-mean correction needed

Sh=kcLDABSh = \dfrac{k_c L}{D_{AB}}

Sherwood number, ratio of convective to diffusive mass transfer, analogous to the Nusselt number

Sc=μρDABSc = \dfrac{\mu}{\rho D_{AB}}

Schmidt number, ratio of momentum to mass diffusivity, analogous to the Prandtl number

Sh=0.023 Re0.8Sc1/3Sh = 0.023\,Re^{0.8}Sc^{1/3}

Mass-transfer analog of the Dittus-Boelter correlation, for turbulent flow in a pipe or duct

kcv=Stm=f2 Sc−2/3\dfrac{k_c}{v} = St_m = \dfrac{f}{2}\,Sc^{-2/3}

Chilton-Colburn analogy connecting the mass transfer Stanton number to the Darcy/Fanning friction factor

Visual Reference

An agar plate showing zones of inhibition around antibiotic discs, a visible diffusion gradient
A different diffusion context (antibiotic diffusing outward from a disc), but the same radial concentration-gradient physics Fick's law describes — visible instead of just calculated. Sommer36, CC BY-SA 4.0, via Wikimedia Commons.

Derivations & Physical Insight

Why Diffusion Through a Stagnant Film Needs a Log-Mean Correction (and Equimolar Counterdiffusion Doesn't)

When component A diffuses through stagnant B (B has zero net flux, NB = 0), A's own diffusion still creates a small net bulk motion of the gas mixture toward the low-concentration end, because as A molecules leave, something has to physically make room, and since B isn't moving, the whole gas mixture drifts slightly in A's diffusion direction. This bulk-flow contribution adds to A's pure-diffusion flux, and because it depends on the local mole fraction of A (which varies with position z), integrating Fick's law with this extra term across the film produces a logarithmic, not linear, mole-fraction dependence.

In equimolar counterdiffusion, by contrast, A and B diffuse in exactly opposite directions at equal molar rates (NA = −NB), so their combined bulk flow contribution is exactly zero everywhere in the film, there's no net convective transport to correct for. That leaves pure Fick's-law diffusion, which integrates to the much simpler linear concentration profile and flux equation. This is the entire reason the two situations need visibly different formulas even though both start from the same Fick's law: whether the bulk-flow term survives integration depends entirely on whether the two fluxes cancel each other (equimolar counterdiffusion) or not (stagnant film).

Why the Heat and Mass Transfer Analogy Works

Fourier's law of heat conduction (q = −k dT/dx) and Fick's law of diffusion (NA = −DAB dCA/dx) have identical mathematical form, both say flux is proportional to the negative gradient of a driving potential (temperature or concentration), differing only in which physical property (thermal conductivity k vs. diffusivity DAB) sets the proportionality constant. Since the underlying transport equations (energy conservation for heat, species conservation for mass) also have the same mathematical structure in a flowing fluid, the dimensionless groups that emerge from non-dimensionalizing each set of equations end up in one-to-one correspondence: Nusselt ↔ Sherwood, Prandtl ↔ Schmidt, Reynolds appears in both unchanged (since momentum transport is common to both).

This structural parallel is why a correlation validated for heat transfer (like Dittus-Boelter, Nu = 0.023Re^0.8Pr^n) can be repurposed for mass transfer almost mechanically, swap Nu for Sh and Pr for Sc, as long as the boundary conditions and flow geometry are analogous. It's not a coincidence or approximation stacked on top of separate physics; it's the same underlying convection-diffusion transport equation solved twice for two different scalar quantities.

Worked Practice Problems

Original practice problems in the GATE CH style, not copied from any question bank. Work them before reading the solution.

1-mark · NAT

Equimolar Counterdiffusion Flux Between Two Bulbs

Problem: Gas A and gas B undergo equimolar counterdiffusion across a stagnant gas film of thickness 2 mm at 298 K, with partial pressures of A being 15 kPa and 5 kPa at the two ends. DAB = 1.5×10⁻⁵ m²/s, total pressure = 101.3 kPa. Find the molar flux of A.

Given: z = 2×10⁻³ m, T = 298 K, pA1 = 15 kPa, pA2 = 5 kPa, DAB = 1.5×10⁻⁵ m²/s, R = 8.314 J/mol·K.

NA=DABRTz(pA1−pA2)=1.5×10−58.314×298×0.002×(15,000−5,000)N_A = \dfrac{D_{AB}}{RTz}(p_{A1}-p_{A2}) = \dfrac{1.5\times10^{-5}}{8.314\times298\times0.002}\times(15{,}000-5{,}000)
=1.5×10−54.955×10,000= \dfrac{1.5\times10^{-5}}{4.955}\times10{,}000

Answer: NA ≈ 3.03×10⁻⁵ mol/(m²·s)

2-mark · NAT

Diffusion of A Through Stagnant B, Log-Mean Correction

Problem: Component A diffuses through a 3 mm stagnant film of B at 320 K and 1 atm (101.3 kPa) total pressure, with mole fractions yA1 = 0.20 at one face and yA2 = 0.05 at the other. DAB = 2.0×10⁻⁵ m²/s. Find the molar flux of A, including the stagnant-film correction.

Given: z = 3×10⁻³ m, T = 320 K, P = 101,300 Pa, yA1 = 0.20, yA2 = 0.05, DAB = 2.0×10⁻⁵ m²/s.

(1−yA)lm=(1−0.05)−(1−0.20)ln⁡[1−0.051−0.20]=0.95−0.80ln⁡(0.95/0.80)=0.150.1739(1-y_A)_{lm} = \dfrac{(1-0.05)-(1-0.20)}{\ln\left[\frac{1-0.05}{1-0.20}\right]} = \dfrac{0.95-0.80}{\ln(0.95/0.80)} = \dfrac{0.15}{0.1739}
(1−yA)lm=0.8626(1-y_A)_{lm} = 0.8626
NA=DABPRTz(1−yA)lm(yA1−yA2)=2.0×10−5×101,3008.314×320×0.003×0.8626×0.15N_A = \dfrac{D_{AB}P}{RTz(1-y_A)_{lm}}(y_{A1}-y_{A2}) = \dfrac{2.0\times10^{-5}\times101{,}300}{8.314\times320\times0.003\times0.8626}\times0.15

Answer: NA ≈ 3.62×10⁻⁴ mol/(m²·s), noticeably higher than the equimolar case would give for the same driving force, due to the bulk-flow enhancement.

Topic-wise PYQ Frequency

Medium

Fick's law & diffusion numericals

GATE 2024 Q17 tested diffusion directly, a recurring numerical style in this subtopic.

Medium

Equimolar counterdiffusion

GATE 2026 Q33 tested equimolar counterdiffusion specifically, a recognizable, formula-driven question type.

Medium

Dimensionless numbers (Sherwood, Schmidt) & mass transfer theories

GATE 2025 Q21 (dimensionless numbers) and Q22 (mass transfer theories) show this is tested regularly, often conceptually.

Recommended Study Order

  1. 1

    1. Fick's first law & diffusivity

    The absolute foundation, every other calculation in this guide is a variant of integrating this one equation under different boundary conditions.

  2. 2

    2. Equimolar counterdiffusion

    The simpler of the two classic 1-D diffusion problems, master this linear case before the stagnant-film log-mean correction.

  3. 3

    3. Diffusion through a stagnant film

    Builds directly on equimolar counterdiffusion, adding the bulk-flow / log-mean correction, a frequent GATE numerical.

  4. 4

    4. Dimensionless numbers & the heat-mass transfer analogy

    Fastest to learn if the Heat Transfer guide's Nusselt/Prandtl material is already solid, recognize the direct correspondence rather than memorizing from scratch.

Common Pitfalls

✗ Applying the simple linear Fick's law flux equation to a stagnant-film problem without the log-mean correction.

✓ Stagnant-film diffusion (NB = 0) always needs the (1−yA)lm correction factor in the denominator, using plain Fick's law here systematically underestimates the flux.

✗ Using the stagnant-film formula for an equimolar counterdiffusion problem, or vice versa.

✓ Check whether the problem states both species are diffusing (equimolar counterdiffusion, simpler linear formula) or only one is net-diffusing while the other is stagnant (needs the log-mean correction).

✗ Confusing the Sherwood number (a result, how effective convective mass transfer is) with the Schmidt number (a fluid property ratio, momentum vs. mass diffusivity).

✓ Sc = μ/(ρDAB) depends only on fluid properties; Sh = kcL/DAB depends on the actual flow/geometry and is what a correlation like Sh = 0.023Re^0.8Sc^(1/3) predicts.

✗ Forgetting that DAB itself depends on temperature and pressure, and using a diffusivity value from different conditions without correction.

✓ Gas-phase diffusivities typically scale roughly as T^1.5/P, check whether a given DAB value needs adjusting to the problem's actual temperature and pressure before using it.

Put It Into Practice

Work the equimolar-counterdiffusion and stagnant-film problems above side by side to internalize when the log-mean correction applies, then apply the same reasoning to the Diffusion questions in the full GATE previous-year test set.

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