Mass transfer: diffusion

In short

Diffusion is the movement of a species through a mixture driven by its own concentration (strictly, chemical potential) gradient, described by Fick's law with the diffusivity D_AB. It is the rate-controlling step behind every separation and most reactors — distillation trays, absorbers, dryers, extractors, catalyst pellets, membranes — so column heights, drying times and catalyst effectiveness all trace back to a diffusivity and a mass transfer coefficient. GATE (Chemical Engineering) asks it every year, usually as equimolar counter-diffusion, diffusion through a stagnant gas film, or two-film overall coefficients. Design and technical-services roles at IOCL, BPCL, HPCL, ONGC, Reliance, GSFC, Tata Chemicals, UPL, Aarti Industries, Deepak Nitrite, Sun Pharma, Dr Reddy's, L&T and Thermax test it in interviews.

Explained

Fick's law, fluxes and reference frames

• Fick's first law (binary, molar basis): J_A = −c·D_AB·(dy_A/dz) — the *diffusive* flux of A relative to the molar-average velocity of the mixture.
• The flux measured against fixed equipment (what a designer needs) is the total flux N_A = J_A + y_A(N_A + N_B): diffusion plus bulk (convective) flow.
• D_AB has units m²/s; it is a property of the *pair* A–B (mutual diffusivity), and in a binary mixture D_AB = D_BA. "Self-diffusivity" (tracer in itself) is a different quantity.
• Typical orders: gases at 1 atm 10⁻⁵ m²/s; liquids 10⁻⁹–10⁻¹⁰ m²/s; solids/polymers 10⁻¹²–10⁻¹⁵ m²/s. Gas diffusion is about 10,000 times faster than liquid.
• Fick's second law (unsteady, constant D, no convection): ∂c_A/∂t = D_AB ∂²c_A/∂z². Solutions use error functions (semi-infinite slab) or series/Gurney–Lurie charts (finite slab, cylinder, sphere).
• Diffusion is really driven by activity, not concentration; in strongly non-ideal liquids D is concentration-dependent and uphill diffusion (against the concentration gradient) can occur near azeotropes.

The two standard steady-state cases (the exam workhorses)

1. Equimolar counter-diffusion (EMCD): N_A = −N_B, so bulk flow cancels and

N_A = D_AB(c_A1 − c_A2)/z = D_AB·P(y_A1 − y_A2)/(R·T·z). Occurs in distillation of components with nearly equal latent heats.

1. Diffusion of A through stagnant, non-diffusing B (UMD / Stefan diffusion): N_B = 0, so the bulk flow that replaces the moved A adds to the flux —

N_A = D_AB·P/(R·T·z) · ln(p_B2/p_B1) = D_AB·P(p_A1 − p_A2)/(R·T·z·p_B,lm). Occurs in evaporation, humidification, absorption of one solute into a solvent, drying.

• The ratio P/p_B,lm is the drift factor (always ≥ 1): it is how much faster UMD is than EMCD at the same driving force. It is near 1 for dilute systems and large for concentrated ones.
• Variants examined: Arnold / Stefan tube with a falling liquid level (Winkelmann's method for measuring D), diffusion through a spherical shell (evaporating drop), diffusion with changing area, and diffusion with chemical reaction at the boundary (e.g. A → 2B gives N_B = −2N_A, so neither case applies — derive from the general flux equation).

Predicting and measuring diffusivity

• Gases — Fuller–Schettler–Giddings (FSG): D_AB ∝ T^1.75/P, using additive atomic diffusion volumes. Accurate to about ±5–10%.
• Hence a gas diffusivity measured at one T, P is corrected by D₂ = D₁(T₂/T₁)^1.75 (P₁/P₂). (Chapman–Enskog kinetic theory gives T^1.5 with a collision-integral correction; both are acceptable in exams if stated.)
• Liquids — Wilke–Chang (dilute solutions): D_AB ∝ T/(μ_B·V_A^0.6), with an association factor φ (2.6 water, 1.9 methanol, 1.5 ethanol, 1.0 unassociated solvents). Error ±10–25%; poor for water as solute and for viscous or electrolyte systems (use Nernst–Haskell for dilute salts).
• Liquid diffusivity rises with temperature mainly because viscosity falls: D·μ/T ≈ constant (Stokes–Einstein behaviour). Liquid D is essentially independent of pressure.
• Measurement: Stefan tube and Loschmidt cell (gases), diaphragm cell, Taylor–Aris dispersion and wetted-wall column (liquids), permeation cells (solids and membranes).
• Solids: D = D₀·exp(−E/RT) (activated). Distinguish Fickian diffusion in a homogeneous solid from porous-solid transport.
• Porous media: effective diffusivity D_eff = D·ε/τ (ε = porosity, τ = tortuosity, typically 2–6). If the pore size approaches the mean free path, Knudsen diffusion dominates: D_K = (d_pore/3)·√(8RT/πM) — pressure-independent and proportional to √(T/M). Combined: 1/D_e = 1/D_AB + 1/D_K. Surface diffusion and viscous (Poiseuille) flow may add.
• Multicomponent gases — Wilke's equation for A diffusing through a stagnant mixture: D_A,m = (1 − y_A)/Σ(y_i/D_A,i). Rigorously, Maxwell–Stefan equations are used; they predict reverse and osmotic diffusion, which Fick's law cannot.

From diffusivity to mass transfer coefficients

• Industrially, the film thickness is unknown, so flux is written as N_A = k·Δ(driving force). Coefficients depend on which driving force is used, and the units follow:

k_c (Δc, m/s), k_G (Δp, kmol/m²·s·kPa), k_y or k_G′ (Δy), k_x or k_L (Δx or Δc, liquid side). For dilute systems k_y = k_c·P/RT and k_x = k_L·c, approximately.

• Primed vs unprimed coefficients: k_y′ (or k_c′) applies to UMD and already contains the drift factor: k_y′ = k_y/y_B,lm. Mixing them is a classic error.
• Dimensionless groups: Schmidt Sc = ν/D = μ/(ρD) (momentum vs mass diffusivity; ≈0.6–2 for gases, 10²–10⁴ for liquids); Sherwood Sh = k_c·L/D (convective vs diffusive transport); Lewis Le = α/D = Sc/Pr; Péclet Pe_m = uL/D; Biot (mass) = k_c·L/D_solid.
• Standard correlations: flat plate laminar Sh = 0.664 Re^0.5 Sc^1/3; turbulent pipe (Gilliland–Sherwood / Linton–Sherwood) Sh = 0.023 Re^0.83 Sc^1/3; single sphere (Ranz–Marshall) Sh = 2 + 0.6 Re^0.5 Sc^1/3 — the limit Sh = 2 is pure diffusion into stagnant fluid around a sphere.
• Analogies (used when mass transfer data are absent but friction or heat data exist): Reynolds (Sc = Pr = 1), Prandtl–Taylor, von Kármán, and the practical Chilton–Colburn: j_D = (k_c/u)·Sc^2/3 = j_H = (h/ρc_p u)·Pr^2/3 = f/2, valid for 0.6 < Sc < 2,500.

Theories of interphase mass transfer

• Film theory (Whitman, Nernst): all resistance lies in a stagnant film of thickness δ; k_c = D/δ, so k ∝ D^1.0. Simple, used for design; δ is fitted, not measured.
• Penetration theory (Higbie): fresh fluid elements sit at the interface for a fixed time t_e; k_L = 2√(D/πt_e), so k ∝ D^0.5. Good for short contact (bubbles, drops, packed-column films).
• Surface renewal theory (Danckwerts): exposure times are distributed with renewal rate s; k_L = √(D·s); also k ∝ D^0.5.
• Boundary-layer theory: k ∝ D^2/3, matching most turbulent correlations and the Chilton–Colburn exponent. Experiments mostly fall between 0.5 and 0.7, so film theory over-predicts the effect of D.
• Two-film theory at an interface: equilibrium is assumed *at the interface only* (y_i = m·x_i for a linear equilibrium / Henry's law); the two film resistances add in series.

1/K_y = 1/k_y + m/k_x and 1/K_x = 1/(m·k_y) + 1/k_x.

• Controlling resistance: very soluble gases (small m ⇒ small m/k_x, e.g. NH₃, HCl in water) are gas-film controlled; sparingly soluble gases (O₂, CO₂ in water) are liquid-film controlled. Interface composition is found graphically by drawing a line of slope −k_x/k_y from (x_b, y_b) to the equilibrium curve.
• Mass transfer with fast chemical reaction raises the liquid coefficient by an enhancement factor (Hatta number theory) — the basis of chemical absorption in amine and caustic scrubbers.

Indian practice, codes and safety

• Diffusion-based designs are reported in SI units; tray and packed-column design in India follows vendor correlations with API/TEMA mechanical codes and IS 2825 (code for unfired pressure vessels) for the shell.
• Fugitive emissions and solvent losses from diffusion-driven evaporation are regulated by CPCB/SPCB consents under the Air (Prevention and Control of Pollution) Act, 1981 and the Environment (Protection) Act, 1986, with industry-specific standards notified under the Environment (Protection) Rules.
• Workplace exposure to diffusing vapours (benzene, toluene, ammonia, chlorine) is governed by the Occupational Safety, Health and Working Conditions Code, 2020, in force from 21 November 2025, which replaced the Factories Act, 1948; permissible exposure limits are in its schedules and corresponding rules — check the current notified values, as the rules are state-framed and were being issued through 2025–26.
• Diffusion coefficients for hazard assessment (vapour dispersion, confined-space ingress) feed HAZOP and QRA studies required for MAH (Major Accident Hazard) units under the chemical-accident rules.

Key points

• Fick's first law - diffusive molar flux J_A = −c·D_AB·dy_A/dz; the minus sign makes flux positive down the gradient.
• Total flux - N_A = J_A + y_A(N_A + N_B); forgetting the bulk term is the commonest exam error.
• EMCD - N_A = −N_B, bulk flow cancels, N_A = D_AB·ΔC/z; the distillation case.
• UMD (stagnant B) - N_A = (D_AB·P/RTz)·ln(p_B2/p_B1); evaporation, absorption, drying.
• Drift factor - P/p_B,lm ≥ 1; the factor by which UMD exceeds EMCD at equal driving force.
• Fick's second law - ∂c_A/∂t = D_AB·∂²c_A/∂z²; penetration depth grows as √(D·t).
• Gas diffusivity - D ∝ T^1.75/P (Fuller); ~10⁻⁵ m²/s at 1 atm, 25 °C.
• Liquid diffusivity - D ∝ T/(μ·V_A^0.6) (Wilke–Chang); ~10⁻⁹ m²/s, pressure-independent.
• Knudsen diffusion - small pores, D_K = (d/3)√(8RT/πM), independent of pressure, ∝ 1/√M.
• Effective diffusivity - D_eff = D·ε/τ in porous catalysts and solids.
• Schmidt number - Sc = μ/ρD; ≈1 for gases, 10²–10⁴ for liquids, which is why liquid films dominate resistance.
• Sherwood number - Sh = k_c·L/D; Sh = 2 is the stagnant-sphere limit.
• Film theory - k = D/δ, k ∝ D; penetration/surface renewal - k ∝ D^0.5; boundary layer - k ∝ D^2/3.
• Two-film theory - interface at equilibrium; 1/K_y = 1/k_y + m/k_x (resistances in series).
• Controlling film - gas film for highly soluble gases (NH₃); liquid film for sparingly soluble gases (O₂, CO₂).
• Chilton–Colburn analogy - j_D = j_H = f/2 links friction, heat and mass transfer data.
• Maxwell–Stefan - needed for multicomponent and non-ideal systems; explains reverse and uphill diffusion.

Formulas and examples

Example 1 — Evaporation of water into stagnant air (UMD), and the drift factor. Water at 25 °C evaporates through a 0.10 m stagnant air column at 101.325 kPa; D_AB = 2.6×10⁻⁵ m²/s. Vapour pressure of water at 25 °C = 3.169 kPa; air above is dry.

• p_A1 = 3.169 kPa, p_A2 = 0 ⇒ p_B1 = 98.156 kPa, p_B2 = 101.325 kPa.
• D·P/(R·T·z) = (2.6×10⁻⁵ × 101325)/(8.314 × 298 × 0.10) = 2.634/247.76 = 0.010633 mol/m²·s.
• ln(101.325/98.156) = ln(1.03229) = 0.031777.
• N_A = 0.010633 × 0.031777 = 3.38×10⁻⁴ mol/m²·s = 6.08×10⁻⁶ kg/m²·s = 0.0219 kg/m²·h.
• EMCD at the same driving force: N_A = 0.010633 × (3.169/101.325) = 3.33×10⁻⁴ mol/m²·s.
• Drift factor: p_B,lm = 3.169/0.031777 = 99.73 kPa; P/p_B,lm = 101.325/99.73 = 1.016 (only 1.6% higher — the vapour is dilute). Check: 3.33×10⁻⁴ × 1.016 = 3.38×10⁻⁴ ✓.

Example 2 — Predicting a gas diffusivity (Fuller) and correcting it for T and P. Water (A) in air (B) at 298 K, 1 bar. Σv: H₂O = 13.1, air = 19.7. M_AB = 2/(1/18 + 1/28.97) = 2/0.090074 = 22.20, √M_AB = 4.712.

• T^1.75 = 298^1.75 = 21,371. (13.1^1/3 + 19.7^1/3)² = (2.357 + 2.701)² = 5.058² = 25.58.
• D = 0.00143 × 21,371/(1 × 4.712 × 25.58) = 30.56/120.53 = 0.254 cm²/s = 2.54×10⁻⁵ m²/s (measured ≈ 2.6×10⁻⁵; within 3%).
• At 373 K and 2 bar: D = 2.54×10⁻⁵ × (373/298)^1.75 × (1/2) = 2.54×10⁻⁵ × 1.481 × 0.5 = 1.88×10⁻⁵ m²/s.

Example 3 — Liquid diffusivity by Wilke–Chang. Ethanol (A) dilute in water (B) at 25 °C: φ = 2.6, M_B = 18, μ_B = 0.894 cP, V_A = 60.2 cm³/mol.

• (φM_B)^0.5 = (46.8)^0.5 = 6.841; V_A^0.6 = 60.2^0.6 = 11.69.
• D = 7.4×10⁻⁸ × 6.841 × 298/(0.894 × 11.69) = 1.509×10⁻⁴/10.45 = 1.44×10⁻⁵ cm²/s = 1.44×10⁻⁹ m²/s (measured ≈ 1.2×10⁻⁹; within the method's ±25%).

Example 4 — Two-film theory: interface composition, overall coefficient, controlling resistance. Bulk gas y = 0.050, bulk liquid x = 0.010; k_y = 1.5×10⁻³ and k_x = 3.0×10⁻³ kmol/m²·s; equilibrium y = 0.8x.

• Interface: k_y(y − y_i) = k_x(x_i − x) with y_i = 0.8x_i ⇒ 1.5(0.050 − 0.8x_i) = 3.0(x_i − 0.010) ⇒ 0.050 − 0.8x_i = 2x_i − 0.020 ⇒ x_i = 0.025, y_i = 0.020.
• Flux: N_A = 1.5×10⁻³(0.050 − 0.020) = 4.50×10⁻⁵ kmol/m²·s.
• Overall: 1/K_y = 1/1.5×10⁻³ + 0.8/3.0×10⁻³ = 666.7 + 266.7 = 933.3 ⇒ K_y = 1.071×10⁻³ kmol/m²·s.
• y* = 0.8 × 0.010 = 0.008; N_A = 1.071×10⁻³ × 0.042 = 4.50×10⁻⁵ ✓.
• Gas-film resistance = 666.7/933.3 = 71.4% — gas-film controlled.

Example 5 — Sherwood correlation, film thickness and penetration theory. A 10 mm naphthalene sphere in air at 1 m/s: ν = 1.6×10⁻⁵ m²/s, D_AB = 6.2×10⁻⁶ m²/s.

• Re = 1 × 0.01/1.6×10⁻⁵ = 625; Sc = 1.6×10⁻⁵/6.2×10⁻⁶ = 2.58; Sc^1/3 = 1.371.
• Sh = 2 + 0.6 × 625^0.5 × 1.371 = 2 + 0.6 × 25 × 1.371 = 22.6.
• k_c = Sh·D/d = 22.6 × 6.2×10⁻⁶/0.01 = 1.40×10⁻² m/s; equivalent film δ = D/k_c = 6.2×10⁻⁶/1.40×10⁻² = 0.44 mm.
• For a liquid film with D = 2×10⁻⁹ m²/s and exposure t_e = 1 s, Higbie gives k_L = 2√(2×10⁻⁹/π) = 2 × 2.52×10⁻⁵ = 5.05×10⁻⁵ m/s — about 280 times smaller than the gas-side value, which is why liquid films usually control.

Where it is used

• Refining and petrochemicals (IOCL, BPCL, HPCL, Reliance, HMEL): tray efficiencies, amine and caustic gas treating, stripper and absorber design — all sized from film coefficients built on diffusivities.
• Fertilisers and inorganics (IFFCO, GSFC, RCF, Tata Chemicals, Coromandel): ammonia and CO₂ absorption, urea prilling, SO₂ scrubbing, where gas-film control decides scrubber height.
• Pharma and agrochemicals (Sun Pharma, Dr Reddy's, UPL, Aarti, Deepak Nitrite): solvent drying, crystal washing, solid-liquid extraction and tablet dissolution (Noyes–Whitney is Fick's law at a solid surface).
• Catalysis and reaction engineering: pore diffusion, Thiele modulus and effectiveness factor in refinery and ammonia catalysts; a diffusion-limited pellet shows an apparent activation energy about half the true value.
• Environment, HSE and safety roles: vapour emission estimates from tanks and spills, dispersion modelling for CPCB/SPCB consents, confined-space and MAH risk assessment.
• Food, metallurgy and electronics: oxygen transfer in fermenters, carburising and nitriding of steels, dopant diffusion in silicon, moisture barriers in packaging.

Interview questions

• Q: What does Fick's first law state and why the negative sign? A: The diffusive molar flux is proportional to the concentration gradient, J_A = −cD_AB·dy_A/dz; the negative sign ensures flux is positive in the direction of decreasing concentration.
• Q: Difference between equimolar counter-diffusion and diffusion through a stagnant film? A: In EMCD the two fluxes cancel so there is no bulk flow; in stagnant-film (Stefan) diffusion only A moves, so bulk flow adds to the flux and the result carries the logarithmic drift factor P/p_B,lm.
• Q: How does gas diffusivity vary with temperature and pressure? A: Roughly as T^1.75 and inversely with pressure (Fuller); kinetic theory gives T^1.5 with a collision correction.
• Q: And liquid diffusivity? A: Essentially independent of pressure and proportional to T/μ — Dμ/T is nearly constant, so most of the temperature effect comes through viscosity.
• Q: What is the Schmidt number and why is it so large in liquids? A: Sc = μ/ρD, the ratio of momentum to mass diffusivity; liquid diffusivities are ~10⁴ times smaller than gas ones while kinematic viscosity falls much less, so Sc reaches 10²–10⁴.
• Q: What does Sh = 2 mean physically? A: It is the limiting Sherwood number for pure molecular diffusion from a sphere into an infinite stagnant medium — no convection at all.
• Q: Which film controls when NH₃ is absorbed in water, and when O₂ is absorbed? A: NH₃ is highly soluble (small Henry constant), so the gas film controls; O₂ is sparingly soluble, so the liquid film controls.
• Q: How do you locate the interface composition in two-film theory? A: Draw a line of slope −k_x/k_y from the bulk point (x_b, y_b) on the operating diagram; its intersection with the equilibrium curve gives (x_i, y_i).
• Q: Film theory predicts k ∝ D, penetration theory k ∝ D^0.5 — which do experiments support? A: Mostly exponents of 0.5–0.7, so penetration, surface-renewal and boundary-layer theories are closer; film theory survives because it is simple and its δ is fitted.
• Q: What is Knudsen diffusion and when does it matter? A: Collisions with pore walls dominate over molecule-molecule collisions when the pore size is comparable to the mean free path — in fine-pore catalysts and at low pressure; D_K is independent of pressure and varies as √(T/M).
• Q: Why do we use effective diffusivity in a catalyst pellet? A: Only the pore volume is available and the path is tortuous, so D_eff = Dε/τ, typically 5–20 times smaller than the bulk value.
• Q: Can a species diffuse against its own concentration gradient? A: Yes — the true driving force is chemical potential, and in non-ideal multicomponent mixtures Maxwell–Stefan analysis predicts uphill and reverse diffusion; Fick's law with a single positive D cannot.

Common mistakes

• Dropping the bulk-flow term y_A(N_A + N_B) and using the EMCD formula for an evaporation or absorption problem.
• Using the arithmetic mean instead of the log mean p_B,lm, or inverting p_B1 and p_B2 inside the logarithm.
• Mixing units: partial pressures in kPa with R = 8.314 J/mol·K, or diffusivity in cm²/s with lengths in metres.
• Confusing primed with unprimed coefficients (k_y′ already contains the drift factor) and quoting k without saying which driving force it belongs to.
• Adding film resistances without the equilibrium slope m — 1/K_y = 1/k_y + m/k_x, not 1/k_y + 1/k_x — and forgetting that K_y and K_x differ by the factor m.
• Using bulk diffusivity for a porous solid, or Fuller/Wilke–Chang outside their range (Wilke–Chang is for dilute solutions and fails for water as the solute and for electrolytes).
• Treating a liquid-phase diffusivity as pressure-dependent, or scaling a liquid D with T^1.75.
• Assuming equilibrium between the two bulk phases instead of only at the interface.

Related topics

Distillation · Absorption and stripping · Extraction and leaching · Drying, humidification and crystallisation · Adsorption and ion exchange · Membrane separations · Design of trays and packed columns