Fick's law and molecular diffusion in gases and liquids

Fick's first and second laws, diffusive versus total flux, equimolar counter-diffusion and diffusion through a stagnant gas, and how diffusivity depends on temperature and pressure.

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Why it matters

Every separation process – absorption, distillation, drying, extraction – ultimately moves molecules across a phase or through a film, and the slowest step is often molecular diffusion. Fick's law tells you how fast that happens, and the difference between equimolar counter-diffusion and diffusion through a stagnant gas is the starting point for every mass transfer coefficient you will use later.

Key ideas

Molecular diffusion is the net movement of a species caused by random molecular motion when its concentration (strictly, its chemical potential) is not uniform. It happens even in a perfectly still fluid. Convective mass transfer adds the effect of bulk fluid motion.

Fluxes and reference frames. Two fluxes must be kept apart:

  • J_A – the diffusive (molar) flux of A relative to the molar-average velocity of the mixture. This is what Fick's law gives.
  • N_A – the total molar flux of A relative to fixed coordinates (the equipment). This is what a design needs.

They are linked by N_A = J_A + x_A (N_A + N_B): the second term is the bulk flow of A carried along by the net molar flow of the mixture.

Fick's first law says the diffusive flux is proportional to the concentration gradient and points down the gradient (hence the minus sign). It holds at every instant and every point, in steady or unsteady problems, for binary mixtures at constant temperature and pressure.

Fick's second law is a mass balance combined with the first law, for a stagnant medium with constant D and no reaction. It gives concentration as a function of position and time (unsteady diffusion) and is the basis of the penetration theory and of diffusion in solids.

Two limiting cases in steady one-dimensional diffusion through a film of thickness z:

  1. Equimolar counter-diffusion (N_A = −N_B): no net molar flow, so N_A = J_A. Typical of distillation of components with similar molar latent heats.
  2. Diffusion of A through stagnant (non-diffusing) B (N_B = 0): e.g. absorption of a solute from air into water, or evaporation of a liquid into still air. The bulk flow term adds to diffusion, so the flux is larger than in case 1 by the factor P / p_BM.

Diffusivity values (orders of magnitude).

  • Gases at 1 atm: about 10⁻⁵ m²/s (1.0–2.0 × 10⁻⁵ is common; H₂ systems up to ~7 × 10⁻⁵).
  • Liquids: about 10⁻⁹ m²/s.
  • Solids: 10⁻¹⁰ down to 10⁻¹⁴ m²/s and smaller.

Effect of T and P. For gases kinetic theory gives D_AB ∝ T^1.5 / P (the Chapman–Enskog equation, with a weakly T-dependent collision integral); the empirical Fuller correlation gives D_AB ∝ T^1.75 / P. D is independent of composition for ideal gases. For dilute liquids the Stokes–Einstein and Wilke–Chang equations give D_AB μ / T ≈ constant, so D rises with temperature mainly because viscosity falls. Liquid D depends on concentration, so data are quoted at infinite dilution. Take molecular parameters and association factors from your data book.

Limits of validity. Fick's law in this simple form is for binary systems; multicomponent diffusion needs the Maxwell–Stefan equations. Very large gradients, pressure diffusion and thermal diffusion are neglected.

Formulas

Fick's first law (binary, molar basis): J_A = −D_AB · dC_A/dz

  • J_A – molar diffusive flux of A, mol/(m²·s); D_AB – diffusivity of A in B, m²/s; C_A – molar concentration, mol/m³; z – distance, m.

Total flux: N_A = −C · D_AB · dx_A/dz + x_A (N_A + N_B)

  • C – total molar concentration, mol/m³ (for an ideal gas C = P/(R·T)); x_A – mole fraction.

Fick's second law (constant D, no reaction, no bulk flow): ∂C_A/∂t = D_AB · ∂²C_A/∂z²

Steady equimolar counter-diffusion in a gas film: N_A = D_AB (p_A1 − p_A2) / (R·T·z)

Steady diffusion of A through stagnant B in a gas film: N_A = (D_AB·P / (R·T·z)) · ln[(P − p_A2)/(P − p_A1)] = D_AB·P (p_A1 − p_A2) / (R·T·z·p_BM) p_BM = (p_B2 − p_B1) / ln(p_B2/p_B1) – log-mean partial pressure of B, Pa.

  • P – total pressure, Pa; p_A – partial pressure, Pa; R = 8.314 J/(mol·K); T – K.

Liquid film, A through non-diffusing B (dilute or not): N_A = D_AB · C_av (x_A1 − x_A2) / (z · x_BM)

Correcting a gas diffusivity: D₂ = D₁ (T₂/T₁)^1.75 (P₁/P₂) (Fuller form; use exponent 1.5 if told to use kinetic theory)

Worked examples

Example 1 – equimolar counter-diffusion (standard). Given: gases A and B at T = 298 K, P = 101.325 kPa, D_AB = 2.0 × 10⁻⁵ m²/s, film thickness z = 0.1 m, p_A1 = 10 kPa, p_A2 = 2 kPa, N_A = −N_B.

  1. Formula: N_A = D_AB (p_A1 − p_A2)/(R·T·z).
  2. Δp_A = 10 000 − 2 000 = 8 000 Pa.
  3. R·T·z = 8.314 × 298 × 0.1 = 247.8 J/mol·m.
  4. N_A = 2.0 × 10⁻⁵ × 8 000 / 247.8 = 6.46 × 10⁻⁴ mol/(m²·s).

N_A = 6.46 × 10⁻⁴ mol/(m²·s), and N_B = −6.46 × 10⁻⁴ mol/(m²·s).

Example 2 – diffusion through stagnant gas (GATE level). Given: A diffuses through stagnant B; T = 298 K, P = 101.325 kPa, D_AB = 2.6 × 10⁻⁵ m²/s, z = 0.05 m, p_A1 = 20 kPa, p_A2 = 0. Find N_A and compare with the equimolar case.

  1. p_B1 = 101.325 − 20 = 81.325 kPa; p_B2 = 101.325 kPa.
  2. p_BM = (101.325 − 81.325)/ln(101.325/81.325) = 20/0.2199 = 90.96 kPa.
  3. N_A = D_AB·P (p_A1 − p_A2)/(R·T·z·p_BM).
  4. R·T·z = 8.314 × 298 × 0.05 = 123.9 J/mol·m.
  5. N_A = 2.6 × 10⁻⁵ × 101 325 × 20 000 / (123.9 × 90 959) = 4.68 × 10⁻³ mol/(m²·s).
  6. Equimolar flux for the same gradient: 2.6 × 10⁻⁵ × 20 000 / 123.9 = 4.20 × 10⁻³ mol/(m²·s). Ratio = P/p_BM = 101.325/90.96 = 1.114.

N_A = 4.68 × 10⁻³ mol/(m²·s), about 11 % higher than equimolar counter-diffusion because of bulk flow.

Common mistakes

  • Using J_A where N_A is needed; forgetting the bulk-flow term when B is stagnant.
  • Using the arithmetic mean of p_B instead of the log mean p_BM (small error only when p_A is small).
  • Mixing kPa and Pa, or °C and K, in R·T. Keep R = 8.314 J/(mol·K) with pressures in Pa.
  • Treating a gas diffusivity as concentration-dependent, or a liquid one as independent of concentration.
  • Scaling a gas D with pressure the wrong way: D ∝ 1/P, so doubling P halves D (but C·D is unchanged).
  • Applying Fick's second law in a form that assumes constant D when D varies strongly with concentration.

For GATE CH

Expect numericals on steady diffusion through a stagnant film (Stefan tube, evaporation from a tube, sublimation of a sphere), equimolar counter-diffusion, conversion between N_A and J_A, and scaling of D with temperature and pressure. Conceptual questions test the sign in Fick's law, the meaning of p_BM, and order-of-magnitude values of D. Practise time-to-evaporate problems where the liquid level falls (quasi-steady diffusion with changing z).

Quick check

  1. What is the difference between J_A and N_A?
  2. For a gas, how does D_AB change if pressure is doubled at constant T?
  3. Is the flux larger for equimolar counter-diffusion or for diffusion through stagnant B, at the same partial-pressure gradient?
  4. Give the order of magnitude of D for a solute in a liquid.

Answers: 1. J_A is relative to the molar-average velocity, N_A to fixed coordinates; 2. it halves; 3. through stagnant B (by the factor P/p_BM); 4. about 10⁻⁹ m²/s.

See it move

All Chemical animations
Visualizing Fick's First Law of Diffusion

Adjust the concentration gradient and diffusion coefficient to see how they affect the diffusion flux. Observe how the flux changes in real-time.

Equations used
  • J = -D (dC/dx) — J diffusion flux (mol/m²·s), D diffusion coefficient (m²/s), dC/dx concentration gradient (mol/m³·m)

Interview questions

All Mass Transfer interview questions

Try answering each one aloud before you open it.

  1. 1.What is Fick's first law of diffusion?Concept

    Fick's first law of diffusion states that the flux of a species is proportional to the concentration gradient. Mathematically, it is expressed as J = -D (dC/dx), where J is the diffusion flux, D is the diffusion coefficient, dC is the change in concentration, and dx is the change in position. The negative sign indicates that diffusion occurs in the direction of decreasing concentration.

  2. 2.Explain the difference between Fick's first and second laws of diffusion.Concept

    Fick's first law, J_A = −D_AB dC_A/dz, relates the diffusive flux to the local concentration gradient at any instant; on its own it solves steady-state problems where the profile does not change with time. Fick's second law, ∂C_A/∂t = D_AB ∂²C_A/∂z², comes from combining the first law with an unsteady mass balance (constant D, no reaction, stagnant medium). It predicts how the concentration profile evolves with time, as in penetration theory, drying of slabs or case hardening of steel.

  3. 3.How does molecular diffusion differ in gases and liquids?Concept

    Gas molecules are far apart and travel a long mean free path between collisions, so gas diffusivities are around 10⁻⁵ m²/s, roughly ten thousand times larger than in liquids (about 10⁻⁹ m²/s), where molecules are closely packed. Gas diffusivity is nearly independent of composition, rises roughly as T^1.5 to T^1.75 and varies as 1/P. Liquid diffusivity depends on concentration and on solvent viscosity, with D·μ/T roughly constant (Stokes–Einstein, Wilke–Chang), and is practically independent of pressure.

  4. 4.Why is the diffusion coefficient important in mass transfer processes?Application

    The diffusion coefficient is a measure of how easily a substance diffuses through a medium. It is crucial in mass transfer processes because it determines the rate at which a species will spread out in a given medium. A higher diffusion coefficient means faster diffusion, which can be critical in designing processes like separation, purification, and chemical reactions where mass transfer is a limiting factor.

  5. 5.What happens to the diffusion rate if the temperature increases?Application

    Diffusivity increases with temperature in every phase. For gases D_AB varies roughly as T^1.5 (kinetic theory) to T^1.75 (Fuller correlation) at constant pressure. For liquids D_AB·μ/T is roughly constant, so D rises both with T and, more strongly, because the solvent viscosity falls. In solids diffusion is activated and follows an Arrhenius law, D = D₀ exp(−Q/RT), so it rises very steeply with temperature.

  6. 6.How does pressure affect molecular diffusion in gases?Application

    For gases at moderate pressures D_AB is inversely proportional to total pressure, because a denser gas means a shorter mean free path. The product C·D_AB = P·D_AB/(RT) is therefore independent of pressure, so the molar flux for a given mole-fraction gradient is unchanged by pressure. In liquids and solids pressure has almost no effect on diffusivity.

  7. 7.Calculate the diffusive flux of a gas with a diffusion coefficient of 2 × 10⁻⁵ m²/s and a concentration gradient of −5 mol/m³ per metre.Numerical

    Using Fick's first law, J_A = −D_AB dC_A/dz = −(2 × 10⁻⁵ m²/s)(−5 mol/m⁴) = 1 × 10⁻⁴ mol/(m²·s). The flux is positive, i.e. in the +z direction, down the concentration gradient. A diffusivity of order 10⁻⁵ m²/s is typical of gases at about 1 atm.

  8. 8.A liquid has a diffusion coefficient of 1 x 10⁻⁹ m²/s. If the concentration gradient is 10 mol/m³ per meter, what is the diffusion flux?Numerical

    Using Fick's first law, J = -D (dC/dx), where D = 1 x 10⁻⁹ m²/s and dC/dx = 10 mol/m³/m. The diffusion flux J = -1 x 10⁻⁹ * 10 = -1 x 10⁻⁸ mol/m²/s. The negative sign indicates the direction of diffusion is opposite to the concentration gradient.

  9. 9.Why is it important to consider both molecular and convective diffusion in industrial processes?Application

    In industrial processes, both molecular and convective diffusion can play significant roles in mass transfer. Molecular diffusion is driven by concentration gradients, while convective diffusion involves the bulk movement of fluid, which can enhance or hinder the overall mass transfer rate. Considering both mechanisms is important for accurately designing and optimizing processes such as mixing, separation, and chemical reactions to ensure efficiency and effectiveness.

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