Diffusion and mass transfer

Molecules move from where they're crowded to where they're scarce, by random jostling: diffusion. It's what spreads a drop of dye in still water, dries a wet cloth and carries reactants into a catalyst. Separation units like absorbers and dryers are all limited by it.

Fick's law is the same shape as Fourier's: the flux J = D × ΔC ÷ L, in mol/(m²·s), where D is the diffusion coefficient (diffusivity) in m²/s and ΔC ÷ L the concentration gradient. Diffusivities in gases are around 10⁻⁵ m²/s; in liquids about 10⁻⁹, ten thousand times smaller; in solids smaller still.

Diffusion is fast over tiny distances and painfully slow over large ones, because the time it takes goes with distance squared: t ≈ L² ÷ D. In water, 1 mm takes about (10⁻³)² ÷ 10⁻⁹ = 1000 s, around 17 minutes; 1 cm takes 100 times longer, more than a day. That's why you stir sugar into tea, and why cells and catalyst particles are small.

In equipment, fluids flow and stir, so mass transfer is written like convection: the flux = k_c × ΔC, where k_c is a mass transfer coefficient. Most of the resistance sits in thin films at the interfaces between phases (the gas side and liquid side of a bubble, say), and stirring or turbulence makes those films thinner.

Momentum, heat and mass move by the same kind of random molecular motion, so their laws look alike and their coefficients are linked through dimensionless numbers: Reynolds for flow, Prandtl (Pr, momentum versus heat diffusing; about 7 for water and 0.7 for air) and Schmidt (Sc, momentum versus mass). Engineers use these analogies to predict a mass transfer coefficient from a heat transfer one measured on the same equipment.