Diffusion

Diffusion is the macroscopic transport of matter resulting from random molecular motion (Brownian motion) without macroscopic bulk flow ¹. It spontaneously dissipates concentration gradients toward thermodynamic equilibrium.

Governing Laws

Fick’s First Law (Steady-State Flux)

Under steady-state conditions, diffusive flux $J$ is directly proportional to the negative gradient of concentration ²:

$J = -D \nabla c$

where:

$J$ is diffusion flux $(\text{mol}\cdot\text{m}^{-2}\cdot\text{s}^{-1})$
$D$ is the diffusion coefficient or diffusivity $(\text{m}^2\cdot\text{s}^{-1})$
$\nabla c$ is the concentration gradient $(\text{mol}\cdot\text{m}^{-4})$

Fick’s Second Law (Non-Steady-State Diffusion)

When concentration varies over time, diffusion is modeled as a parabolic partial differential equation ²:

$\frac{\partial c}{\partial t} = D \nabla^2 c$

In one dimension along the $x$ -axis with constant diffusivity:

$\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$

Key Factors Influencing Diffusivity

Particle Mass: The rate of diffusion is inversely related to particle mass ³. Particles with lower relative molecular mass ( $M_r$ ) move at higher average speeds and diffuse faster than heavier particles at the same temperature (e.g., $\text{NH}_3$ vs. $\text{HCl}$ ) ³.
Temperature & Viscosity: For spherical particles in a continuum fluid, diffusivity is modeled by the Stokes-Einstein relation ²: $D = \frac{k_B T}{6 \pi \eta r}$ where $k_B$ is the Boltzmann constant, $T$ is absolute temperature, $\eta$ is dynamic viscosity, and $r$ is hydrodynamic radius.
State of Matter: Diffusion rates in gases are orders of magnitude faster than in liquids, which in turn are faster than in solids ( $\text{Gas} \gg \text{Liquid} \gg \text{Solid}$ ), governed by molecular mean free paths and intermolecular forces.

Mass - Fundamental quantity of matter governing particle speed and diffusion rate.
Chemistry - Area governing molecular science and physical transport phenomena.
State of Matter - Physical forms of matter determining particle mean free paths and transport mobility.

IUPAC Gold Book - Diffusion - Definition of diffusion as the movement of molecules driven by thermal motion and concentration gradients. ↩
LibreTexts Chemistry - Diffusion and Transport Properties - Mathematical formulation of Fick’s laws and molecular transport models. ↩ ↩² ↩³
Cambridge IGCSE Chemistry (0620) - States of matter, diffusion rate, and relative molecular mass. ↩ ↩²