Fick's First Law of Diffusion

Adolf Fick wrote this in 1855 by direct analogy with Fourier's law of heat conduction, and the parallel is exact: flux is proportional to gradient, with a transport coefficient in front. Across a 2 mm membrane holding 500 mol/m³ on one face and 100 on the other, with a typical aqueous diffusivity of $10^{-9}\ \text{m}^2\text{/s}$, the flux is $10^{-9}\times 400/0.002 = 2\times 10^{-4}$ mol per square metre per second.

The sign convention causes endless grief. The textbook form carries a minus sign, $J = -D\,dC/dx$, because flux runs down the gradient and $dC/dx$ is negative in the direction of flow. This page writes it as $D(C_1 - C_2)/L$ with $C_1$ the high side, so the minus sign is already absorbed and a positive answer means flow from face 1 to face 2. Enter them the other way round and the answer simply comes back negative, which is the page telling you the flow runs the other way.

What is worth internalising is how slow molecular diffusion actually is. The characteristic time to diffuse a distance $L$ is roughly $L^2/D$, and that square is brutal: a small molecule crosses 10 μm of cytoplasm in about a tenth of a second, but crossing 10 cm of still water takes around 10⁷ seconds, four months. Nature and industry both solve this by never relying on diffusion over distance. Stirring, convection, and very thin membranes are all the same trick, which is shrinking $L$ so the square stops hurting.