If diffusion changes the concentration stored inside a region, flux and conservation must be combined.

In one dimension, conservation of the diffusing species gives

$$\frac{\partial C}{\partial t}=-\frac{\partial J}{\partial x}.$$

With Fick's first law $J=-D,\partial C/\partial x$, and constant $D$,

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

This is Fick's second law, mathematically the same diffusion equation that appears in heat conduction.

Diffusion length scale

Dimensional reasoning shows that a disturbance spreads over a characteristic distance

$$\ell\sim \sqrt{Dt}.$$

Equivalently, reaching depth $L$ requires a time of order

$$t\sim \frac{L^2}{D}.$$

This square dependence is crucial: doubling a required diffusion depth takes roughly four times as long at the same diffusivity.

Example

For $D=10^{-12},\mathrm{m^2/s}$ and $t=10^4,\mathrm{s}$,

$$\ell\sim\sqrt{10^{-12}\times10^4}=10^{-4},\mathrm m=0.1,\mathrm{mm}.$$

Exact concentration profiles depend on initial and boundary conditions. For many geometries, solutions collapse naturally onto distances scaled by $\sqrt{Dt}$, which is why this length scale appears throughout diffusion problems.

Fick's second law therefore turns diffusivity into predictions of homogenization, carburizing depth, dopant penetration and many other time-dependent transport processes.