Let us first consider two containers filled with different gases, for example hydrogen and helium. If the two containers are connected, one observes that after some time the gases have completely mixed.
Since both gases have the same pressure and the same temperature, the mixing occurs solely due to the concentration difference \(\mathrm{d}c / \mathrm{d}x\) between the gases, where for simplicity we restrict ourselves to the one-dimensional case.
To better understand this process, we introduce an imaginary reference plane between the containers. Due to the statistical distribution of velocities, the gas particles move equally in all directions with the same average speed.
Thus, per unit time, some gas molecules diffuse from one container through the reference plane into the other container. At the same time, some particles migrate back into the original container.
Diffusion therefore always refers to the net flux from a container with higher concentration to another with lower concentration.
Even when the concentration differences have equalized after some time, particles still move back and forth between the containers.
However, the net flux is then zero, so the diffusion process stops.
If we regard concentration differences as the driving force, we can, analogous to the relation between volume flux and pressure difference in Hagen-Poiseuille’s law, assume the following linear relationship between the net flux density \(j\) and the concentration gradient \(\mathrm{d}c / \mathrm{d}x\):
$$
j = -D \frac{\mathrm{d}c}{\mathrm{d}x}
$$
This is the first law of Fick, which can easily be confirmed experimentally. The proportionality constant \(D\) is called the diffusion coefficient.
The particle flux density is defined as the particle flux divided by the cross-sectional area, and thus has the unit \(1/(\mathrm{s}\,\mathrm{m}^2)\).
Concentration, on the other hand, has the unit \(1/\mathrm{m}^3\).
Consequently, the diffusion coefficient has the unit \(\mathrm{m}^2/\mathrm{s}\), ensuring dimensional consistency.
Since the concentration decreases along the direction of diffusion, the slope in the direction of the diffusion flux is negative.
For this reason, the minus sign appears in the equation above.
So far, we can only express the diffusion flux for a concentration difference that remains constant over time. However, diffusion continuously transports particles, which in turn changes the concentration gradient.
The relation between the time dependence of the concentration and the flux density is:
$$
\frac{\mathrm{d}c}{\mathrm{d}t} = -\frac{\mathrm{d}j}{\mathrm{d}x}
$$
Here, the derivative with respect to \(x\) is required to ensure dimensional consistency.
If we now substitute Fick’s first law into this equation, we obtain Fick’s second law:
$$
\frac{\mathrm{d}c}{\mathrm{d}t} = D \frac{\mathrm{d}^2 c}{\mathrm{d}x^2}
$$
This means that the time rate of change of concentration is directly proportional to the second derivative of the concentration with respect to position.
Fick’s laws provide a mathematical description of why even very small amounts of CO\(_2\) (about 0.1%) in inhaled air can be dangerous, whereas the 80% nitrogen in the atmosphere causes no problems. The reason is that carbon dioxide produced in the human body can diffuse out of the cells all the more poorly the more CO\(_2\) is already present in the inhaled air.