Diffusion

We first consider two vessels that have been filled with different gases, such as hydrogen and helium. If you connect the two vessels with each other, you will find that the gases have mixed completely after a long period of time. Since both gases have the same pressure and temperature, the mixing takes place only due to the concentration difference $\mathrm{d} c/\mathrm{d} x$ between the gases, whereby for the sake of simplicity we restrict ourselves to the one-dimensional case. In order to better understand this process, we put an imaginary reference plane between the vessels. Due to the statistical velocity distribution, the gas particles move with the same velocity in all directions. A few gas molecules per unit of time diffuse from one vessel through the reference plane into the other vessel. At the same time, however, some of the particles migrate back into the original vessel. Diffusion thus always refers to the net flow from a container with a high concentration to another container with a low concentration. Even if the concentration differences have evened out after some time, the particles continue to migrate back and forth between the vessels. On the other hand, the net current is 0, so that the diffusion comes to a standstill. If one assumes the differences in concentration as the causal force, then one can assume the following linear relationship between the net flow density $j$ and the change in concentration $\mathrm{d} c/\mathrm{d} x$ analogously to the volume flow and the pressure difference using the Hagen-Poiseuille law \begin{equation} \boxed{j = -D\frac{\mathrm{d} c}{\mathrm{d} x}} \end{equation} This is Fick's first law, which can be easily confirmed experimentally. The constant of proportionality $D$ is called the diffusion coefficient. The particle flux density is the quotient of particle flux and cross-sectional area, which is why it has the unit $1/(\mathrm{s}\,\mathrm{m}^2)$. The concentration, on the other hand, has the unit $1/\mathrm{m}^3$. Thus the diffusion coefficient has the unit $\mathrm{m}^2/\mathrm{s}$ so that the left and right sides are identical. The concentration gradient decreases along the moving particles, i.e. the slope in the direction of the diffusion flow is negative. For this reason, the minus sign was introduced in the equation presented above. Up to now we can only state the diffusion current for a time-constant concentration difference. On the other hand, particles are continuously transported by the diffusion current, which results in a change in the concentration gradient. The relationship between the change in concentration over time and the current density is simple \begin{equation} \frac{\mathrm{d} c}{\mathrm{d} t} = -\frac{\mathrm{d} j}{\mathrm{d} x} \end{equation} The derivation in terms of $x$ must be performed so that the units on both sides match. If you use Fick's first law here, you get Fick's second law: \begin{equation} \boxed{\frac{\mathrm{d} c}{\mathrm{d} t} = D \frac{\mathrm{d}^2 c}{\mathrm{d} x^2}} \end{equation} That is, the change in concentration over time is directly proportional to the second derivative of concentration with respect to location. Fick's laws provide a mathematical description of the fact that even the smallest amounts of CO${}_2$ ($\approx 0.1\%$) in the inhaled air can be dangerous, whereas the $80\%$ nitrogen in the atomosphere poses no problems because the carbon dioxide produced in the human body is less able to diffuse out of the cells, the more there is already in the inhaled air.