Diffusion

Abstract

Molecules move by random Brownian motion so that over time all concentration differences are evened out and the entropy of the system is maximised. Fick’s first law of diffusion (for simplicity here for the 1-D case) states that the net flux J is proportional to the concentration difference and the pathlength x:

$$J = -D\left (\frac{\partial c} {\partial l}\right )$$

(27.1)

where D is the diffusion coefficient of the substance in question. Because of mass conservation, a molecule entering a given volume element must either leave again, or increase the concentration in that element, this is expressed in the continuity equation:

$$\left (\frac{\partial c} {\partial t}\right ) = -\left (\frac{\partial J} {\partial l} \right )$$

(27.2)

Fick’s second law states:

$$\left (\frac{\partial c} {\partial t}\right ) = D\left (\frac{{\partial }^{2}c} {\partial {l}^{2}} \right )$$

(27.3)

For multi-dimensional diffusion the equations are similar, if more complex. The average displacement < l > of molecules, which spread from an infinitesimal thin starting zone, is time-dependent:

$$< {l}^{2} >= 2Dt$$

(27.4)

which is the equation for a random-walk from the origin. The width of the starting zone increases with the square-root of time.