Fick's Laws of Diffusion - Fick's First Law

Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, the law is

where

• is the "diffusion flux", example . measures the amount of substance that will flow through a small area during a small time interval.

• is the diffusion coefficient or diffusivity in dimensions of, example

• (for ideal mixtures) is the concentration in dimensions of, example

• is the position, example

is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10−9 to 2x10−9 m2/s. For biological molecules the diffusion coefficients normally range from 10−11 to 10−10 m2/s.

In two or more dimensions we must use, the del or gradient operator, which generalises the first derivative, obtaining

.

The driving force for the one-dimensional diffusion is the quantity

which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written as:

where the index i denotes the ith species, c is the concentration (mol/m3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).

If the primary variable is mass fraction (, given, for example, in ), then the equation changes to:

where is the fluid density (for example, in ). Note that the density is outside the gradient operator.