Diffusion

July 26, 2020April 18, 2012 by Anand Bhumkar

Diffusion describes the spread of particles through random motion from regions of higherconcentration to regions of lower concentration. The time dependence of the statistical distribution in space is given by the diffusion equation. The concept of diffusion is tied to that of mass transferdriven by a concentration gradient. Diffusion is invoked in the social sciences to describe the spread of ideas.

Fick’s laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.

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

$\bigg. J = - D \frac{\partial \phi}{\partial x} \bigg.$

where

$J$ is the “diffusion flux” [(amount of substance) per unit area per unit time], example $(\tfrac{\mathrm{mol}}{ \mathrm m^2\cdot \mathrm s})$ . $J$ measures the amount of substance that will flow through a small area during a small time interval.

$\, D$ is the diffusion coefficient or diffusivity in dimensions of [length² time⁻¹], example $(\tfrac{\mathrm m^2}{\mathrm s})$

$\, \phi$ (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length⁻³], example $(\tfrac\mathrm{mol}{\mathrm m^3})$

$\, x$ is the position [length], example $\,\mathrm m$

Fick’s second law predicts how diffusion causes the concentration to change with time:

$\frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}\,\!$

Where

$\,\phi$ is the concentration in dimensions of [(amount of substance) length⁻³], example $(\tfrac\mathrm{mol}{m^3})$
$\, t$ is time [s]
$\, D$ is the diffusion coefficient in dimensions of [length² time⁻¹], example $(\tfrac{m^2}{s})$
$\, x$ is the position [length], example $\,m$

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