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.
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).
Fick's second law predicts how diffusion causes the concentration field to change with time:
It can be derived from Fick's First law and the mass balance:
Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying by the constant:
For the case of diffusion in two or more dimensions the Second Fick's Law is:
which is analogous to the heat equation.
If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:
An important example is the case where is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain
A simple case of diffusion with time t in one dimension (taken as the x-axis) of a density from a boundary located at position where the density is maintained at a value is
where erfc is the complementary error function. The length is called the diffusion length and provides a measure of how far the density has propagated in the x-direction by diffusion in time t.
For more detail on diffusion length, see these examples
Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, semiconductor doping process, etc. A large amount of experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian". See also non-diagonal coupled transport processes (Onsager relationship).
The diffusion coefficient at different temperatures is often found to be well predicted by
An equation of this form is known as the Arrhenius equation.
Typically, a compound's diffusion coefficient is ~10,000x greater in air than in water. Carbon dioxide in air has a diffusion coefficient of 16 mm²/s, and in water, its coefficient is 0.0016 mm²/s.
An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using Stokes-Einstein equation, which predicts that:
The first law gives rise to the following formula:
Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.
The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.
In certain cases, the solutions are obtained for boundary conditions such as constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate).
Diffusion Measurement Software Module aids live cell imaging.(Introducing the New Diffusion Measurement Package for the Olympus FluoView FV1000 Confocal Microscope)
Jul 22, 2010; Diffusion Measurement module for Olympus ASW v2.1 software is designed for Olympus FluoView FV1000 confocal microscope for live...