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Introduction to Diffusion: Physics and Math

Expressions for Flux

Diffusion is the macroscopic result of random thermal motion on a micoscopic scale. For example, in the diagram at right, oxygen and nitrogen molecules move in random directions, with kinetic energy on the order of kT. If there are more oxygen molecules on the left side of the plane A-A than on the right, more molecules will cross to the right than to the left: there will be a net flux even though the motion of each individual molecule is completely random. The flux is proportional to the gradient in concentration (molecular or molar):


("Fick's Law)

A simple estimate of the diffusion constant D is obtained by assuming that the molecules striking the plane A-A last collided one mean free path to each side, and move with mean thermal velocity (see kinetic theory for these terms). For "self-diffusion" (e.g. O[18] in O[16]) the form on the right is a reasonable approximation.

Typical values of D:
0.1 to 1 cm2/s @ 1 atm
75 to 750 cm2/s @ 1 Torr

The expressions above are fine for components present in small concentrations (infinitely dilute case). When the concentration is not negligible we must recall the constraint that the pressure (and thus total species concentrations) are constant everywhere. As a consequence, in for example a two-component mixture, if species 1 diffuses to the left, species 2 must diffuse to the right. D becomes the "binary diffusivity". A very rough estimate of binary diffusivity is obtained by using the RMS molecular diameter and the effective mass as shown at right.

 

The Diffusion Equation

 

Consider a volume from (x) to (x+dx) with transport only by diffusion. The change in the number of molecules with time is the difference between the flux going in and the flux going out, assuming molecules are not generated or lost in the volume.

 

 

By using the Ficke's law expression for the flux, we can relate the change in concentration to the derivatives at the edges of the region.

 

By expanding the derivative at x+dx to first order in the derivative at x, we can derive the Diffusion Equation, which describes the relationship between the spatial and temporal behavior of concentration:

 

 

Some Solutions to the Diffusion Equation

Many practical situations can be approximated by two extreme cases:

 

1: fixed total amount of stuff, semi-infinite region:

solutions are of the form:

 

2: fixed concentration at one boundary, semi-infinite region:

solutions are of the form:

 

 

 

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