Ficks Laws and the Diffusion Equations

Fick s (continuum) laws of diffusion can be related to the discrete atomic processes of the random walk, and the diffusion coefficient defined in terms of Fick s law can be equated to the random-walk displacement of the atoms. Again it is easiest to use a one-dimensional random walk in which an atom is constrained to jump from one 

If the process is random, the jump frequency is independent of direction and we can set T12 equal to r2i- In addition, when a large number of jumps are taken into account, the number of jumps in either direction will be equal, so  

The number of mobile atoms on plane 1 is N per unit area, so that the concentration per unit volume at plane 1 is N /a, or c2. Similarly, the number of mobile 

The concentration gradient, dc/dx, is given by the change in concentration between planes 1 and 2 divided by the distance between planes 1 and 2, that is 

The units of the term Ta2 are m2 s 1, the units of the diffusion coefficient, and writing  

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