Derivation of Fick's 1st Law in 1 Dimension
The following derivation is based on a similar argument made in Berg 1977 (see references).
Consider a collection of particles performing a random walk in one dimension with length scale and time scale . Let be the number of particles at position at time .
At a given time step, half of the particles would move left and half would move right. Since half of the particles at point move right and half of the particles at point move left, the net movement to the right is:
The flux, J, is this net movement of particles across some area element of area a, normal to the random walk during a time interval . Hence we may write:
Multiplying the top and bottom of the righthand side by and rewriting, we obtain:
We note that concentration is defined as particles per unit volume, and hence .
In addition, is the definition of the diffusion constant in one dimension, . Thus our expression simplifies to:
In the limit where is infinitesimal, the righthand side becomes a space derivative:
Read more about this topic: Fick's Laws Of Diffusion
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