A familiar equation for Fick's first law
Fick's Law again: Flux is directly proportional to the steepness of the gradient.
So now we know that the gradient is represented by dC/dx, but what does “directly proportional” mean? Use your mouse to see how the steepness of the gradient affects the amount of diffusion:
When we have a small gradient, there is not much diffusion going on, but when the gradient is BIG, then diffusion can be substantial. But how can we represent this mathematically? Well, if two quantities are directly proportional, they are related to each other through an equation like y = mx. Hmmm.... this equation should sound familiar -- it's the equation for a line which goes through (0,0).
So, FINALLY, a mathematical equivalent to Fick's law, in both a continuous (calculus) and discrete version. Notice that they both have the same basic form that is directly analogous to the basic equation for a straight line that intersects the origin (y=mx):
So for both equations, "y" is the flux (or more correctly, the flux density), and is dependent (and therefore called the dependent variable) on two quantities: 1) the steepness of the gradient (in red) and 2) a coefficient based on the particular substance being measured (the Diffusion coefficient, "D" - more on that later).
One last note ... Why the negative sign in front of the diffusion coefficient??? Remember that diffusion always goes DOWN the concentration gradient -- its direction is opposite the concentration gradient. So the the diffusion coefficient has a Movement always occurs in a direction OPPOSITE to that of the gradient.
Copyright University of Maryland, 2007
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