# Diffusion From a Micropipette

In Chapter 4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I solve the diffusion equation. We consider the classic one-dimensional example of particles released at a point ( x = 0) and at one instant ( t = 0). The particles diffuse, and the concentration C( x, t) has a Gaussian distribution

where D is the diffusion constant and N is the number of particles per unit area (assuming diffusion along a tube of fixed cross-sectional area).

Often this concentration distribution is drawn as a function of x at a fixed time t (a snapshot). Russ and I include such an illustration in IPMB’s Figure 4.13. Below is a modified version of that figure, showing the Gaussian distribution at three times. The concentration C(x,t) as a function of x at three times t, 2t, and 3t.

Alternatively, we could plot the concentration as a function of t, for a particular location x. Such a plot illustrates how a wave of particles diffuses outward, so at any point x the concentration starts at zero, rises quickly to a peak, and then slowly decays. The concentration C(x,t) as a function of t at three locations x, 2x, and 3x.

We can calculate the time when the concentration reaches its peak, tpeak, by setting the time derivative of C( x, t) equal to zero and solving for t. The result is tpeak = x2/2 D. To find the maximum value of the concentration, Cmax, at any location we plug tpeak into the expression for C( x, t) and find Cmax = 0.242 N/ x.

I was motivated to draw the concentration as a function of time by the discussion of diffusion in Howard Berg’s book Random Walks in Biology. He also analyzes the three-dimensional version of this problem.

I’ll let the reader analyze this case by writing a new homework problem. Enjoy!

Originally published at http://hobbieroth.blogspot.com.

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