Table of Contents

• Prequiz...
• Just Plain Movement
• 1: Measuring Movement
• 2: Directed vs. Undirected Movement
• 3: Measuring Movement Using Flux
• Introducing Gradients
• 4: What is a gradient?
• 5: Watching diffusion happen (applet)
• 6: Visualizing the Gradient
• Fick's Laws, Part I
• 7: Fick's First Law
• 8: The gradient in Fick's First Law
• 9: A familiar equation for Fick's first law
• 10: The diffusion coefficient is the slope
• 11: What are the units for D?
• 12: How does flux depend on distance?
• 13: A fragrant example
• 14: General Transport Equations and Fick's First Law
• Fick's Laws, Part II
• 15: Fick's Second Law ...
• 16: ... is about "time to diffuse"
• 17: A Very Useful Formula
• 18: Time-to-diffuse "explodes"
• 19: General Functions and Diffusion
• Biological Applications
• 20: Biological Applications
• 21: Why do rhinos have lungs and amoebas don't?
• 22: Diffusion is efficient in small organisms, but not big ones
• 23: Diffusion within lungs
• 24: How many macrophages does it take to kill a virus?
• 25: You need a lot of macrophages - or one smart one!
• Review
• 26: Final Review

The gradient in Fick's First Law

Here are our two main points again. If you understand and memorize these two sentences, you will have half of diffusion down cold:

• Diffusion is the net flux of particles down a concentration gradient due to random movement, and
• Diffusive flux is directly proportional to the steepness of the gradient (Fick’s first law).

Let’s make it look more technical. It's easiest to think of a gradient in one dimension (so there is a left and right side). The picture below shows a gradient from left (high) to right (low).

How can we measure this gradient? The easiest thing to do would be to find out the difference between the concentrations on the left and on the right, and the distance between the left and the right. The gradient is the difference between concentrations divided by the distance between the ends:

 

You could also measure the concentrations closer and closer together to get more exact information. For example you could measure the gradient every centimeter.

Or every millimeter. Or every micrometer. Here is where calculus comes in handy (yes, Virginia, calculus is actually useful sometimes!). Calculus allows you to calculate infinitely many gradients that are each infinitely short. Instead of taking the difference between concentrations on the left and right side of some distance, we take the difference between concentrations that are infinitely close together, and call it dC. Instead of using some large distance like 1 centimeter or 1 millimeter, we use an infinitely small distance, dx. So the gradient is dC/dx (which, you may recall, is a "derivative" in calculus). dC/dx tells you how much the concentration changes as you move

We can't directly measure the concentration of particles in infinitely small slices!! But on the graph, we can interpret the gradient as the slope of the line -- which tells us how concentration changes when distance changes very slightly.

We introduce the calculus version of the gradient not to confuse you, but because the flux equation is almost always written in its calculus form, using dC/dx as the gradient. In fact, you can think of dC/dx simply as a single variable that could equally well be called the "gradient".