Table of Contents

• Prequiz...
• Just Plain Movement
• 1: Measuring movement
• 2: Directed vs. undirected movement
• 3: Measuring movement using flux
• Introducing Gradients
• 4: Gradients and diffusion
• 5: Watching diffusion happen (applet)
• 6: Visualising 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: Graphing Fick's First Law
• 11: How does flux depend on distance?
• 12: A fragrant example
• 13: General transport equations and Fick's First Law
• Review
• 14: Summary

The gradient in Fick's First Law

Here are our two main points again. If you understand these two sentences, you will have half of diffusion sorted out:

• Diffusion is the net flux of particles down a concentration gradient due to random movement, and
• Flux is directly proportional to 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 are a left and a right side). The picture below shows a gradient from left (high) to right (low). While it is convenient to represent the gradient in one dimension to keep things simple, remember that we are really talking about concentrations of substances in a volume i.e. in three dimensions, and these substances move through a cross-sectional area.

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 centimetre.

You could also measure the gradient every millimetre, or every micrometre. Here is where calculus comes in handy. Calculus allows you to calculate infinitely many gradients that are each infinitesimally 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 infinitesimally close together, and call it dC. Instead of using some large distance like 1 centimetre or 1 millimetre, we use an infinitesimally small distance, dx. So the gradient is dC/dx (which, you may recall, is a "derivative") . dC/dx tells you how much the concentration changes as you move.

The gradient is measured in units of M m^-1 or M cm^-1.

We can't directly measure the concentration of particles in many infinitesimally 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 have introduced the calculus version of the gradient not to confuse you, but because the flux equation is almost always written using the notation dC/dx for the gradient. In fact, you can think of dC/dx simply as a single symbol or quantity that represents the "gradient".