Fick's First Law: Definition, Derivation & Examples

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  • 0:03 Introduction to Fick's Laws
  • 3:22 Fick's First Law
  • 5:36 Lesson Summary
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Lesson Transcript
Instructor: Kip Ingram

Kip holds a PhD in Engineering from The University of Texas at Austin and was an occasional substitute lecturer in engineering classes at that institution.

Fick's First Law states that diffusion in gases or fluids is driven by local concentration variations. Even though individual molecules move in a very erratic, random way, their overall diffusion is extremely predictable.

Introduction to Fick's Laws

Fick's laws (the first and second ones) describe the phenomenon of diffusion, which is how gases and fluids spread and mix. This may seem like a complicated, involved topic, but it's really not very complicated at all. You can think of a volume of gas or fluid as an enormous number of incredibly tiny objects (molecules) that are moving around at random, in all directions. They move in straight lines until they bump into a wall or one another. Some move faster than others, but there's a rule that says their average speed depends on the temperature of the gas or fluid. By studying the implications of this totally random motion, we can derive Fick's First Law fairly easily.

Let's do an example, something easy to imagine. To keep things simple, we'll assume that movement occurs in only the x direction (instead of x, y, and z directions). Picture a fish pond that's divided into two equal-size sections by a removable barrier. We'll call these sections left (L) and right (R). Let F(L) and F(R) denote how many fish are in each section. We like fish a lot, so F(L) and F(R) have large numbers of fish, and the fish constantly bump into each other and into the walls as they swim around. When they bump into something they change direction, but we'll assume they all swim at the same speed all the time. It's important to remember that gas molecules flying around in a container don't all move at the same speed, but if we use their average speed everything we discuss will work out.

Let's remove the barrier and watch a single fish for a while as he zigzags around. We're interested in how long it takes him to move from one section to the other. This is the elapsed time from when he crosses the boundary in one direction until he crosses it in the other. This will vary from crossing to crossing because the fish is swimming in random directions. But, we can watch him for a couple of hours and take an average, which we'll call T. Assume we measure T in seconds. Then we can say our one fish produces 1/T crossings per second. So, if he changes direction, on average, every 30 seconds, he produces 1/30 crossings per second.

This already takes into account our fish bumping into other fish, and he has no other interactions with them. Since our fish all have the same speed, each one will produce 1/T crossings per second. If we watched two fish, we'd see 2/T crossings per second, and so on. So, we can say that all F(L) fish in the section L will produce F(L)/T crossings per second, and their next crossing has to be from left to right. Similarly, the F(R) fish in section R produce F(R)/T crossings per second, and their next crossing has to be right to left.

We're usually interested in the net crossings in some reference direction - say left-to-right. We can write that like this:

Rate = F(L)/T - F(R)/T

Factoring out 1/T:

Rate = (1/T) * ( F(L) - F(R) )

Factoring out -1:

Rate = -(1/T) * ( F(R) - F(L) )

We wrote this the final way, with the negative sign because we've now basically arrived at Fick's First Law!

Fick's First Law

Scientists write Fick's First Law like this:

C = -D * dF/dx

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