# Fick’s into other fish, and he has

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.

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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)/TFactoring 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

• C represents the flow (or flux), of whatever material we’re studying (rate is the flux in our fish pond example).
• D is a constant (called the diffusivity) that depends on the system being studied – in our case that’s 1/T.
• F is the number of molecules in a unit volume (like per cubic meter) – that’s completely analogous to our ‘fish per section’ F(L) and F(R). Molecules per cubic meter is often called concentration.

• d/dx is the derivative (rate of change) with respect to x. We don’t really have x in our fish pond, but we do have the two sections, and F(L)-F(R) is our ‘change per section.’

So our fish pond equation matches up term-by-term with Fick’s First Law.The law tells us that even though each molecule moves randomly, the overall motion arises from non-uniform concentration. In our pond, if we see F(L) go down and F(R) go up, it’s because F(L) is larger than F(R), not because of how each fish is moving.

It’s the same with molecules – molecules move from where they are more crowded toward where they are less crowded. Since the law says lower concentrations always go up, and higher concentrations always go down, eventually we’ll get the same concentration everywhere. That’s called steady state.Because molecules are so small and so numerous, even in small volumes, we can use tiny ‘sections’ and yet still have a huge number of molecules in each one. So in common applications randomness gets averaged out thoroughly and Fick’s First Law becomes very precise.

Fick’s First Law works in 3D as well – it just looks fancier. Basically, the flux in the x direction is driven by dF/dx, the flux in the y direction by dF/dy, and the flux in the z direction by dF/dz. The 3D flux is thus a vector that points in the direction of ‘steepest descent’ of the concentration. So overall molecule motion is such that the molecules follow the shortest path they can to a less crowded region.

## Lesson Summary

Fick’s First Law tells us that molecules tend to move from more crowded regions to less crowded regions, which you can see either with our fish pond example or with something as simple as combining one type of liquid with another.

By averaging the random behavior of individual molecules over very large molecule counts, the law provides a precise description of a gas volume in spite of the underlying randomness.Even though molecules themselves never stop moving, a large volume of gas will eventually reach a steady state where the concentration is everywhere the same, and the difference between the highest and the lowest concentrations is zero. Fick’s First Law applies in one, two, or three dimensions. The flux in each direction is proportional to the rate of concentration change in that direction. Always remember that there are even several molecules even in small volumes, making Fick’s First Law great for everyday applications in the lab.

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