Welcome to the Molecular

Here the molecules are bouncing back and forth between the top and bottom of a container. We've arranged them so that the rate of impact on the lid is uniform. Each impact gives a little nudge to the lid from recoil (or Newton's Third Law, if you want to be technical). The cumulative, averaged effect of these impacts is precisely the pressure of the (molecular) gas.

We can measure this pressure very simply: it's basically the weight of the lid. To see this, think what happens to the lid: during an impact it gets a jolt upwards: it moves up a little, like a ball thrown upwards, until the force of gravity slows it down to a stop at its peak, then moves it down. If everything is balanced, the lid just reaches it starting position when the next molecule hits, starting the process all over. The lid weight is thus the time average of the force generated by the molecular impacts, because the weight exactly balances the impacts.

Go ahead, put the Amazing Trained Molecules through their paces. If your browser supports Java, click on the green weights to move them from the reserve pile to on top of the lid, or vice versa. More weights mean more pressure on the molecules, and you'll see the position of the lid change in response. One thing you'll notice is that the lid will slowly bounce (or oscillate, in physics terms) up and down like it's on a spring. One of the first physics papers on this subject was titled A Defence of the Doctrine Touching the Spring And Weight of the Air, written by the gentleman pictured. In real gas, there are so many molecules moving randomly that this is a very small tremble, but in the circus, it's a large motion and sorta cool.

It does make it hard, though, to measure the average size of the gas volume (the average height of the lid). So we've provided a trick: if you click anywhere in the molecule area, you'll momentarily stop the lid movement. It'll restart, but with a smaller bounce, just like a stretched spring released at that point. In fact, if you notice carefully, the new bounce will just reach the spot where the lid was when you clicked and stopped it. It'll have the same center of oscillation, though, so if you click when the lid is close to the center you can (almost) stop the bounce. Play with this to see what I mean. (To start an oscillation, add or remove a weight from the lid, and a few seconds later change the weight back)

Since this is a physics circus, you can plot a graph. First make the bounce very small, and then click the Plot button. This puts a point on the graph. The vertical axis is P, pressure, which is the number of weights at that point. The horizontal axis is V, volume of the gas, which is the height of the lid at that point.

Try plotting several points, each with a different number of weights on the lid. If you get a bad point, just click on it to erase it. Click most anywhere during the plot animation to cancel it. If you click again on the plot button during the animation, it'll finish plotting the point immediately (yes sir!) If you plot at least three points, we'll draw the theoretical line thru them so you can check how well you're doing. If you're careful with stabilizing the lid before plotting a point, the points should fall on the line (or if you just fool around and plot randomly, not). It's a little off the tour, but if you'd like to take a side trip, you can change the speed of the molecules and come back.

The goal is to see how the pressure (weight) and volume (height) are related. If the weight is twice as great, the impact rate (impacts per second) must be twice as great to balance. Since we're keeping the speed and number of molecules the same, the only way to change the impact rate is to reduce the distance they have to travel between impacts: if we halve the distance, each one hits twice as often. If we triple the pressure, we have to triple the impact rate and so reduce the volume to a third.

If you're handy with algebraic formulae, you can see that this relationship is

Pressure times Volume = constant

But does this work with wild molecules, not just these circus ones? The short answer is, yes! Even if the molecules are going in all directions at different speeds, each one has this behavior and so does the average of all of them. Here's a Java simulation of the wild case.

You may have seen this before ;) It's Boyle's Law in the textbooks (at least in English speaking countries...) That's the Right Honourable Robert Boyle in the wig above. History note: look up Edme Mariotte if you're interested why it's Loi de Mariotte in Francophone places and Boyle-Mariotte elsewhere...the abbé built the first windtunnel and is also credited in France with the device sold as Newton's Cradle)

One personal situation where Boyle's Law is useful is Scuba Diving: hold your breath on ascent and your lungs explode! That'll get your attention faster than an equation!

What Boyle's Law means is that as pressure goes up, volume goes down, and vice versa. Put that way, this isn't very surprising for almost any substance: if you squeeze it it gets smaller; if you release, it springs back. What's special is the exact numerical relationship: in a gas, if you double the pressure, the volume is exactly halved. If you halve the pressure, the volume doubles.

It's interesting that Boyle discovered this law "touching the spring of the air," while working with another physicist you may have heard of, Robert Hooke. Hooke's Law is a somewhat similar result relating force and stretch of springs: remember the bounce in the experiment above? An important difference is that the constant in Hooke's Law is different for each kind of spring, but the gas law has a universal constant for all gases (up to temperature and number of molecules). Boyle's Law is deeper because it points to molecule motion, what later became "the kinetic theory of gases." Unfortunately, Boyle's Law too often winds up as a natural set of test questions to see if you can handle formulas, rather than a source of physical insight.

Of course, for engineering purposes, like car suspensions, sometimes you don't want a nice simple behavior and instead go to some effort to make it more complex: automotive shock absorbers

Extra Credit Trick: Another way of seeing the same inverse relationship of P and V is to plot P vs 1/V, which is a straight line. This shows the accuracy of the the law better, since our eyes can tell slight departures from straight lines much better than slight departures from hyperbolae. You can see this done with Boyle's original data (There's a whole armamentarium of plotting tricks like this that you'll see: a popular one is logarithmic scales.)

The Flying Bernoulli Brothers have a lot of other acts, too. There's a molecular explanation of the other gas laws from the textbooks, one that really defines "before it's time". Daniel Bernoulli (of the Bernoulli Effect, which causes airplanes to fly) derived the whole shebang, the Ideal Gas Law, from basic molecule mechanics in 1738, but no one took it seriously until much later, because the existence of molecules wasn't accepted until generations later. (Newton had previously done the simple basics of Boyle's Law with particles, as we've looked at in the Circus) He and the whole family had a few other successes in mathematical physics to keep them satisfied, though.

There are other Gas Laws too, named after other dead guys, that involve what happens when you change temperature and gas, which we've held constant here. You can see what happens with changing temperature, which turns out to be how fast the molecules are moving, here if you didn't take the opportunity above.

Uh, OK, but what does this have to do with liquids and gases? I'm glad you asked that question! Boyle's law is a lead-in to a more general approach:
follow the yellow brick road!

Note to sticklers: yes, the pressure is the force per unit area on the lid. Imagine two identical setups like this, side by side, then fuse them together: nothing changes. There's twice the weight, twice the volume, twice the number of molecules, etc, but the behavior is the same. Pressure is a concept invented to take that scaling out of the description. We can consider the lid to have unit area "in the appropriate units" since we aren't going to consider other containers or calculate values to compare with experiment. (A real physicist's trick.) Don't try this on your exam, but it's ok here.

Return to CPO Home Page