Beyond Boyle

Beyond Boyle!

If you've played with the Molecular Circus Boyle's Law simulator and thought, "Gee, how can I change the molecule speeds? I bet that would do something cool!" here's your chance.

There's a scrollbar to make the molecules fast or slow. You can plot points as before, but if you change the speed between points, they won't fall on the hyperbola anymore. A nice official thing to do is plot several points at one speed, note the hyperbola, then erase the points by clicking on them, change the speed, and plot some more points. You'll see what mathematicians call a family of hyperbolae, each one being an instance of Boyle's Law.

If you think about what's mechanically happening to the lid, you can segue into some of the other Gas Laws. There are two issues with molecule speed:

Faster molecules cover the distance back and forth faster, so they'll hit the lid more frequently if the volume doesn't change. If the molecules are twice as fast, they'll have twice the impact rate; three times as fast, three times the impact rate, and so on.
Faster molecules deliver a bigger recoil kick to the lid when they bounce off. Newton says that this is momentum which is proportional to speed, so twice the speed, twice the kick in each impact.

Putting these two issues together, twice the speed means twice the impact rate with each impact being twice the kick: the pressure is then four times as great. Three times the speed, three times three or nine times the pressure. Mathwise, pressure is proportional to speed squared if volume is constant. As before, even if molecules are moving in all directions at different speeds, this still works for the average, but now we're talking average square speed. You may know that kinetic energy goes as speed squared, so we can restate that as: pressure is proportional to average kinetic energy of a molecule.

What you may not know yet is that temperature is just the average molecule energy! Taking that on faith for now, we can make one last restatement and say that

Pressure is proportional to Temperature, with Volume held constant

Yes! Whatzisname's Law! Well, some people might get excited. The point is that all those named Gas Laws connect to molecule mechanics: they aren't just formulas to torture students. This is the obscure gas law: points if you know that it's Amontons' Law, after Guillaume Amontons who did many things, including pioneer in data communications (interestingly, he was deaf).

If instead you hold the pressure constant and let the volume change with temperature, there's another Gas Law: did you know that both Jacques Charles and Joseph Louis Gay-Lussac were balloonists?

Extra credit: can you deduce that the rectangular area in the Boyle's Law hyberbola on the previous page is constant because it's the energy of the gas?

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