Topic 3.5: Kinetic Molecular Theory

Kinetic molecular theory: rain on Titan

This is Titan, Saturn’s biggest moon, a little bigger than the planet Mercury, and wrapped in a sky so thick and orange that from out here you can’t see the ground at all. In January 2005, a probe called Huygens fell into that sky on a parachute, and it’s still the farthest from home anything we’ve built has ever landed. The Cassini spacecraft overhead found clouds and rain, and whole seas near the north pole, the biggest one larger than the Caspian Sea.

But the rain on Titan isn’t water. It’s methane, the gas that burns blue on a kitchen stove, because down here it’s about minus one hundred seventy-nine degrees Celsius. The drops are big and slow, and they drift down about as gently as snowflakes. It’s minus one hundred seventy-nine degrees down here, so what do you think the methane molecules in this air are doing?

Right next to a drop, the methane molecules are anything but gentle, flying at about three hundred fifty meters a second, about as fast as the fastest car ever driven on Earth. So how do molecules that fast ever settle into a drop? The ideal gas law you learned last time treats a gas as tiny particles that never pull on each other, and particles like that never turn into a liquid, however cold they get.

So if methane really behaved like an ideal gas, Titan would look like this. No clouds, no rain, no seas, just an empty basin. The real Titan is wet, because real molecules aren’t ideal. By the end of this lesson, you’ll be able to explain what temperature and pressure really are for the molecules in a gas, and how molecules as fast as the fastest car on Earth end up as slow rain on a frozen moon.

And you’ll know when the ideal gas law stops telling the truth. And which way it gets it wrong. And at the very end, that switch, the one that dried this moon out, is going to be yours. First we need the model behind the ideal gas law, the kinetic molecular theory, and the best place to watch it is a desert morning on Earth, next to the fastest car ever built.

Fast, but going nowhere

This is the Black Rock Desert in Nevada, and on the morning of October fifteenth, 1997, a British team came out here to set a record. That’s ThrustSSC, pushed by two jet engines from a fighter plane, with Andy Green of Britain’s Royal Air Force at the wheel. He averaged seven hundred sixty-three miles per hour, three hundred forty-one meters every second, in the first car ever to officially go faster than sound.

Let’s freeze it right here. Titan’s methane flies about as fast as this car, and here the car isn’t even the fastest thing in the picture. Let’s dive into the air in front of its nose, down to the size of the molecules, and slow time down about a hundred billion times.

That’s the air, mostly nitrogen, and if we take it at three hundred kelvin, about room temperature, the most common nitrogen speed is about four hundred twenty meters a second, so most of these molecules are faster than the fastest car ever built. And yet on a morning with no wind, all that air just sits there. So how can air that fast stay still? Explaining what you see, like calm air, from what its molecules are doing is exactly the kind of question the AP exam asks.

Let’s follow one nitrogen molecule. It flies straight until it hits another molecule, then bounces off in a new direction at a new speed, billions of times a second. Now let’s give every molecule an arrow for its speed and direction. For every molecule heading one way, another heads the opposite way, so all that motion cancels out.

Behind the car it’s different. Those molecules still zip around at random, but they also share one drift, and a shared drift is what you feel as wind. So still air is fast, but it’s heading every which way at once. As a whole, it goes nowhere.

This picture is called the kinetic molecular theory, and kinetic just means motion. First, gas particles never stop moving, in straight lines between collisions, in random directions. Second, they’re tiny compared with the space between them, about ten molecule-widths apart in air. Third, in this ideal model they don’t pull on each other at all.

And fourth, when two of them collide they can trade energy, one speeding up while the other slows down, but their total stays the same, so we call those collisions elastic. Keep an eye on lines two and three, because Titan breaks one of them and a hydrogen tank breaks the other. Particle drawings like these are a classic AP question, so let’s try one. Which box shows the molecules in still air?

Pressure: molecules hitting the walls

So what happens when these fast molecules hit something solid, like the canopy over Andy Green’s head? Each hit gives the canopy a tiny push, and every square centimeter gets about three times ten to the twenty-third hits every second. All those pushes blur into one steady push, and the push on each bit of wall is what we call pressure.

The canopy doesn’t cave in, because the cockpit air hits its inside face too, pushing back from the other side. So to explain any pressure, talk about molecules hitting the walls, which is exactly how the AP exam wants it.

Pressure comes down to two things, how often the molecules hit the walls, and how hard. Heat a gas in a sealed, rigid container and you change both, because faster molecules hit more often and harder.

Squeeze it into half the space at the same temperature, and the molecules are no faster, but they hit the walls more often. Now let half the molecules out, and keep the volume and temperature the same. The rest move just as fast on average. They just hit the walls half as often. Which change doubles the pressure without changing how fast the molecules move?

The Maxwell–Boltzmann distribution

Our molecule’s speed never sat still, and neither does any other’s, so the air has a whole spread of speeds. Let’s freeze this box of air and drop each nitrogen molecule’s speed onto a line across the desert floor, with ThrustSSC parked at three forty-one.

Every bead is one molecule, slow ones near zero, fast ones way past the car, and most in between. Make the bins narrower and the pile becomes a smooth curve, with its peak, about four hundred twenty meters a second, just past the car. It’s called the Maxwell–Boltzmann distribution, and the rest of this topic keeps coming back to it.

Let’s put it on the board. Speed goes along the bottom. The fraction of molecules goes up the side. The peak is the most probable speed, four hundred twenty-two meters a second, the speed more molecules have than any other.

The average is a bit higher, about four seventy-six, because the long tail of fast molecules pulls it up. That tail is also why this isn’t a bell curve. It starts at zero, rises steeply and trails off slowly toward high speeds.

The whole area under the curve is every molecule, and the area over any range of speeds is the fraction in that range. So if we mark the car’s speed, the area to its right is the fraction faster than ThrustSSC, about seventy-three percent. Hold on to that seventy-three percent, because soon you’ll change it yourself.

Temperature is average kinetic energy

So what does temperature do to this curve? Let’s heat this air from three hundred kelvin all the way to six hundred. When we double the temperature from three hundred to six hundred kelvin, what happens to the curve’s peak?

To see what temperature really measures, we need kinetic energy, and for one molecule it’s one half m v squared. Let’s do one nitrogen molecule at five hundred meters a second. First I need its mass in kilograms, so I divide twenty-eight point oh two grams per mole by Avogadro’s number, and then by a thousand to turn grams into kilograms.

Grams cancel, moles cancel, and one molecule is four point six five three times ten to the negative twenty-sixth kilograms. Then into the formula, one half times that mass times five hundred squared. A kilogram meter squared per second squared is a joule, so it’s five point eight two times ten to the negative twenty-first joules, tiny, which makes sense for one molecule.

Now what if it were going twice as fast? Before you reach for a calculator, think about that square. Twice the speed means two squared, four times the energy, about two point three three times ten to the negative twentieth joules, and now I want you to try one. What’s the kinetic energy of this helium atom? It’s moving at twelve hundred meters a second.

Every particle has its own kinetic energy, but their average is proportional to the temperature in kelvins, and that’s the last piece of the kinetic molecular theory. Double the kelvins and you double the average kinetic energy, so air at twenty-seven degrees Celsius would have to reach three hundred twenty-seven.

But the speeds don’t double. The peak only goes from four twenty-two to five ninety-seven, about forty percent faster, because energy goes with speed squared. It’s the square again, run backwards, so when v squared doubles, the speed only goes up by the square root of two, about one point four one times.

Back in the desert, the peak is still about eighty meters a second ahead of the car. Now, the thermostat is yours. Drag the thermostat until the most common nitrogen speed is exactly the car’s speed.

But look at the area to the right of the car. Even this cold, about fifty-seven percent of the molecules still beat ThrustSSC. At three hundred kelvin it’s seventy-three percent. The car never really had a chance.

Same temperature, different speeds

Now, keep the desert at three hundred kelvin, and swap the air for helium. Look at that. Helium runs past a thousand meters a second, more than two and a half times faster than nitrogen. And heavier argon peaks at about three fifty, so ThrustSSC could almost keep up with a typical argon atom.

Same temperature, completely different speeds, so does a helium atom have more energy than an argon atom? Let’s switch the line on the ground from speed to kinetic energy. The two curves land right on top of each other. At the same temperature, every gas has the same average kinetic energy. Helium atoms are lighter, so they have to move faster to carry the same energy.

Now back to Titan, where the air holds two gases at one temperature, nitrogen and methane, both at about ninety-four kelvin. These two curves are the nitrogen and the methane in Titan’s air. Which one is methane, and how do their average kinetic energies compare? Titan’s methane outruns its nitrogen only because it’s lighter. Any two gases at one temperature work the same way.

Kinetic molecular theory on the AP exam

Here’s a real question from the 2019 AP Chemistry exam. Carbon dioxide in a rigid container at two hundred ninety-nine kelvin and point seven oh atmospheres is heated to four hundred twenty-five kelvin. Part a asks how heating changes the motion of the molecules, and the answer is that their average kinetic energy goes up, so they move faster on average.

Part b wants the new pressure. The container is rigid and sealed, so n and V are fixed, which means P over T stays the same. So point seven oh atmospheres times four twenty-five over two ninety-nine, the kelvins cancel, and it’s point nine nine atmospheres, to two significant figures, like the point seven oh we started with.

Part c trips people up, because it asks you to explain the higher pressure with the kinetic molecular theory. I’d write it like this. At the higher temperature the molecules move faster, so they hit the walls of the container more often and with more force, and the pressure increases.

The point went for collisions with the walls, and either more often or more force was enough, so writing both is the safe answer. Two answers miss it. Molecules hitting each other more doesn’t push on the container, and molecules never get bigger when you heat them. So the model handles parts a, b and c beautifully, and then part d asks something it can’t explain on its own.