Topic 3.6: Deviation from Ideal Gas Law

Deviation from ideal gas law: when molecules attract

In part d the student measures the actual pressure at four twenty-five kelvin, and it comes out lower than the point nine nine atmospheres we predicted. Our model missed, and the reason is in the two lines I asked you to watch, which real molecules break. Real molecules do pull on each other, even nonpolar ones like carbon dioxide, through London dispersion forces, so let’s start with the pulls.

To make the gap big enough to see, here’s a much fuller rigid tank, one mole of carbon dioxide in two liters at three hundred kelvin, and the blue needle shows what P V equals n R T predicts, about twelve point three atmospheres. Now let’s turn on the attractions and follow one molecule to the wall. The molecules behind it pull it back, so it hits more gently, and less often, too. So where do you think the real needle lands? It lands below the blue one, at about eleven point six atmospheres, six percent low, and that’s the real value, from the reference data of NIST, the U.S. national standards lab.

Here’s the same tank at three temperatures, with what P V equals n R T predicts next to NIST’s real values. At four hundred kelvin the real pressure comes out about three percent low, and cooling the tank makes the gap grow, to ten percent low at two hundred forty. Slower molecules spend longer near each other, so the pulls have more time to act, and that’s why attractions matter most when a gas is close to condensing.

And Titan’s methane lives near that edge, close enough to condensing that in its cold clouds the attractions win and it falls as rain. So, back to part d. It’s tempting to write that the molecules take up space, but that would mean more wall hits and a higher pressure, and this one came out low.

A low pressure points to the attractions, because the carbon dioxide molecules pull on each other, so they hit the walls less hard and less often, and the pressure comes out below the prediction. That’s exactly the idea the scoring guidelines gave the point for. Which of these explanations would have earned the point for part d?

Real gases: when molecules crowd

Now for the other line of the model, the one that only breaks in a really crowded gas, like the hydrogen going into this fuel-cell car in California. It goes in at seven hundred bar, about seven hundred times the air’s pressure, pre-cooled so hard that frost grows on the nozzle. Under the floor of this Toyota Mirai are three carbon-fiber tanks, a hundred forty-two point two liters in all, rated by Toyota to hold five point six kilograms of hydrogen.

So how much hydrogen does the ideal gas law say should fit in there? This time I want you to do the whole calculation yourself. How many kilograms of hydrogen does the ideal gas law say these tanks hold at seven hundred bar and fifteen degrees Celsius?

P V equals n R T promises eight point three eight kilograms. Real hydrogen at this pressure fills the tanks with only five point seven one, close to Toyota’s own rating of five point six. Almost three kilograms of it only exists on paper.

Look inside a tank. The hydrogen molecules sit only about one and a half molecule-widths apart. In ordinary air they’re about ten apart. Here their own volume fills a real share of the tank.

The molecules never get squashed. They keep their size. Only the space between them shrinks. Each molecule has less empty space to fly through, so it gets back to a wall sooner. It hits the walls more often than the ideal gas law expects.

More hits from each molecule means the gauge reaches seven hundred bar with fewer molecules. That’s why only five point seven one kilograms fit. Now run it the other way. Take the real five point seven one kilograms and divide by hydrogen’s molar mass. That’s about two thousand eight hundred thirty moles.

Put those moles into P V equals n R T, with the same volume and temperature as before. The law predicts only about four hundred seventy atmospheres. The real gauge reads six hundred ninety-one.

This is the sentence that earns the point. At extremely high pressure, the molecules’ own volume is a significant part of the container’s volume, so the real pressure is higher than the ideal gas law predicts. It’s the carbon dioxide answer in a mirror, opposite cause and opposite sign.

The gauge reads six hundred ninety-one atmospheres, but P V equals n R T says four hundred seventy. Which line of the model breaks in this tank?

Which effect wins?

Let’s put both effects on one graph, pressure across the bottom and real over ideal up the side, so an ideal gas sits on the dashed line at one. Hydrogen sits above the line the whole way. Its attractions are so weak that its size wins from the start.

Nitrogen stays almost on the line at first. Then it climbs above it. Methane dips much further, to about point eight two near a hundred seventy bar, before the crowding takes over and it climbs back through the line.

And carbon dioxide plunges, until at sixty-seven bar the line just stops, because the gas condenses. Below the line the attractions win, above it the molecules’ own volume wins, and at low pressure every gas comes back to one, which is why the ideal gas law works so well at everyday pressures. And where a curve crosses the line, the molecules haven’t stopped interacting. The two effects are just cancelling out. So under which conditions does a real gas behave most like an ideal gas?

Back to Titan

So, let’s go back to Titan. Now you can explain that rain. Inside one of these low clouds the methane flies at about three hundred fifty meters a second, a little over half its speed in a warm kitchen.

If it were warmer, they’d bounce off each other. Then they’d just keep on going. But in the cloud the methane is crowded enough and slow enough that when two molecules meet, the attraction holds them, and more join, and a droplet grows. Switch the attractions off, and the same molecules never stick, however slow. Switch them on, and they stick again, and a new drop starts to grow.

So that’s the answer to our puzzle. These molecules really are as fast as the fastest car on Earth, but that’s slow for methane, slow enough for their attractions to win, and that’s the rain.

Here’s the switch I promised you. Turn the attractions off and see what happens to Titan. So here’s the whole lesson on one shore. Gas molecules never stop moving, their hits on the walls are the pressure, and the kelvins tell you their average kinetic energy.

Below the ideal gas law’s prediction, the attractions are winning, in a gas close to condensing like this methane, and above it, the molecules’ own volume is winning, at extreme pressure like that hydrogen. In 2034, if all goes to plan, a NASA drone called Dragonfly will start flying through this sky, and now you know why there’s weather here for it to fly through.