It’s been very cold this week, even where I live in Louisiana, thanks to the outbreak of a polar vortex. This frigid air is bad for all kinds of things, including soccer helmets, apparently. But it’s actually a good time to demonstrate one of the basic ideas of science: the ideal gas law.

You probably have some balloons somewhere around the house, maybe left over from New Year’s. Try this: Blow up a balloon and tie it very tightly. I understand? Now put on the warmest jacket you have and take the balloon outside. What happens? Yes, with the drop in temperature. the balloon shrinks—the interior volume decreases—even though it still contains the *the same amount* of air!

How can it be? Well, according to the ideal gas law, there is a relationship between the temperature, volume and pressure of a gas in a closed container, so if you know two of them you can calculate the third. The famous equation is **PV = nRT**. The pressure says (**P**) multiplied by the volume (**V**) is equal to the product of the amount of gas (**north**), a proportionality constant (**R**), and the temperature (**t**). Oh, by “amount of gas” we mean the mass of all the molecules it contains.

There are a lot of things to go over here, but let me get to the main point. There are two ways to consider a gas. The one I just gave is actually the chemistry method. This treats a gas as a continuous medium, in the same way you would consider water as a fluid, and has the properties we just mentioned.

But in physics we like to think of a gas as a collection of discrete particles that move. In air, these would be nitrogen molecules (N_{2}) or oxygen (O_{2}); In the model, they are simply tiny balls bouncing around in a container. An individual gas particle has no pressure or temperature. Instead, it has mass and speed.

But here is the important point. If we have two ways of modeling a gas (as a continuum or as particles), these two models should agree in their predictions. In particular, I should be able to explain pressure and temperature using my particle model. Ah, but what about the other properties of the ideal gas law? Well, we have the volume of a continuous gas. But since a gas occupies all the space in a container, it is equal to the volume of the container. If I put a bunch of tiny particles in a box of volume **V**, that would be the same as the volume of the continuous gas. Then we have the “amount” of gas designated by the variable **north** in the ideal gas law. This is actually the number of moles of that gas. Basically it is another way of counting the number of particles. So the continuum and particle model have to coincide here as well. (Do you want to know more about moles? Here you go an explanation for you.)

Particle model for the ideal gas law

Well, if you take an inflated balloon, it will have a LOT of air molecules, maybe around 10^{22} particles. There’s no way you can count them. But we can build a physical model of a gas using a much smaller number of particles. In fact, let’s start with a single particle. Well, I can easily model a single object moving with a constant velocity, but that’s not a gas. At least I need to put it in a container. To keep it simple, let’s use a sphere.

The particle will move inside the sphere, but will have to interact with the wall at some point. When that happens, the wall will exert a force on the particle in a direction perpendicular to the surface. To see how this force changes the motion of the particle, we can use the momentum principle. This says that a moving particle has momentum (**p**) which is equal to the mass of the particle (**meter**) multiplied by its speed (**v**). So a net force (**F**) will produce a certain change in momentum (symbolized by **Δp**) per unit of time. Does it look like this: