### Video Transcript

A gas has a volume of five cubic meters. If it is heated where the volume is kept constant such that the pressure increases by a factor of four, by what factor does the temperature change?

So the question is asking us about a gas that’s heated at a constant volume. And we could imagine that as a gas inside a box like this one, where the volume of that box is fixed. The particles in a gas are free to move around. And so we’ve got particles flying around in all directions inside this box, and those particles can collide with each other. And importantly, they can also collide with the walls of the box. When a particle collides with one of the walls, for example, like this one here is about to, then, since the volume of the box is fixed, that wall isn’t going to move. The particle is simply going to bounce off of the wall, changing the direction of its velocity and exerting a force on the wall with an outward component.

Now, there are absolutely loads of particles flying around inside this box, far more of them, in fact, than we’ve drawn in this sketch. That means that there are continually particles colliding with and bouncing off all of the walls of the box. Each of these collisions results in a force on the wall with an outward component. And all these forces acting across the area of the box’s walls results in a pressure on those walls.

In this question, we’re told that our gas is heated at a constant volume and this causes the pressure to increase by a factor of four. So let’s think about why the pressure might be related to the temperature of the gas.

If a particular particle is moving faster when it collides with one of the walls, then the force that it exerts on that wall will be greater. Since all of these forces from these individual collisions result in a pressure on the walls of the box, then if the particles have a greater average speed, there’ll be a greater pressure. We can recall that the average speed of the particles in a gas indicates the temperature of that gas. Specifically, a higher temperature means a greater average speed. This means that we can say that a greater or higher temperature means a greater pressure. The relationship can be expressed mathematically as Gay-Lussac’s law, which says that pressure 𝑃 is directly proportional to temperature 𝑇.

We should keep in mind that Gay-Lussac’s law only applies for gases that are held at a constant volume. Since we’re told in this question that the volume is kept constant, we know that this law will indeed apply here.

The law says that 𝑃 is directly proportional to 𝑇. And directly proportional means that if one quantity increases by some factor, then the other increases by that same factor. So for example, if the pressure of some gas doubled from an initial value of 𝑃 to twice that value, two times 𝑃, then Gay-Lussac’s law tells us that the temperature of the gas must also double. So if it started out with a value of 𝑇, it will end up with a temperature of two times 𝑇.

Now in this question, the pressure of the gas doesn’t double, but rather it increases by a factor of four. That means that the final pressure is four times the initial pressure. From Gay-Lussac’s law, we know that the temperature of the gas must increase in proportion to its pressure. Since the pressure gets multiplied by four, the temperature must also get multiplied by four. So we can say that the final temperature of the gas is equal to four times its initial temperature. In other words, our answer is that when the pressure of the gas increases by a factor of four, the temperature also increases by a factor of four.