Whenever you're performing homework questions in chemistry, don't think of them as just math problems where you plug in values into a particular equation. Chemistry is an experimental science. All of these equations are derived from experimental evidence. We're going to go through some experimental evidence to relate pressure, volume, temperature, and the number of moles to see how we derive the ideal gas equation. To do this, I'm going to take this balloon right here, and the first relationship we're going to look at is how the number of moles of a gas and the volume of the gas are related. Note that right now, there isn't much of the gas in the balloon. The volume is very small, and the number of moles are very small. So now I'm going to blow up this balloon by adding more moles of gas into it. [ Blowing up Balloon ] We can see that when I blow up the balloon and added more moles, the volume got larger. How can we express this relationship mathematically? What we just saw here is a proportion between volume and the number of moles of gas. This is a direct proportion, and we can express that by saying that the volume of a gas is directly proportional to the number of moles. This relationship is referred to as Avogadro's law. Now let's look what will happen when we look at the temperature of the gas versus the volume of the same gas. I'm going to use the same balloon, and in this Dewar right here, I have liquid nitrogen. Liquid nitrogen is very, very cold, so when I pour this over the balloon, we're lowering the temperature. [ Chemical Rumbling ] As you can see, when the temperature gets really, really cold, the volume of gas inside the balloon got much smaller. Now if I stop pouring the liquid nitrogen on the balloon, it's slowly reaching room temperature, and the volume of the gas got much, much larger. Now, how can we express this relationship mathematically in the form of a chemical equation? When the temperature was very, very high, the balloon was very large. When we lowered the temperature, the balloon got much smaller. This shows us that we have a direct proportion between the volume and the temperature. This relationship is known as Charles's law. Now let's look at the relationship between pressure and volume by taking a small balloon here and placing it in a bell jar. I have a vacuum pump right here, and what we're going to do is we're going to reduce the pressure inside of this bell jar to see what will happen to the volume. As we can see, as the pressure is decreasing, the volume of the balloon is increasing. If I remove the pressure, the volume of the balloon gets smaller again. We remove pressure, volume gets larger. And if the pressure goes up, the volume of the balloon goes back to a smaller size. So how can we express this experimental observation as a mathematical relationship? As we can see from this relationship, if we look at the volume of the balloon and see how it was proportional to the pressure, as we removed the pressure, the volume got larger. So this is an inverse proportion. So we can say that volume is proportional to 1 over the pressure. So now we have three relationships. Volume is proportional to the number of moles. Volume proportional to temperature. And volume is proportional to 1 over the pressure. We can combine all of these relationships to say that the pressure times the volume is going to be proportional to n, which is the number of moles, times T, or the temperature. Whenever we solve problems involving these relationships, we don't want a proportion. We want an equality. So we can put some kind of constant in here, or we can say that PV equals n times R, which is a constant, times T. For an ideal gas, we can use this particular equation right here to solve all of our homework problems involving gases. So when you see this equation, PV=nRT, don't just memorize it. Think of these experimental observations that we went through, and think of all the work that Avogadro, Charles, and Boyle put together to come up with this particular equation.