 Now we're going to talk about a different way to use the ideal gas law in ratio problems. So our ideal gas law, the basic equation we use in physics, is pV equals nRT. I have a separate video covering exactly what all these terms mean. But this particular equation is used for the state of a gas sample at one moment in time. So you have to have the pressure, volume, number of moles of gas and temperature of the gas all at the same moment. But sometimes you have a single sample at two different times and you want to compare what's going on as the sample changes. So you've got a p1v1n1t1 for the first time and the same equation for a second time. So we're specifying what the values are at two different times. So now we can construct our ratio equation. And basically we just take one equation and divide it by the other. And it doesn't actually matter which one of the two we put on top. Depending on what you're solving for, you might want to use one or the other of these versions. They're exactly equivalent to each other. So now let's see how we can simplify these equations. So here's our ratio equation. But notice that the universal gas constant, R, is the same all the time. And that means you can actually cancel it out of the top and bottom. Now another common simplification is if we have a closed system. A closed system refers to one where no gas is allowed to escape the system or enter the system. And that means necessarily that n1 and n2, the amount of gas, has to stay the same. So again, that can be divided out. So now we've got an equation where we just have to look at the relationship between the pressures, the volumes, and the temperatures. So let's have an example. So let's say we start with a closed system. So we've got our equation here that relates the pressure, volume, and temperatures. And let's say we also hold the pressure constant. So somehow or another, we're making sure maybe it's kept at normal atmospheric pressure. That means my pressures also can divide out because the pressure didn't change. So my equation simplifies even more. Now if I have any three of these values, I can find the fourth one. So for example, if I wanted to solve for the second volume, it depends on both temperatures and the original volume. So we can have less than full information. I don't actually know the number of moles of gas. And I may not actually know the pressure, but I know those things didn't change. Then I can find out how the other variables are related to each other. So that's a brief introduction to how we can use ratio problems with the ideal gas law. We'll see more examples in class.