Gas mixtures and partial pressures

- [Instructor] In this video, we're going to introduce ourselves to the idea of partial pressure due to ideal gases. And the way to think about it is imagine some type of a container, and you don't just have one type of gas in that container. You have more than one type of gas. So let's say you have gas one that is in this white color. And obviously, I'm not drawing it to scale, and I'm just drawing those gas molecules moving around. You have gas two in this yellow color. You have gas three in this blue color. It turns out that people have been able to observe that the total pressure in this system and you could imagine that's being exerted on the inside of the wall, or if you put anything in this container, the pressure, the force per area that would be exerted on that thing is equal to the sum of the pressures contributed from each of these gases or the pressure that each gas would exert on its own. So this is going to be equal to the partial pressure due to gas one plus the partial pressure due to gas two plus the partial pressure due to gas three. And this makes sense mathematically from the ideal gas law that we have seen before. Remember, the ideal gas law tells us that pressure times volume is equal to the number of moles times the ideal gas constant times the temperature. And so if you were to solve for pressure here, just divide both sides by volume. You'd get pressure is equal to nR times T over volume. And so we can express both sides of this equation that way. Our total pressure, that would be our total number of moles. So let me write it this way, n total times the ideal gas constant times our temperature in kelvin divided by the volume of our container. And that's going to be equal to, so the pressure due to gas one, that's going to be the number of moles of gas one, times the ideal gas constant times the temperature, the temperature is not going to be different for each gas, we're assuming they're all in the same environment, divided by the volume. And once again, the volume is going to be the same. They're all in the same container in this situation. And then we would add that to the number of moles of gas two times the ideal gas constant, which once again is going to be the same for all of the gases, times the temperature divided by the volume. And then to that, we could add the number of moles of gas three times the ideal gas constant times the temperature divided by the volume. Now, I just happen to have three gases here, but you could clearly keep going and keep adding more gases into this container. But when you look at it mathematically like this, you can see that the right-hand side, we can factor out the RT over V. And if you do that, you are going to get n one plus n two plus n three, let me close those parentheses, times RT, RT over V. And this right over here is the exact same thing as our total number of moles. If you say the number of moles of gas one plus the number of moles of gas two plus the number of moles of gas three, that's going to give you the total number of moles of gas that you have in the container. So this makes sense mathematically and logically. And we can use these mathematical ideas to answer other questions or to come up with other ways of thinking about it. For example, let's say that we knew that the total pressure in our container due to all of the gases is four atmospheres. And let's say we know that the total number of moles in the container is equal to eight moles. And let's say we know that the number of moles of gas three is equal to two moles. Can we use this information to figure out what is going to be the partial pressure due to gas three? Pause this video, and try to think about that. Well, one way you could think about it is the partial pressure due to gas three over the total pressure, over the total pressure is going to be equal to, if we just look at this piece right over here, it's going to be this. It's going to be the number of moles of gas three times the ideal gas constant times the temperature divided by the volume. And then the total pressure, well, that's just going to be this expression. So the total number of moles times the ideal gas constant times that same temperature, 'cause they're all in the same environment, divided by that same volume. They're in the same container. And you can see very clearly that the RT over V is in the numerator and the denominator, so they're going to cancel out. And we get this idea that the, I'll write it down here, the partial pressure due to gas three over the total pressure is equal to the number of moles of gas three divided by the total, total number of moles. And this quantity right over here, this is known as the mole fraction. Let me just write that down. It's a useful concept. And you can see the mole fraction can help you figure out what the partial pressure is going to be. So for this example, if we just substitute the numbers, we know that the total pressure is four. We know that the total number of moles is eight. We know that the moles, the number of moles of gas three is two. And then we can just solve. We get, let me just do it, write it over here, I'll write it in one color, that the partial pressure due to gas three over four is equal to two over eight, is equal to 1/4. And so you can just pattern match this, or you can multiply both sides by four to figure out that the partial pressure due to gas three is going to be one. And since we were dealing with units of atmosphere for the total pressure, this is going to be one atmosphere. And we'd be done.