Gas mixtures and partial pressures

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 obviously I'm not drawing it to scale and I'm just drawing those gas molecules moving around you have gas too 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 get pressure is equal to n R 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 and 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 1 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 of the same can 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 n1 plus n2 plus n3 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 and 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 4 atmospheres and let's say we know that the total number of moles in the container is equal to 8 moles and let's say we know that the number of moles of gas 3 is equal to 2 moles can we use this information to figure out what is going to be the partial pressure due to gas 3 pause this video and try to think about that well one way you could think about it is the partial pressure due to gas 3 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 because 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 3 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 4 we know that the total number of moles is 8 we know that the moles the number of moles of gas 3 is 2 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 3 over 4 is equal to 2 over 8 is equal to 1/4 and so you can just pattern match this or you can multiply both sides by 4 to figure out that the partial pressure due to gas 3 is going to be 1 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