hello and welcome back in this tutorial we're going to take a look at the ideal gas law and the density of gases previously we saw different relationships of the properties of pressure volume moles and temperature and specifically we saw that pressure and volume are inversely proportional and so the relationship that comes out is pressure times volume is some kind of constant volume and temperature are inversely excuse me directly proportional and so when they're brought to the same side you get volume over temperature is equal to some constant and now since I'm putting putting all of those listening all together I'm just saying K Prime some some other constants since we're holding other variables constant it doesn't matter what we represent here I'm just saying some constant okay and then volume is directly proportional to moles the more of more more moles the greater volume that they'll take up at constant temperature and pressure so you get V over N is equal to some constant and I've distinguished with K Prime and K double Prime now what happens if we hold nothing constant no moles no temperature no pressure no volume constant at all so everything is allowed to move well what we simply would do is just combine all of these and that means nothing was constant so the relationship we get would be pressure times volume we have volume over temperature so volume over temperature and volume over moles so volume over moles and all of that is equal to some big old constant we'll call it R all right so if we rearrange this equation we then get PV we bring it so that it's not a quotient will multiply both sides by n T we're going to get n PV equals n times R times T and it's usually written like this because it's alphabetically listed on the other side the value of our if we come back since it is a constant and this is equal this side is equal to this side the units for R is going to have to be exactly the same as all the units that come out on the other side so it's a pretty complex set of units the value of R is 0.08206 and we have to have pressure times volume so it's liters times atmospheres per mole per Kelvin all of these all of these units are part of the constant so that the units on this side will equal the units on that side so this is called the ideal gas constant now this equation that we have derived or come up with from all of arias relationships this is called the ideal gas law and this is what we will be using when we have only one set of conditions for each of our variables so one pressure one volume one mass or number of moles and one temperature okay and in this circumstance the units for each one of these variables is going to be dictated by what units are included in R which volume must be now in liters pressure must be in atmospheres moles of course are still moles and temperature is still in Kelvin all right previously you saw the gas laws PV 1 equals PDE 2 P 1 P von with P p1 times v1 is equal to p2 times v2 so you had two sets of pressure and two sets of volume or V over T v1 over t1 equals v2 over to t2 2 sets of volume two sets of temperatures etc in those particular equations the only thing that has to be exactly the same thing as what was stated previously in the properties of gases video is temperature must always be in Kelvin irrespective of what information is given other than temperature pressure has to be the same unit on both sides so you could use you could you even use inches of mercury if you wanted you have you could use tor you can use atmosphere you can use Pascal's you can use whatever pressure unit as long as both of them are the same unit you're good volume if you use milliliters that's fine as long as both volumes are in milliliters because they'll cancel out in this scenario where you don't have two sets of anything you have only one pressure one volume 1 number of moles one temperature now this is when it becomes particular rigid if you will where the units cannot be anything other than these specific units because it's dictated by the units for R that we have all right so now this is the ideal gas law and it's pretty useful very very useful let me show you how all the other laws that you saw in the previous video where you have two constant two properties held constant you can derive every one of those from the ideal gas law let's see how we would do them so we start with the ideal gas law so if you remember this one equation you can derive all the others if you like and whatever set of conditions you have that are constant you will bring to the sidewith are the constant side everything else will go to the other side all right so let's say that we're going to hold let's see the first one that we had looked at is number of moles was constant and temperatures constant right so these are the conditions that are held constant we would need to bring moles well it's already here right with our we would need to bring temperature with art well look it's already there so this is the constant side we could just say this is constant and this side here PV is going to equal PV 1 and 2 so we can derive this relationship from the ideal gas law if we take something else let's say that temperature and pressure are held constant well if we bring pressure over here and keep temperature over here we would have V over N equals V over N 1 and 2 so it doesn't matter which set of conditions are held constant you bring that if it's just one or those if it's two conditions are held constant bring anything that's constant with are bring all the other pieces on the other side and see what you're left with and whatever's here is going to equal to the other the same thing on the other side 1 & 2 so if we hold only number of moles constant what do we get well moles with ours already there we would divide both sides by T and we'd end up with PV over T right so it would be p1 v1 over t1 equals p2 v2 over t2 and by the way this is called the combined gas law all right so very useful not only for itself but if we remember this one equation we can derive all the other relationships that we have seen up until this point all right now let's use the ideal gas law to to derive another relationship and that is the density of gases and take starting again with clean slate of PV equals NRT and I'm going to manipulate this equation where we have n over V and n over V will equal P over RT I'm just manipulated so that I have n over V and what the units are right now here our moles per liter and that's the same on both sides if we multiply this by the molar mass grams per mole look what happens we end up with mass over volume which this now gives us density so let's do that if we then take this this form of the ideal gas law and we multiply both sides by molar mass and I'm going to represent it now with MU this is density so density of gas is pressure times molar mass divided by the gas constant divided by absolute temperature and that's how we determine the density of gases all right let's take a look very quickly now at how we deal with how we deal let's say that you have a problem that says calculate the density of carbon dioxide at STP all right so we are given standard temperature and pressure so we know that the temperature is equal to 273 Kelvin our pressure is one atmosphere that's what standard STP U stands for standard temperature and pressure we have carbon dioxide and we're looking for density well density we have a relationship density is equal to pressure times molar mass over RT so that's one way we can calculate the density all right we can plug in and we would get plug all this in so the density is equal to pressure one atmosphere times the molar mass which is 44 grams per mole divided by our point zero eight two zero six liter atmosphere per mole per Kelvin divided by temperature which is 273 Kelvin and we would get our answer now let's check to make sure our unit's cancel atmospheres over atmospheres cancel we have in the denominator Kelvin per Kelvin that cancels in the numerator we have per mole and a denominator we have per mole the moles cancel and we're left with grams per liter which density okay and you can plug that into your calculator and get the value for density the units will be in grams per liter now let me show you because we're dealing with standard temperature and pressure we know that twenty 2.4 liters per mole for anything any gas at STP and for density to calculate density there's a perhaps shorter way of calculating it at STP by using the standard molar volume density remember is mass over volume so we need to start with the molar mass 44 grams per mole and if we multiply we'll get rid of moles and divided by liters to get grams per liter this is standard molar volume twenty two point four liters per mole the moles cancel and you get grams per liter so this this might be a little bit shorter if you can think to to go this this route and it will be a little bit less work to plug into a calculator or write out however these will give you the same answers and that wraps up our lesson on the ideal gas law and the density of gases