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Derivation of gas constants using molar volume and STP

Video transcript

alright so we just put together an equation based off some basic observations of gas and we called it the ideal gas equation or PV equals in our T and for the most part the equation is pretty intuitive but I remember that when I was learning it for the first time the part that confused me the most was this constant value R so that's what I want to try to clear up for you now so for starters R is a constant and what that means is that as the rest of the variables change with whatever situation we're looking at the R value is going to stay the same so let me show you what I'm talking about let's isolate R so when we divide both sides by the moles and the temperature we're going to get R is equal to PV over NT which means that for any ideal gas if you multiply the pressure and the volume and you divide that product by the number of moles and by the temperature in the system you're going to get the same number again for any ideal gas you'll always get the value R so I'm going to show you how this works but first I need to make sure that we're on the same page about a couple different things first I need you to know that when I say STP I mean standard temperature and pressure in the standard temperature is 273 Kelvin which is the same thing as zero degrees Celsius and the standard pressure is one atmosphere so standard temperature and pressure is just kind of a nice theoretical condition that we can perform kind of situational experiments with and so the second thing that I need to clarify is that we find experimentally that for any ideal gas one mole of gas takes up a volume of approximately twenty 2.4 liters and we're going to use these conditions to find the ideal gas constant so let's start with R equals PV over NT and for our conditions we're going to talk about one mole at standard temperature and pressure so for our pressure we have one atmosphere and then we have one mole and we know that one mole is twenty 2.4 liters so that's our volume and again at standard temperature and pressure which talking about 273 Kelvin and this should equal our so if we solve this out our ones are going to cancel and we really just need to divide 20 2.4 by 273 and that's going to give us point zero eight two one and that our units are going to be atmospheres times leaders so atmospheres times leaders / moles times Kelvin so the constant R is equal to point zero eight two one atmospheres times liters / moles Kelvin so this is the ideal gas constant that's going to be the same for all ideal gases as long as we're dealing with pressure and atmospheres and volume and liters and it's probably the one most often used in general chemistry but and I'm and I mentioned this earlier sometimes we deal with pressure in a unit called a Pascal and we deal with volume in cubic meters and we use both of these because they're both based off of SI units and so we might have another value for R which looks like this we're still going to start with R equals PV over NT and we're still going to use standard temperature and pressure but for our for our centre pressure instead of using atmospheres we're going to use Pascal's and so one atmosphere is equal to 101,325 pascals so 101,325 pascals is going to be our pressure and we're still going to use one mole of gas but instead of measuring the volume of one mole and liters instead of saying that it's equal to 20 2.4 liters I want to do it in cubic meters because that's another SI unit so one cubic meter is equal to one thousand liters and this means that if we take 20 2.4 liters and and we do a dimensional analysis we're going to get point zero two to four meters cubed and so that's the value that we're going to use in our formula because we're still talking about one mole so point zero two to four and then last but not least we're still talking about under temperature and pressure which in and Kelvin the temperature would be 273 so we have our pressure we have our volume we have our moles and we have our temperature and so if we run this through a dimensional analysis to see if we can cancel any units we would start with 101,325 pascals but keep in mind that a Pascal is the same thing as saying a Newton per meter squared because it's a SI unit of pressure so we'd have 101,325 Newton's per meter squared and then we would take our volume and that's point zero 2 to 4 meters cubed and so we would put that in there and our meters most of our meters would cancel two of the meters on top and two of the meters on the bottom leaving us with just a meter on top and then we would add in a division by 1 mole because we're dividing by 1 mole and we'll put that on the bottom and then we'll finish it off by putting 273 Kelvin on the bottom and so if we simplify this down we'd say the 101,325 times point 0 2 to 4 divided by 273 and that would give us a value of R equals 8.314 newton meters per mole Kelvin and we could simplify the unit's just to take more because newton meters are the same thing as joules so we could just insert a Joule and we'd have joules per mole Kelvin now to be sure the constants look different here but keep in mind that the only thing changing are the units the values here are so same and it's always the same for any ideal gas so for any ideal gas system the product of the pressure and the volume divided by the number of moles and the temperature is equal to R which might look like eight point two one times 10 to the negative 2 atmosphere liters per mole Kelvin or it might look like eight point three one for joules per mole Kelvin but again this is the same value so think about what this means if R is constant then R is equal to PV over NT for the initial state of a gas so for the initial state of a gas but R is also equal to PV over NT for the final state of a gas so for a final state of gas and this is true because PV over NT always equals R for any ideal gas and so what we're really saying here is that the initial PV over NT is equal to the final PV over NT which opens up all sorts of neat kind of predictive possibilities say if we were to change one of the variables and we're going to explore these possibilities in the next few videos