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robert boyle was an Irish scientist in the 1600s in it and it's actually from his experiments that we get Boyle's law which actually preceded the ideal gas equation and we've already shown that but we're going to work backwards and we'll use Boyle's law to prove part of the ideal gas equation and we get a little bit of history along the way which is always fun so Boyle was experimenting with gases and he had a big Jay tube set up in the entrance of his house which I'm sure his wife was thrilled about and so he trapped some gas in that Jay tube and he filled the bottom of the tube but with a little bit of mercury which trapped that gas on the closed side because mercury is a pretty dense liquid and gas has a hard time kind of moving through it so it trapped a little bit of that gas on the other side and so this left of the open side exposed to the atmosphere here so you have the pressure of the gas on one side and the pressure of the atmosphere on the other and we know that they're pressing down with it with the same amount of pressure because as he started the height of the mercury was the same on both sides now things got really interesting when he added a bit more mercury because now the two levels didn't equalize instead they were offset and what this meant was that the trapped gas pressure over here it was greater than the atmospheric pressure such that the gas pressure was equal to the atmospheric pressure plus the fluid pressure of the height difference and you can think that the gas is pushing down on this part of the mercury with the same force or with the same pressure as the atmosphere pushing down over here Plus this this bit of fluid over here pushing down now he added a bit more mercury which compressed the gas even more making its volume less and he found that there was an even greater offset and the two fluid heights and he correctly took this to mean that the gas was exerting even more pressure because now the gas pressure is equal to the atmospheric pressure plus even more fluid height and so Robert Boyle plotted this data and these are the values that he got in the middle of the 17th century he plotted the volume in big inches and he plotted the pressure in inches of mercury and he was measuring that height difference for the pressure and so starting off his volume was a hundred and seventeen point five cubic inches and his pressure was 12 inches of mercury and as he filled the the Jay tube with a little bit more mercury he had a volume of 87 point 2 and a pressure of 16 inches of mercury and as he continued to fill it he got a volume of 70 point 7 with a pressure of 20 inches and continued for a volume of 58.8 and a pressure of 24 inches and continued and got forty four point two and thirty two and as he kept going he got thirty five point three cubic inches and forty inches of mercury and then the last value was twenty nine point one inches of mercury after that's that's the volume that he had compressed it down to and 48 inches of mercury for the pressure and so what he did is he plotted this data and he graphed pressure as a function of volume so he had a graph with pressure as a function of volume and if we look at our pressure about the highest it gets is 48 inches so we'll make the top 50 and the middle part we can say is 25 as a kind of benchmark and if you look at volume the highest we have is 117 so we'll make it a hundred and we'll go a little bit over and we can kind of fill in our graph so 50 25 and 75 cubic inches for volume and so we see that when our volume is a hundred and seventeen point five our pressure is twelve so that'd be right about here and we see that as our as our volume is 87 point to our pressure goes up a little bit to 16 and we when we have our volume is 70 point seven our pressure is about 20 so that'd be about right there and when our volume is just about 60 here we have a volume of Tony I'm sorry a pressure of 24 and then as our volume goes to 44 point 2 we have 32 ish that'd be about right there and then 35 point three for the volume is about 40 for the pressure and then right under a pressure of 58 our volume would be about 29 about right there and so what we have when we plot pressure as a function of volume is we have a hyperbola and what we see is that as the volume drops by half from about 50 to 100 the the pressure essentially doubles and as we go from 50 to 25 for the volume we go from 25 to 50 for the pressure about so we have an inverse relationship for the pressure and the volume so if we graph volume then as a function of the inverse of pressure and we get this graph we've got volume as a function of the inverse of pressure so we're going to need the inverse values of all of our pressures so one over twelve the inverse of 12 would be 0.08 and one over 16 for the inverse of 16 would be about point zero six to five and if we continue finding the inverse values of these pressures we would get Oh point zero five for 20 and then 24 would be point zero four to about and one over thirty-two would be point zero three one two five and forty would be point zero two five one divided by 40 is point zero two five and then one over forty eight is point zero two zero eight and so we can populate our graph with with these values and about the highest inverse pressure value we have is point zero eight about the lowest is point zero two so we can kind of fill that in here and we're still working with the same values for for volume so the highest is a little bit over hundred and then we can put in 50 and 25 and 75 so when our volume is a hundred and seventeen point five cubic inches the inverse pressure would be about point zero eight and then as we go down eighty seven point two would be point zero six two five and seventy point seven would be point zero five right in the middle here and then 58.8 just about sixty would be point zero four two and forty four point two would be point zero three one two five and then thirty five point three would be point zero two five and twenty nine point one would be point zero two zero eight and this isn't a perfectly clean graph but but we do see that when we graph volume as a function of the inverse of pressure we get a straight line and if we write this a straight line graph as an equation it would be y is equal to MX plus B that's the equation for for this graph where M is our slope and B is our y-intercept but our y-intercept here is zero so all we all we really need is Y is equal to MX well in our graph Y our y-value is our volume and our x-value is the inverse of our pressure so that's let's fill that in here if we we call our slope K instead of in if we just use a different letter then we'll get V is equal to K times one over P and multiplying both sides by P would give us PV is equal to K or in other words the product of the volume and the pressure for a gas is a constant value just like we see in the ideal gas equation so let's test this out by going back to those original values that Robert Boyle plotted if we measure the product of the pressure and the volume here we'll see that 117 times 12 is just about 1400 and 87 times 16 is just about 1400 and in fact all of these volumes multiplied by the pressure the product is always almost exactly 1,400 and so one great application of this concept is that if the number of moles and in the temperature of an ideal gas are constant then the initial product of P and B will equal the final product of P and B so PF and VF or final and so let's try to use this in an example if the pressure of a gas and a 1.25 liter container is initially 0.87 2 atmospheres what is the pressure if the volume of the container is increased to 1.5 liters assuming that the temperature doesn't change and we know that if this is a closed container the number of particles isn't going to change so our moles are also constant and so let's use this idea that p1 v1 is equal to p2 v2 and our initial pressure is point 8 7 2 atmospheres and our initial volume is 1.25 liters and we're looking for the final pressure when the when the final volume is 1.5 liters and so the first thing that we're going to need to do is divide both sides by 1.5 liters to isolate our final pressure and so on this side we completely cancel out 1.5 liters and on this side we cancel out our units of liters and we get 0.87 2 times 1.25 divided by 1.5 then we'll retain our unit of atmosphere here and that will give us our final pressure which happens to be 0.72 7 atmospheres and just kind of as a final kind of common-sense check this result follows Boyle's law because we increase the volume from 1.25 to 1.5 and so we decreased the pressure from 0.87 to 2 0.727