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okay so continuing the history of the ideal gas equation here we get to the 19th century with an Italian chemist named Amedeo Avogadro and actually his name was Lorenzo Romano Amedeo Carlo Avogadro de quoi regne add that sheddeth though but we're going to call him Amedeo and Amedeo spent a bit of his time experimenting with tiny particles and in honor of his experiments the number of particles in one mole of something was named Avogadro's number which is approximately 6.02 times 10 to the 23 and one thing that abogado postulated was that equal volumes of gas at the same temperature and pressure contain equal number of particles whether those particles are atoms or molecules so for instance if you filled up four balloons to exactly one liter at 25 degrees Celsius with different gases so let's have a green balloon and and we can say that this green balloon is argon and we'll have a pink balloon and I'll try to make it the exact same size as that as the previous one will say it's nitrogen and then we've got a blue balloon that will say as hydrogen and again I'm off a little bit in the sizes but we're saying that all of these have exactly one liter volume and then our fourth balloon will be yellow and we'll say it's filled with methane and so what our God Rho was saying was that at 1 liter each one of these four balloons would have the same number of particles so I'm using five particles but the point is that all of these balloons would have the same number of particles and so at 1 liter each of these balloons would contain point zero for one moles so point zero for one moles and if we use Avogadro's number here that's the same thing as saying that each of these balloons would be filled with two point five times ten to the twenty two particles and so if you think about it this is the exact same thought behind our value for the molar volume at STP we said that for any gas one mole at stp would take up a volume of about 20 2.4 liters and the fact that we know that this is true for any gas not just kind of a particular gases is credited to the work of Amedeo Avogadro and Avogadro used this idea and some intuition to develop his law which is V is equal to a a constant a times the number of moles which means that the moles of gas the number of particles of gas in the system varies directly with the volume or the quotient of V and n is equal to a constant and I'm sure that you've used this same intuition when you blown up a balloon as you as you put in more air particles as you blow more air into the balloon that the balloon gets larger so for example if we put in one mole of air particles the balloon would be about twenty 2.4 liters and as we blew more air into it say another third of a mole we would increase the volume to twenty nine point eight eight liters so another third of a mole of air gets us up to twenty nine point eight eight liters our volume is getting bigger and if we were to blow even more air into this system we would increase the volume even more so another third of a mole of air would bring our volume up to thirty seven point three five liters our volume has expanded proportionally to the the amount of air that we've increased in the system and so Avogadro's law validates the V in part of the ideal gas equation because the volume in the moles are directly proportional to each other so let's put this principle to work in an example so when point one five moles of helium gas are added to a piston containing point eight two moles of another gas by what percent does the total volume increase assuming isothermal and isobaric condition so isothermal same temperature and isobaric is same pressure so we have same temperature same pressure and we're looking for the the sent a volume increase related to the molar increase and so this is a perfect opportunity to use Avogadro's law so let's start with v1 divided by n1 is equal to v2 divided by into and we know this is true because Avogadro's law says that the quotient of the volume and the number of moles is constant for an ideal gas so the initial conditions would equal the final conditions and so we have a ratio set up here and we can rearrange this ratio to say that v1 / / v2 is equal to n1 over n2 and here we've got kind of an interesting part that we can actually start using to solve this problem because what we see is that the increase in volume is directly proportional to the increase in moles so for that reason the percent change in volume looks like the final volume minus the initial volume divided by the initial volume this is just the the formula kind of four percent change which is pretty intuitive the the final volume minus the initial volume the change divided by the initial volume is going to give us the percent change and so we know that this is equal to the percent change in the number of moles so we can say into minus n1 divided by n1 and this really gets us to a point where we can use the values in the formula because it gives us an initial number of particles and initial moles and it gives us the change in moles and so we have the we had the final moles as well so to find the final moles because it just gives it just says that point one five moles are added to 0.82 we'll need to add those together so point eight two plus 0.15 plus 0.15 is equal to 0.9 seven so 0.97 is is our final number of moles that would be our into so let's substitute these values in our final moles is point nine seven minus our initial moles which is point eight two divided by our initial moles and that's going to give us point one five divided by 0.8 to now we're at a point where we can solve this and we can just kind of evaluate this expression and that would give us the total or the percent of the total volume increase but I want to take a second I want to think about how we could think about this kind of with some common sense and use rounding to give us an approximate answer so 0.15 divided by 0.8 - is between 0.15 divided by 1 and 0.15 divided by 0.75 and 0.15 divided by 1 is 15% and point 1 5 divided by 0.75 it goes into 0.75 five times and so that would give us 20 percent because one-fifth is 20 percent and so if we had to think about this expression in a hurry say on a test and we didn't have a calculator we could use rounding to get us a pretty approximate answer we know that the answer is between 15 and 20 percent and it turns out that that actual answer if you do use a calculator is really close to 18 point three percent so when 0.15 moles of helium gas are added to a piston containing point eight two moles of another gas by what percent does the total volume increase what it's just about exactly 18 point three percent and we were able to solve this problem with a little bit of critical thinking based on the principles of Avogadro's law