The mole and Avogadro's number

in a previous video we introduced ourselves to the idea of average atomic mass which we began to realize could be a very useful way of thinking about a mass at an atomic level or at a molecular level but we're to do in this video is connect it to the masses that we might actually see in a chemistry lab you're very unlikely to just be dealing with one atom or just a few atoms or just a few molecules you're more likely to deal with several grams of an actual substance so how do we go from the masses at an atomic scale to the masses masses of samples that you see in an actual chemistry lab or and I guess you could say our scale of the world well the chemistry community has come up with a useful tool they said alright let's think about a given element so say lithium we know it's average atomic mass is six point nine four six point nine for unified atomic mass units per atom atom of lithium what if there were a certain number of atoms of lithium such that if I have that number so times certain certain number of atoms then I will actually end up with six point nine four grams grams of lithium and this number of atoms is six point zero two to one four zero seven six times ten to the 23rd power so if you have a sample with this number of lithium atoms that sample is going to have a mass of six point nine four grams whatever its average atomic mass is in terms of unified atomic mass units if you have that number of the atom you will have a mass of that same number in terms of grams now you might be saying is there a name for this number and there is indeed a name and it is called Avogadro's number named in honor of the early 19th century Italian chemist amedeo avogadro and in most contexts because you're not normally dealing with data with this many significant digits we will usually approximate it as 6.0 to 2 times 10 to the 23rd power now there's another word that it's very useful to familiarize yourself with in chemistry and that's the idea of a mole now what is a mole it is not a little mark on your cheek it is not a burrowing animal actually it is both of those things but in a chemistry context a mole is just saying you have this much of something the word mole was first used by the German chemist Wilhelm Ostwald at the end of the 19th century and he came up with the word because of its relation to molecule now what does that mean well think about the word dozen if I say I've got a dozen of eggs how many eggs do I have well if I have a dozen of eggs that means I have 12 eggs so if I say I have a mole of lithium atoms how many lithium atoms do I have that means that I have 6.022 one four zero seven six times ten to the 23rd lithium atoms exact same idea it's just that Avogadro's number is a much hairier of a number than a dozen so let's use our newfound powers of the mole and Avogadro's number to start doing some useful things let's say that someone were to walk up to you and say hey you I have a fifteen point four milligram sample of germanium how many atoms of germanium am i dealing with pause this video and try to think about that so let me clear out some space the periodic table of elements was taking up all right so we started off with fifteen point four milligrams of germanium the first step might be hey let's convert this to gray of germanium and so we can do a little bit of dimensional analysis we can just multiply this for every one gram of germanium that is equivalent to one thousand milligrams milligrams of germanium and so if you essentially multiply by one thousandth or divide by a thousand we're gonna get the grams of germanium and you could see that in the dimensional analysis by seeing that that is going to cancel out with that leaving us with just the grams of germanium and now that we have an expression for grams of germanium we could think about moles of germanium so how do we do that well we're going to multiply by some quantity and in the denominator we're going to want grams of germanium for the dimensional analysis to work out grams of germanium and in the numerator we want the new expression to be in terms of moles of germanium so one mole of germanium is equal to how many grams of germanium well we see it right over here germanium molar mass is seventy two point six three grams per mole so for every mole we have seventy two point six three grams of germanium and you can see that the units work out these grams of germanium are going to cancel with the grams of germanium just leaving us with moles of germanium and in actual chemistry practice finding out the moles of a substance might actually be the most useful thing but if you wanted to find out the actual atoms of germanium that we're dealing with we will just multiply by the number of atoms you have per mole and this is going to be true for any element for every mole you have Avogadro's number of atoms and we're going to approximate that as 6.022 times ten to the 23rd atoms atoms of germanium for every one mole mole of germanium and so just to review what we just did we had milligrams of germanium you multiply these two together you'll have grams of germanium which makes sense you're essentially just dividing by a thousand if you were to multiply your grams of germanium times the mole per gram which is really just the reciprocal of this molar mass we got here and just to make sure where it makes sense the units work out nice through the dimensional analysis this right over here tells you your moles moles of germanium and then if you take your moles and then you multiply it by Avogadro's number it tells you how many atoms of germanium we have and that makes sense if if I told you I had a certain number of dozen of eggs if I want to know how many eggs that is I would multiply by 12 so this whole expression is the number of atoms atoms of germanium so we have 15 point 4 milligrams if we want to figure out how many grams we have we then divide that by a thousand that's what our dimensional analysis tells us and it also makes logical sense divided by a thousand so this is how many grams we have and then if we want to figure out how many moles and it's going to be a small fraction of a mole because a mole is 72 point 6 3 grams per mole we have a small fraction of a gram much less 72 point 6 3 grams and so we saw from our analysis to figure out the number of moles we're now we're going to essentially divide by seventy two point six 3 so divided by 72 point six 3 is equal to this is the number of moles of germanium we have and if we want to figure out the number of atoms of germanium well then multiply that times Avogadro's number so times 6.022 times 10 to the 23rd and this ie button means times 10 to the OEE 23rd power so that's how you do it on a calculator and then that gives us this many atoms and let's see just to get our significant digits here our significant figures out of all of the things we multiplied see we had 4 significant digits here the four significant digits here but we only had 3 over here so I'm going to round to three significant digits so I'll go to one point two eight times 10 to the 20th atoms so we have approximately one point two eight times 10 to the 20th atoms of germanium which is a lot