The mole and Avogadro's number | Atomic structure and properties | AP Chemistry | Khan Academy

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 mass at an atomic level or at a molecular level. But what we're going 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 of samples that you see in an actual chemistry lab, or in, I guess you could say, our scale of the world?

Well, the chemistry community has come up with a useful tool. They've said, "All right, let's think about a given element." So, say lithium. We know its average atomic mass is 6.94 unified atomic mass units per atom of lithium. What if there were a certain number of atoms of lithium such that if I have that number—so times a certain number of atoms—then I will actually end up with 6.94 grams of lithium?

And this number of atoms is 6.02214076 times 10 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 6.94 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 Amadeo Avogadro. In most contexts, because you're not normally dealing with data with this many significant digits, we will usually approximate it as 6.022 times 10 to the 23rd power.

Now there's another word that is 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. Doesn't, 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 6.02214076 times 10 to the 23rd lithium atoms. Exact same idea. It's just that Avogadro's number is a much heavier 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 15.4 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 15.4 milligrams of germanium. The first step might be, "Hey, let's convert this to grams 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 1000 milligrams of germanium.

And so if you essentially multiply by 1000 or divide by a thousand, we're going to 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 can 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's molar mass is 72.63 grams per mole. So for every mole, we have 72.63 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.

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 10 to the 23rd atoms of germanium for every one 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 moles 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 with the dimensional analysis. This right over here tells you your 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 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 of germanium. So we have 15.4 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. 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 72.63 grams per mole, we have a small fraction of a gram, much less than 72.63 grams.

And so we saw from our analysis to figure out the number of moles, we're now going to essentially divide by 72.63. So divided by 72.63 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, we'll then multiply that times Avogadro's number. So times 6.022 times 10 to the 23rd. And this e button means times 10 to the 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 four significant digits here, four significant digits here, but we only had three over here.

So I'm going to round to three significant digits, so I'll go to 1.28 times 10 to the 20th atoms. So we have approximately 1.28 times 10 to the 20th atoms of germanium, which is a lot.