Calculations using Avogadro's number (part 1) | Chemistry | Khan Academy

I have about 3.21 grams of sulfur powder over here. My question to you is, how many atoms of sulfur are there? At first, this question sounds ridiculous. I mean, there's going to be lots and lots of atoms. How in the world are we going to count that? That's what we're going to find out in this video. We're going to do that by introducing the idea of the mole.

So let's begin. To come up with the idea of moles, we first need a new unit of mass to deal with the masses of atoms. See, atoms are very tiny. Their masses are going to be incredibly tiny. So kilograms or grams are going to be very inconvenient to use. So we come up with a new unit called the atomic mass unit, AMU or U. It's a very tiny unit of mass, just like grams or kilograms. It's a unit of mass, but of course, whenever we learn about a new unit, we need to ask ourselves, how big is that unit? What is the definition of that unit?

How big is one U? Well, here's how we define what a U is. Okay, you take a single atom of carbon-12. Now, its mass, by definition, is 12 U. This is not something that we have measured; this is something that we fixed. We fixed the mass of a carbon-12 atom to be 12 U exactly. Okay, now what is 1 U? Well, if the mass of a carbon-12 atom is 12 U, 1 U is one-twelfth of its mass, right? So we define one atomic mass unit, one U, as one-twelfth of the mass of a single atom of the carbon-12 isotope. Does that make sense?

Well, I'm sure at this point you may be having some questions, like why did we decide to use carbon as a reference and not any other elements? Well, it turns out that we actually started with hydrogen because it's one of the lightest elements. Then we ran into some problems, and then we switched to oxygen because, again, it's extremely abundant. Then again, we ran into some other problems, and then finally, we decided to go with carbon, which is also abundant. We'll not delve into the histories and details of what really happened, but yeah, we have to choose some element as a reference, and we ended up choosing carbon as a reference.

Another question you could be having is why do we fix the mass of a single atom of this carbon-12 to be 12 U? Why not any other number? Why 12? Well, for that, you can see that over here carbon has how many protons and neutrons in it? Well, it has a total of, I mean, it has six protons and six neutrons. So it has a total of 12 protons and neutrons, 12 particles. I think of protons and neutrons together over here because they have pretty much similar mass. I mean, a neutron is actually slightly heavier than a proton, but for our purposes to get an intuition over here, their masses we can pretty much think of them to be almost equal to each other. So it has a total of 12 particles, right?

Now, by fixing the mass of those 12 particles to be 12 U, look at what we're doing. We are basically saying, "Hey, let's fix the mass of a single proton or a neutron to be about one U." That was the whole intention! Okay, so you can also think of one U as kind of a representation of the mass of a single proton or a neutron. But again, this is not exact because the masses of protons and neutrons are not exactly equal to each other. So a proton and neutron will have a mass very close to one U, but it's not exactly one U. But it's a good way to think about what a U represents. It represents sort of the mass of a proton or a neutron.

Anyways, now that we understand this, here's a question: What do you think is the mass of a single atom of the oxygen-16 isotope? A single atom of this? What would be its mass in U, atomic mass unit? Well, it has a total of 16 particles, 16 protons and neutrons together. And since each particle, each proton or neutron, has a mass of one U and there are a total of 16, the oxygen mass will be about 16 U. Again, you can see it's not going to be exactly 16 U because the mass of each proton or neutron is not exactly one U, but it's going to be very close to that.

Similarly, if you take an isotope of, say, chlorine, a particular isotope, the most abundant isotope of fluorine, which has 35 protons and neutrons together in it, well then its mass will be close to 35 U. Makes sense, right?

Okay, now here's a question we're going to ask ourselves. Let's go back to carbon. Each carbon has a mass of 12 U, right? By definition, now how many carbon atoms do I need to take together such that the total mass of all of those carbon atoms together becomes 12 g? You can imagine it's going to be lots and lots of atoms, right? Because each atom has a very tiny mass and we want together 12 g. So we probably need to take billions and billions and billions of atoms. But the big question is, how many atoms do I need to take so that they all add up to give me 12 g of mass?

Well, it turns out we figured it out. Again, we'll not get into the details of how we figured it out. Okay, the history is actually pretty interesting, but again, we will not talk about that over here. But we figured it out, and it turns out to be this number: you need to take about 6.022—and there are some other decimals over here, some numbers here—times 10^23, which is a huge number. Okay, if you take these many carbon-12 atoms together, they will together have a mass of 12 g. This number is what we call the Avogadro number, named after the scientist A.O. Avogadro, who worked a lot on this idea.

But anyways, you can now see the significance of this number. I can now count the number of atoms in a carbon isotope. If you give me 12 grams of carbon, I know it has these many number of carbon atoms in it—carbon-12. Okay, these many number of carbon-12 atoms in it. If you give me 24 g of carbon, there must be twice the amount. If you give me 6 g of carbon, then there must be half the amount. You tell me the mass of the carbon-12 isotope that I'm holding in my hand, and I can now use this number to tell you how many atoms there are. Beautiful, isn't it?

In other words, this becomes the conversion factor for our tiny unit of mass, from our tiny unit of mass, U, to our more familiar big unit of mass, grams. If you take U and you multiply it with this number, you get grams. And whenever you have an Avogadro number of things with you, we call it a mole. Just like how when you have 12 things with you, we call it a dozen. These many things, if you have together, it could be things of— it could be anything. It could be these many atoms, then we'll call it a mole of atoms. Or it could be these many babies, then we'll say we have a mole of babies.

It's a ridiculous number, but you get the idea. This word "mole" actually comes from the Latin "molecule," which translates to a very tiny amount of something. But anyways, what is a mole? A mole represents an Avogadro number, these many number of things. It could be the number of atoms, molecules, particles, anything. And what's so special about this number? It's a conversion factor from the tiny unit of mass U to grams. You take this number, multiply it by this number, and you will now get the mass in grams.

Okay, now let's see if you understand this. What do you think would be the mass of one mole of oxygen-16 atoms? If I had an Avogadro number of oxygen-16 atoms together, what do you think collectively would its mass be? Well, an Avogadro number of two values will give me a mass of 12 g, so an Avogadro number of 16 U will give me a mass of 16 g. That's what we mean by a conversion factor, okay? It works for any atom which has any mass. You just multiply by this, and now you will get the mass in grams.

Similarly, if I had an Avogadro number of chlorine-35, if I had one mole of chlorine-35 atoms with me, then it will have 35 g of mass. Make sense?

And so another way to talk about all of these things, whatever I just said now, another way to talk about this is we say the molar mass of carbon-12 is 12 g. Carbon-12 has a mass of 12 g per mole. Makes sense, right? We would say oxygen-16 will have 16 g per mole. I mention oxygen-16 because remember there are other isotopes as well. Different isotopes will have different masses, so their molar mass would be different.

So oxygen-16 isotope has a molar mass of 16 g per mole, and chlorine-35 has a molar mass of 35 g per mole. Okay, same thing. Whatever I just said, it's a technical way of stating the same thing over here.

All right, the last thing we need to do to make this more practical is to remember that over here, we considered pure cases. I took a pure carbon-12 isotope where every single atom was carbon-12, or I took a pure chlorine isotope where every single atom was chlorine-35. But in reality, that's not the case. If I take a chunk of chlorine, a lot of it will be chlorine-35, but there will be some other isotopes as well. Like another abundant isotope next to chlorine-35 is chlorine-37. And that sounds really complicated, but what's important and powerful is that that doesn't matter to us. This whole idea still works.

Okay, here's what I mean. Let me take an example. If you look at our periodic table, then you can see that the atomic mass of chlorine is given to be not 35; it's 35.45. So significant deviation from 35. Why? Because this also accounts for the fact that if you take a chunk of chlorine, it'll also contain a lot of chlorine-37 in it. So what we do is sort of like take an average. This is an weighted average, we say. So this is the average atomic mass of chlorine.

Since I know the average atomic mass of chlorine is 35.45, if I now take one mole of chlorine, not pure as it exists as a mixture in nature, then its one mole will have a mass of 35.45 g. That's it! Similarly, if I take one mole of carbon, which you know it's not exactly 12 g because there are other isotopes, it'll be 12.01 g. You see what I mean? A mole is a conversion factor. Take one mole of anything, it'll be this number in grams.

And so now we can try and answer the original question we asked ourselves: if you have 3.21 g of sulfur, how many atoms are there? Why don't you pause the video and see if you can now answer this question yourself? If I take one mole of sulfur, if I take an Avogadro number of sulfur atoms, it will have a mass of 32.1 g, roughly 32.1 g. So 32.1 g represents 1 mole of sulfur. But how much sulfur do I have? I have not 32.1; I have 3.21 g, which is just one-tenth of a mole.

That's why I took 3.21 to just keep the calculation simpler. We can do it in our head. This is one-tenth of a mole, so how many atoms must we be having? One-tenth of a mole, so one-tenth of the Avogadro number. So the answer would be the Avogadro number, which is 6.022 * 10^23, divided by 10, one-tenth of it. So it'll be 6.022 * 10^22.

Okay, here's our final question: If I take one mole of carbon dioxide, what do you think will its mass be? What is the molar mass of one mole of carbon dioxide? Can you pause the video and try to think about this?

Okay, let's do this step by step. I know if I have one mole of carbon dioxide, then it must be having an Avogadro number of molecules of carbon dioxide, right? Remember, if I had half a mole of carbon dioxide, it means that I would have half the Avogadro number of carbon dioxide. Make sense, right?

Okay, anyways, now comes the question: How many carbon atoms must be there, and how many oxygen atoms must be there? What do you think? Well, a single carbon dioxide molecule has one atom of carbon. If I have five molecules of carbon dioxide, I have five carbon atoms.

Which means if I have these many molecules of carbon dioxide, I should have exactly that many amount of carbon atoms, meaning I have one mole of carbon atoms with me. Okay, what about the number of oxygen atoms? Well, each carbon dioxide molecule has two atoms of oxygen. And so if I had five, for example, molecules of carbon dioxide, I would have twice the amount, 10 atoms of oxygen. Therefore, if I have these many molecules of carbon dioxide, I would have twice the amount.

Okay, so which means I have 2 moles of oxygen atoms with me, and I can now look at the periodic table to find the mass of one mole of carbon: it's 12.0107 g, and for oxygen, it would be 15.9994 g. That will be the mass of one mole of oxygen. But then we have to multiply it by two because over here we have 2 moles.

Simplifying this will give me the molar mass of carbon dioxide. So one mole of carbon dioxide will have this much mass, or we can also say that carbon dioxide has a mass of 44.95 g per mole. Same thing—it's the same thing. Okay, they all mean the same thing. Of course, we can round it off, and we are actually doing a numerical.