Hey, guys. So a lot of times in our physics equations we usually use the mass of a material. For example, our most important equation, F equals ma, we use the mass to calculate the force. But, a lot of times in thermodynamics, you're going to need to know the number of moles or the number of particles that make up a material like a gas. So, I'm going to introduce to you the moles and also Avogadro's number in this video, and I'm also going to show you a systematic way to convert between these three related ideas: the mass, the mole, and the number of particles. If you've ever taken a chemistry class, you're probably familiar with the mole, but if not, I'm going to go ahead and introduce it to you. Let's check this out. The mole, which is given by the letter little *n*, is really just the amount of material. It's an amount of stuff that is equal to a very specific number of particles, 6.022×1023. This really big number here is known as Avogadro's number, has a special name, and we also give it the letter big N_{a}, *a* for Avogadro. So a mole is really just a shortcut way of saying it's an amount that's equal to this many particles. It's kind of like if you said if you had a dozen donuts, that dozen just means you have 12. A mole just means that you have this many of whatever it is that you're talking about, and it actually applies to any substance that you talk about. For example, if you have a mole of nitrogen, that just means 6.022×1023 nitrogen atoms, it's Avogadro's number. If you have a mole of H_{2}O, it just means that you have an Avogadro's number of H_{2}O molecules. If you have a mole of dollars, you'd be an extremely rich person because you would have an Avogadro's number of dollars. That's just a, you know, a mole just means a shortcut way of saying this many particles or whatever it is. Now as we said before, in a lot of our thermodynamics discussions and equations, we're gonna be talking about these sort of 3 related ideas, the mass, the mole, and also the number of particles, and specifically you're going to need to know how to convert between them. So unfortunately, what happens is that a lot of these terms have sort of similar letters. The number of particles is given by big *N*, the number of moles is given by little *n*, and then the mass of something is given by little *m*. They're all sort of like, they are sort of similar. Now fortunately, the equations are very straightforward. So let's check this out. If you wanted to connect to the number of particles to the number of moles, we're just gonna use this equation over here. Little *n* equals big N, the number of particles divided by Avogadro's number. If you have some amount of particles, you wanna figure out how many moles, you just have to divide it by this really really big number here,6.022×1023, the number of particles, and it'll get you that many moles. So let's go ahead and check out our first example here. So we want to calculate how many moles are in this many carbon atoms. So what are we trying to do? We're trying to go from a number of moles, which is little *n*, and we're trying to get I'm sorry. We're trying to get go from a number of particles, big *N*, and we're trying to get to the number of moles. So we're really just trying to go from here to here, and therefore, we're just gonna use this equation. Alright? So let's do it. So we have little *n* is equal to big N over Avogadro's number. So this is gonna be 8.33×1037. This is gonna be particles, and then we're gonna divide it by Avogadro's number 6.022×1023. The units for this are basically particles per mole. So what you'll see here is the particles and over particles will cancel out and you're just left with moles. Now if you go ahead and plug this into your calculator, you're gonna get 1.38×1014, and this is how many moles of carbon atoms you have. That's the answer. Right? So you just figure out on this flowchart here which two variables you're trying to connect and then use the equation. Let's check out the second example. We're trying to calculate how many grams of aluminum is in this many moles of aluminum. So here we're going from moles and now we want to calculate a gram, which is a mass. So here on our flowchart, we're actually trying to go from here to here, so we're gonna need another equation to do this. And that equation is little *m*, the mass, divided by big *M*. So this big *M* here is called the molar mass of a material. It's also known for those of you who've taken chemistry as the atomic mass, which you can find in any periodic table. So fortunately here at Clutch, we have our own periodic table you can find at the bottom of this page here. And if you go ahead and click on aluminum, what you're going to get is that the atomic mass is 26.98. That's the molar mass. What that means is that there are that many grams per 1 mole of aluminum. All right? Now one thing I want to mention here is that one way to kind of remember these equations is that little *n* is always equal to the two letters divided. So big *N* over little, big *N* over Avogadro's number, there are 2 big *N*'s and also little *m* over big *M*. That might be one way that you remember that. Right? So let's go ahead and use this equation here. We have *n* equals little *m* divided by big *M*, but we want to calculate this little *m* here. That's the mass. So we rearrange this equation and we're going to get *m* is equal to *n* times, this is going to be big *M*. So the number of moles that we have is 2.35 and the molar mass of aluminum we were given is 26.98. So this is 26.98 like this, and if you go ahead and work this out, you're gonna get a mass of 63.4, and that's grams of aluminum. Alright. So that's the answer, that's 63.4 grams. Let's check out the third one. Now we want to calculate how many particles are in 24 grams of H_{2}O. So here what we have is we have a mass and we're actually trying to get actually let me lay it out this way. So we have a mass, and we want to get to the number of particles. Remember, that is big *N*. So in our diagram, we're actually trying to go from this little *m* and we're trying to go all the way back to the number of particles. But in order to do that, we're going to actually have to go through moles. Moles are kind of the bridge between these variables. So if you kind of think about this, if I'm trying to get from here to here and I actually have to first go through moles and then convert it to the number of particles. So that's what we're going to do here. So in this first step, we're just gonna use the equation *n* equals little *m* over big *M*. I want to calculate the number of moles, so I'm just gonna go ahead and plug in my mass and divide it by the molar mass. I'm just gonna so this is gonna be a little *m* and that is gonna be a big *M*. So this is my mass which is 24 grams, and this is my molar mass which I'm given is 18.02. So this is 18.02 like this. And if you go ahead and work this out, what you're going to get is 1.33 moles. That's how many that's so many moles of H_{2}O that I have. The second step here is now I actually want to go from moles to number of particles, and I'm actually gonna use the other equation. I'm gonna use my, my *n* over *n _{a}* equation. So this is,

*n*equals and I have

*n*over

*n*. Now what I'm actually trying to do is calculate

_{a}*n*. So I'm just gonna rearrange this equation. This is gonna be

*n*is equal to little

*n*times Avogadro's number. So I have little

*n*, that's the number of moles. Right? It's 1.33 times, and then I've got, 6.022×1023. That's the number of particles. Let me go ahead and move, and I've got the number of particles is equal to 8.01×1023, and that's gonna be particles. Right? That's how many molecules or particles, if you will, of H

_{2}O that you will have. Alright? So that's the answer. That's how you sort of navigate this flowchart here. You just figure out where you're trying to go, what are your given and target variables, and use the right equation. Let me know if you guys have any questions.