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Before I continue, I want to introduce you to what might be an unfamiliar concept, although if you've taken chemistry, you might know a little bit about it-- it's called a mole. This isn't the thing that grows on your face with the hair in it, or the animal that digs in your backyard, although those are also called moles. We're talking about the SI unit called a mole. A mole is just a number. It's like saying a mole of something means a certain number of something, just like a dozen-- it's like saying that a dozen eggs is 12 eggs. Just like that, a mole would be 6-- I always forget the exact number, but it's 6.023, or something of that nature. You could look it up-- I think it's 6.023 times 10 to the twentieth. Let me look up the exact number, just because I think that 23 I'm misremembering. The mole is 6.022 times 10 to the 23 of something, so it's a very number of something. Normally, we don't deal in moles of eggs-- I don't think there has been a mole of eggs ever produced in the history of the universe. 10 to the 23 is a very, very, very large number. Where is it useful? A mole is useful for counting atoms, and so what is a mole of atoms or molecules? What's that many molecules? It's 6 followed by roughly 23 0's of molecules-- a very, very big number. What's interesting about a mole is that when I have a mole of something, its mass-- let's say its mass in grams. A mole of carbon: its mass in grams is going to be equal to-- so if I have this many carbon molecules, its mass in grams is x grams. It'll have a mass of x grams, where x is the atomic mass number of an atom of carbon, although if I was talking about a mole of a molecule, I would figure out the atomic mass of the entire molecule. What's an atomic mass number? Let me see if I can do a Web search on a periodic table. I'm showing you what I do here, and it's not fancy. Let me go to Google, and let's look up periodic table, and let's see if we can find a good one-- this one looks good. If we go to carbon, which is right here, we see that its atomic number is 6, and that's the number of protons that it has. The atomic mass number-- it's the mass of the entire atom. And just so you know-- we're delving into a little bit of chemistry here-- but most of the mass of an atom is the protons and the neutrons. The neutrons and the protons weigh roughly the same thing, and then the electrons are much, much, much smaller. If you factor in the mass of the protons and the neutrons, you pretty much have the mass of the particle. Just a little more chemistry here is that although on average most of the most of the atoms have roughly the same number of protons and neutrons-- some don't. Some, you could have a carbon atom that has seven neutrons, another one with five, another one with six, and those are actually all called isotopes, and I won't go into all of that-- they're just the same atom with different numbers of neutrons. In general, the atomic mass is equal to the mass of the protons and the neutrons, and they tend to be equal. So if the atomic number is 6, the atomic mass tends to be 12. So why is this useful? We can say if we have niobium-- let's say I have a mole of niobium. If we look here on the periodic table, it has an atomic mass number of 41, and its average atomic mass-- if we were to average all of the isotopes based on the weighting of how they exist in nature-- it's 92.9, so roughly 93. It's actually a little bit more than double its atomic number, but let's say 93. If we had a mole of niobium-- if we had 6.022 times 10 to the 23 of niobium, it would have a mass of 92 grams. That's pretty easy-- look at any element. Let's say chromium: we see its atomic mass number is roughly 52, and we see that there. If I have a mole of it-- if I have roughly 6 times 10 the 23 three of it, that much will have a mass of 52 grams. That's how we think about a mole. If I tell you I have a mole of something, I'm also telling you how many of that molecule I have, and I'm also telling you what the mass of that mole that quantity will be, assuming that you have a periodic table in front of you. With that said, and with that out of the way, let's make some more progress with our thermodynamics. We said in the last several videos that pressure times volume is somehow proportional-- let's call that K. And this is an arbitrary number, it's not some special constant-- to the total kinetic energy of a system. We also said that that is roughly proportional-- that's another constant-- times the number of molecules we have times the temperature, because temperature we viewed as kinetic energy per molecule. In general, we could also say that this is proportional to this, which is proportional to this, that pressure times volume is proportional. We'll use R, because you'll see where that's coming from in a second. It's proportional, it's equal to some constant times the number of molecules n. Here I just take the absolute numbers-- if I had five molecules, I'd put a five here, but now this n, I'm counting in moles. If this n is 1, that means that I have 6.022 times 10 to the 23 molecules. One mole equals 6.022 times 10 to the 23, so I'm just saying that this is another way to write the number of molecules, and then that's times temperature. Then if we rearrange it-- PV equals nRT. We have a relationship, that if I know the pressure, the volume, and the number of molecules, I can figure out the temperature, or if I know the number of molecules, the temperature, and the pressure, I can figure out the volume, assuming I know what R is. I'm about to tell you what that is. R is called the universal gas constant, and it is R is 8.31 joules per mole-Kelvin. That kind of tells you what you need in this formula. This should end up being joules, so if you have pressure in pascals and volume in meters cubed, you'll end up with joules there. This should be in moles-- this is 8.31 joules per mole-Kelvin. And then this, as we always said, should be in Kelvin. Honestly, if you just memorize two things in all of thermodynamics, you'll probably be able to do 95% of problems, but you actually should have the intuition of how they work. Just remember that PV over T is equal to a constant, or that if you change them, they relate to each other in that they all equal a constant, so P1 times V1 divided by T1 is equal to P2 times V2 divided by T2. You also should just need to memorize PV is equal to nRT, where R is equal to 8.31 joules per mole Kelvin. I know you might not have a lot of intuition of this formula yet, because I haven't used it, but I'm going to do that in the next video. These are literally the two most important things you should know in thermodynamics, and hopefully you have a little intuition at this point of what they mean. See you soon.