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## Temperature, kinetic theory, and the ideal gas law

Current time:0:00Total duration:10:14

# Thermodynamics part 4: Moles and the ideal gas law

## Video transcript

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.