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

# Thermodynamics part 1: Molecular theory of gases

## Video transcript

After all the work we've been
doing with fluids, you probably have a pretty good
sense of what pressure is. Now let's think a little bit
about what it really means, especially when we think
about it in terms of a gas in a volume. Remember, what was
the difference between a gas and a liquid? They're both fluids, they both
take the shape of their containers, but a gas is
compressible, while a liquid is incompressible. Let's start focusing on gases. Let's say I have a container,
and I have a bunch of gas in it. What is a gas made of? It's just made up of a whole
bunch of the molecules of the gas itself, and I'll draw each
of the molecules with a little dot-- it's just going to have
a bunch of molecules in it. There's many, many, many more
than what I've drawn, but that's indicative, and they'll
all be going in random directions-- this one might be
going really fast in that direction, and that one might
be going a little bit slower in that direction. They all have their own little
velocity vectors, and they're always constantly bumping into
each other, and bumping into the sides of the container, and
ricocheting here and there and changing velocity. In general, especially at this
level of physics, we assume that this is an ideal gas, that
all of the bumps that occur, there's no
loss of energy. Or essentially that they're all
elastic bumps between the different molecules. There's no loss of momentum. Let's keep that in mind, and
everything you're going to see in high school and on the AP
test is going to deal with ideal gases. Let's think about what pressure means in this context. A lot of what we think about
pressure is something pushing on an area. If we think about pressure
here-- let's pick an arbitrary area. Let's take this side. Let's take this surface
of its container. Where's the pressure
going to be generated onto this surface? It's going to be generated by
just the millions and billions and trillions of little
bumps every time-- let me draw a side view. If this is the side view of the
container, that same side, every second there's always
these little molecules of gas moving around. If we pick an arbitrary period
of time, they're always ricocheting off of the side. We're looking at time over a
super-small fraction of time. And over that period of time,
this one might end up here, this one maybe bumped into it
right after it ricocheted and came here, this one changes
momentum and goes like that. This one might have already been
going in that direction, and that one might ricochet. But what's happening is, at
any given moment, since there's so many molecules,
there's always going to be some molecules that
are bumping into the side of the wall. When they bump, they have
a change in momentum. All force is change in
momentum over time. What I'm saying is that in any
interval of time, over any period or any change in time,
there's just going to be a bunch of particles that are
changing their momentum on the side of this wall. That is going to generate force,
and so if we think about how many on average--
because it's hard to keep track of each particle
individually, and when we did kinematics and stuff, we'd keep
track of the individual object at play. But when we're dealing with
gases and things on a macro level, you can't keep track of
any individual one, unless you have some kind of unbelievable
supercomputer. We can say, on average, this
many particles are changing momentum on this wall in
this amount of time. And so the force exerted on this
wall or this surface is going to be x. If we know what that force is,
and we you know the area of the wall, we can figure out
pressure, because pressure is equal to force divided
by area. What does this help us with? I wanted to give you that
intuition first, and now I'm just going to give you the one
formula that you really just need to know in thermodynamics. And then as we go into the next
few videos, I'll prove to you why it works, and
hopefully give you more of an intuition. Now you understand, hopefully,
what pressure means in the context of a gas
in a container. With that out of the way, let
me give you a formula. I hope by the end of this video
you have the intuition for why this formula works. In general, if I have an ideal
gas in a container, the pressure exerted on the gas-- on
the side of the container, or actually even at any point
within the gas, because it will all become homogeneous at
some point-- and we'll talk about entropy in future videos--
but the pressure in the container and on its
surface, times the volume of the container, is equal
to some constant. We'll see in future videos that
that constant is actually proportional to the average
kinetic energy of the molecules bouncing around. That should make sense to you. If the molecules were moving
around a lot faster, then you would have more kinetic energy,
and then they would be changing momentum on the sides
of the surface a lot more, so you would have more pressure. Let's see if we can get a little
bit more intuition onto why pressure times volume
is a constant. Let's say I have a container
now, and it's got a bunch of molecules of gas in it. Just like I showed you in that
last bit right before I erased, these are bouncing
off of the sides at a certain rate. Each of the molecules might
have a different kinetic energy-- it's always changing,
because they're always transferring momentum
to each other. But on average, they all have
a given kinetic energy, they keep bumping at a certain rate
into the wall, and that determines the pressure. What happens if I were able to
squeeze the box, and if I were able to decrease the
volume of the box? I just take that same box with
the same number of molecules in it, but I squeeze. I make the volume of
the box smaller-- what's going to happen? I have the same number of
molecules in there, with the same kinetic energy, and on
average, they're moving with the same velocities. So now what's going to happen? They're going to be hitting the
sides more often-- at the same time here that this
particle went bam, bam, now it could go bam, bam, bam. They're going to be hitting
the sides more often, so you're going to have more
changes in momentum, and so you're actually going to have
each particle exert more force on each surface. Because it's going to be hitting
them more often in a given amount of time. The surfaces themselves
are smaller. You have more force on a
surface, and on a smaller surface, you're going to
have higher pressure. Hopefully, that gives you an
intuition that if I had some amount of pressure in this
situation-- if I squeeze the volume, the pressure
increases. Another intuition-- if
I have a balloon, what blows up a balloon? It's the internal air pressure
of the helium, or your own exhales that you put
into the balloon. The more and more you try to
squeeze a balloon-- if you squeeze it from all directions,
it gets harder and harder to do it, and that's
because the pressure within the balloon increases as you
decrease the volume. If volume goes down, pressure
goes up, and that makes sense. That follows that when they
multiply each other, you have to have a constant. Let's take the same example
again, and what happens if you make the volume bigger? Let's say I have-- it's huge
like that, and I should have done it more proportionally, but
I think you get the idea. You have the same number of
particles, and if I had a particle here, in some period
of time it could have gone bam, bam, bam-- it could have
hit the walls twice. Now, in this situation, with
larger walls, it might just go bam, and in that same amount
of time, it will maybe get here and won't even hit
the other wall. The particles, on average, are
going to be colliding with the wall less often, and the walls
are going to have a larger area, as well. So in this case, when our volume
goes up, the average pressure or the pressure in
the container goes down. Hopefully, that gives you a
little intuition, and so you'll never forget
that pressure times volume is constant. And then we can use that to do
some pretty common problems, which I'll do in
the next video. I'm about to run out of time. See you soon.

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