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# Ideal gas equation: PV = nRT

Intuition behind the ideal gas equation: PV=nRT. Created by Sal Khan.

Video transcript

Let's say I have a balloon. And in that balloon
I have a bunch of particles bouncing around. They're gas particles, so
they're floating freely. And they each have some
velocity, some kinetic energy. And what I care about, let me
just draw a few more, what I care about is the pressure that
is exerted on the surface of the balloon. So I care about the pressure. And what's pressure? It's force per area. So the area here, you can think
of it as the inside surface of the balloon. And what's going to apply
force to that? Well any given moment-- I only
drew six particles here, but in a real balloon you would have
gazillions of particles, and we could talk about how
large, but more particles than you can really probably
imagine-- but at any given moment, some of those particles
are bouncing off the wall of the container. That particle is bouncing
there, this particle is bouncing there, this guy's
bouncing like that. And when they bounce, they apply
force to the container. An outward force, that's what
keeps the balloon blown up. So think about what the pressure
is going to be dependent on. So first of all, the faster
these particles move, the higher the pressure. Slower particles, you're going
to be bouncing into the container less, and when you do
bounce into the container, it's going to be less of a
ricochet, or less of a change in momentum. So slower particles,
you're going to have pressure go down. Now, it's practically impossible
to measure the kinetic energy, or the velocity,
or the direction of each individual particle. Especially when you
have gazillions of them in a balloon. So we do is we think
of the average energy of the particles. And the average energy of the
particles, you might say oh, Sal is about to introduce
us to a new concept. It's a new way of looking at
probably a very familiar concept to you. And that's temperature. Temperature can and should be
viewed as the average energy of the particles
in the system. So I'll put a little squiggly
line, because there's a lot of ways to think about it. Average energy. And mostly kinetic energy,
because these particles are moving and bouncing. The higher the temperature,
the faster that these particles move. And the more that they're going
to bounce into the side of the container. But temperature is
average energy. It tells us energy
per particle. So obviously, if we only had
one particle in there with super high temperature, that's
going to have less pressure than if we have a million
particles in there. Let me draw that. If I have, let's take two
cases right here. One is, I have a bunch of
particles with a certain temperature, moving in their
different directions. And the other example,
I have one particle. And maybe they have the
same temperature. That on average, they have
the same kinetic energy. The kinetic energy per
particle is the same. Clearly, this one is going to
be applying more pressure to its container, because at any
given moment more of these particles are going to be
bouncing off the side than in this example. This guy's going to bounce,
bam, then going to go and move, bounce, bam. So he's going to be applying
less pressure, even though his temperature might be the same. Because temperature is kinetic
energy, or you can view it as kinetic energy per particles. Or it's a way of looking at
kinetic energy per particle. So if we wanted to look at the
total energy in the system, we would want to multiply the
temperature times the number of particles. And just since we're dealing
on the molecular scale, the number of particles can often
be represented as moles. Remember, moles is just
a number of particles. So we're saying that that
pressure-- well, I'll say it's proportional, so it's equal
to some constant, let's call that R. Because we've got to make all
the units work out in the end. I mean temperature is in Kelvin
but we eventually want to get back to joules. So let's just say it's equal
to some constant, or it's proportional to temperature
times the number of particles. And we can do that
a bunch of ways. But let's think of
that in moles. If I say there are 5 mole
particles there, you know that's 5 times 6 times 10
to the 23 particles. So, this is the number
of particles. This is the temperature. And this is just
some constant. Now, what else is the pressure
dependent on? We gave these two examples. Obviously, it is dependent on
the temperature; the faster each of these particles move,
the higher pressure we'll have. It's also dependent on the
number of particles, the more particles we have, the more
pressure we'll have. What about the size of
the container? The volume of the container. If we took this example, but
we shrunk the container somehow, maybe by pressing
on the outside. So if this container looked like
this, but we still had the same four particles in it,
with the same average kinetic energy, or the same
temperature. So the number of particles
stays the same, the temperature is the same, but
the volume has gone down. Now, these guys are going to
bump into the sides of the container more frequently
and there's less area. So at any given moment, you have
more force and less area. So when you have more force and
less area, your pressure is going to go up. So when the volume went down,
your pressure went up. So we could say that pressure
is inversely proportional to volume. So let's think about that. Let's put that into
our equation. We said that pressure is
proportional-- and I'm just saying some proportionality
constant, let's call that R, to the number of particles times
the temperature, this gives us the total energy. And it's inversely proportional
to the volume. And if we multiply both sides
of this times the volume, we get the pressure times the
volume is proportional to the number of particles times
the temperature. So PV is equal to RnT. And just to switch this around
a little bit, so it's in a form that you're more likely to
see in your chemistry book, if we just switch the
n and the R term. You get pressure times volume
is equal to n, the number of particles you have, times some
constant times temperature. And this right here is the
ideal gas equation. Hopefully, it makes
some sense to you. When they say ideal gas, it's
based on this little mental exercise I did to come
up with this. I made some implicit assumptions
when I did this. One is I assumed that we're
dealing with an ideal gas. And so you say what, Sal,
is an ideal gas? An ideal gas is one where the
molecules are not too concerned with each other. They're just concerned with
their own kinetic energy and bouncing off the wall. So they don't attract
or repel each other. Let's say they attracted each
other, then as you increased the number of particles
maybe they'd want to not go to the side. Maybe they'd all gravitate
towards the center a little bit more if they did
attract each other. And if they did that, they would
bounce into the walls less and the pressure would
be a little bit lower. So we're assuming that
they don't attract or repel each other. And we're also assuming that
the actual volume of the individual particles are
inconsequential. Which is a pretty good
assumption, because they're pretty small. Although, if you start putting
a ton of particles into a certain volume, then at some
point, especially if they're big molecules, it'll start to
matter in terms of their size. But we're assuming for the
purposes of our little mental exercise that the molecules have
inconsequential volumes and they don't attract
or repel each other. And in that situation, we can
apply the ideal gas equation right here. Now, we've established the
ideal gas equation. But you're like, well what's R,
how do I deal with it, and how do I do math problems,
and solve chemistry problems with it? And how do the units
all work out? We'll do all of that in the next
video where we'll solve a ton of equations, or a
ton of exercises with the ideal gas equation. The important takeaway from this
video is just to have the intuition as to why this
actually does make sense. And frankly, once you have
this intution, you should never forget it. You should be able to maybe even
derive it on your own.