- [Sal] In other videos, we touched on the notion
of kinetic molecular theory, which I'll just shorten as KMT.
And it's just this idea that if you imagine a container,
I'll just draw it in 2-dimensions here,
that it contains some gas, you can imagine the gas
as being these particles where their collective volume
is much smaller than the volume of the container.
And the temperature we're dealing with
is related to the average kinetic energy of the particles.
These particles are all moving around, zooming around,
and they each would have some kinetic energy.
Remember, kinetic energy, you calculate that
as MV squared over two.
So, each of these particles would have some mass
and some velocity,
but they could all have different velocities for sure,
even if they're the same type of particle.
And if they're different types of particles,
they can have different masses as well.
But, the average of these kinetic energies
across all of these particles,
that is proportional to temperature when measured in Kelvin.
And pressure, the pressure, remember,
pressure is nothing but force per unit area.
And so, you can imagine this surface of our container,
this could be some type of a cube
so I can draw it in 3-dimensions here,
so there's some area over here.
And you have your particles,
let me do this in a different color,
these particles are constantly bouncing off of it
and there's way more particles then what I have drawn here,
so at any given moment, you're having some particles
that are bouncing off of this side of the container,
actually all sides of the container.
And these are perfectly elastic collisions,
they're preserving kinetic energy.
And so, they're applying some force, collectively,
on this area, so the pressure is because
of these particle collisions on the surface.
Now, what I wanna do in this video is take these ideas
that we conceptualize in kinetic molecular theory
and to understand why the ideal gas law, PV is equal to nRT,
make sense when we conceptualize the world here.
Just a reminder, P is pressure, V is volume,
n is the number of moles of whatever gas we're dealing with,
the amount of that gas,
and then T is the temperature in Kelvin,
and R is just the ideal gas constant,
that's just whatever constant you're doing,
so that the units all work out together.
So, let's first think about how pressure relates to volume
if we were to old everything else constant.
Well, the idea gas law tells us that pressure times volume
is going to be equal to this, if we hold it at constant,
I can even just write a K here for constant,
but that would also mean we could divide,
let's say both sides by V,
we can say that pressure is equal to some constant over V.
Another way to think about it is is that pressure
is proportional to the inverse of volume.
You could also write this, if we divide both sides by P,
is that volume is proportional to the inverse of pressure.
Does that make sense
from a kinetic molecular theory point of view?
Pause this video and think about it.
Well, imagine we have our original cube right over here.
And I had the same number of particles,
they have the same average kinetic energy,
but let's say I were to increase the volume.
So, if I were to make the volume go up,
so I were to some how expand this,
or maybe put the exact number,
the same particles with the same temperature,
in a larger container, then at any given moment,
you're just gonna have fewer bounces of particles
off of the container.
Because they just have more room to go in that volume,
and even the surface area of the container
is going to be high as well.
So, it makes sense that if the volume goes up,
the pressure is going to go down.
And you can think about it the other way.
If you make this smaller, that same number of particles
with the same average kinetic energy,
they're just gonna bump into the container
that much more often.
And that's going to increase the pressure,
so volume goes down, pressure goes up.
And this relationship, that pressure
is inversely proportional to volume, or vice versa,
if you hold everything else constant,
that's often known as Boyle's law.
Now, another relationship, what if we were to hold volume
and the number of moles constant,
and we wanna think about the relationship
between pressure and temperature.
Well, this is constant, this constant, and this is constant,
the ideal gas law would say that pressure
is going to be proportional to temperature
or that temperature's proportional to pressure.
Does that make sense?
Well, let's go back to our original container.
If you were to increase the temperature,
that means that the average kinetic energy is increased.
That means that these particles,
when they hit the side of the container,
they're going to hit it with more velocity.
That means that you're going to have, at any given moment,
you're gonna have more pressure
exerted on the side of the container.
And you could go the other way.
Think about lowering the temperature,
so the kinetic energy goes really low,
then these particles are just slowly drifting.
And the speed with which they are hitting
the side of the container is going to go down
and so the pressure would go down.
So, it completely makes sense,
if temperature goes up, pressure goes up,
if temperature goes down, pressure goes down,
and this is often known as Gay-Lussac's law.
Now, another relationship, and I'm really just going through
all of the combinations over here,
what if we were to hold pressure
and the number of molecules constant?
So, we're really looking at the relationship
between volume and temperature.
So, once again, if P, n, and R is always constant,
if those are constant, the ideal gas law would tell us
that the volume is proportional to the temperature,
once again, holding everything else constant.
Well, to think about that you can go through
that same thought experiment we just had.
If we increase the temperature,
if these things are moving around faster,
if you want to have the same amount of force
per area on the container, on the side of the container,
you're going to have to increase the volume.
So, this relationship, which is completely consistent
with kinetic molecular theory,
is often known as Charles's law.
Now, another one is the relationship between volume
and the number of moles.
If everything else is held constant,
the ideal gas law would tell us that volume
is going to be proportional to the number of moles
of our particle, or of our gas that we are dealing with.
And that makes sense 'cause, once again,
you're holding everything else constant,
you want pressure to be constant,
temperature to be constant.
If I were to double the number of particles here,
but I don't wanna change the pressure or the temperature,
makes sense that I would have to double the volume.
Likewise, if I wanted to double the volume here
and I didn't wanna change the pressure or the temperature,
I would have to put twice as many particles in there,
so I still have sufficient number of interactions
of bouncing of the particles
with the sides of the container,
so that I have sufficient pressure.
And this notion is called Avogadro's law.
Last but not least,
let's say I have two identical containers.
I have two identical containers,
that's one there, that's one over here,
actually I'm gonna draw that same container a third time.
And let's say over here, I have gas one
and in this case, it has some pressure due to gas one,
we're gonna assume the volume and the temperatures
are the same across all three of these.
And let's say we have gas two
and it is exerting pressure too,
if I were to take all of the gas in both of them,
and put them both into this third container,
so this third container is gonna have all of the original
of gas one and all of the original of gas two,
but we aren't changing the volume
and we aren't changing the temperature.
In any given unit area on the surface of the container,
you're gonna get the collisions from particle one,
which would give you P1 in that force per unit area
and you're going to get the collisions from particle two,
which would give you that force per unit area.
So, it makes sense that the partial pressures would add up
to be equal to the total pressure in the container.
And this is known as Dalton's law.
But, the whole point of this video is just appreciate
that everything we've talked about
with the ideal gas law, actually makes a lot of sense,
I would argue it makes the most sense,
when you think about it
in terms of kinetic molecular theory.