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### Course: AP®︎/College Physics 2 > Unit 1

Lesson 1: Gases- The ideal gas law (PV = nRT)
- What is the ideal gas law?
- Calculations using the ideal gas equation
- The kinetic molecular theory of gases
- Kinetic molecular theory and the gas laws
- Kinetic molecular theory
- Boltzmann's constant
- What is the Maxwell-Boltzmann distribution?
- Worked example: RMS speed and average KE of gas molecules
- RMS speed and average kinetic energy of gas molecules

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# Kinetic molecular theory and the gas laws

The kinetic molecular theory (KMT) can be used to explain the macroscopic behavior of ideal gases. In this video, we'll see how the KMT accounts for the properties of gases as described by the various gas laws (Boyle's law, Gay-Lussac's law, Charles's law, Avogadro's law, and Dalton's law of partial pressures). Created by Sal Khan.

## Want to join the conversation?

- A few questions about pressure:

1. Is pressure = force /**total surface area**of the container?

2. Force is a vector quantity, so if we want the total force, shouldn't force to the right cancel with forces to the left?

(Sorry if this is more focusing on physics)

Thanx!:)(5 votes)- Pressure is force per unit of area. We're looking at each individual square of area (of whichever unit) and seeing how much force is being applied to each of those squares. If it's a gas then we assume the pressure is being exerted equally on the entire surface.

Force is a vector but all of these forces are directed outward against the surface of the container so they're not opposing each other and therefore not canceling each other out.

Hope that helps.(8 votes)

- do the molecules of a gas exert an attractive force between each other(5 votes)
- Good question! This is where things start getting interesting. You are right in saying that there is an attractive force between gas molecules, but for ideal gasses we ignore them and assume they don't impact anything (that's why they're "ideal"). Of course, sometimes, this causes problems. Deviations from the ideal gas law like these are covered in the next unit.(5 votes)

- At4:18Sal explains that Pressure is proportional to Temperature because as the temperature increases, the velocity of the particles hitting the wall also speeds up, thereby increasing the pressure on the container. However, since the temperature is proportional to (mv^2)/2, wouldn't the velocity of the particles have a square root relationship to temperature. Thus shouldn't the pressure of the container also have a squared rooted relationship to temperature?(2 votes)
- Pressure is directly proportional to temperature, or P α T. And temperature is directly proportional to kinetic energy where kinetic energy has the formula K.E. = (1/2)mv^(2). So we can say that temperature is directly proportional to the square of the particle's velocity, or T α v^(2). So therefore we can say that pressure is also directly proportional to the square of the particle's velocity through the transitive property, or P α v^(2).

For some reason you're taking the square root of the temperature-velocity relationship, essentially getting sqrt(T) α v, and then saying that now the square root of temperature is directly proportional to pressure, or sqrt(T) α P. Which doesn't follow since there is no way to link the square root of temperature with pressure, except through velocity. We can do the same operation with the pressure-velocity relationship and say that sqrt(P) α v, and only know we can relate pressure and temperature together through the transitive property using velocity. This yields sqrt(P) α sqrt(T), where the square roots can be removed by squaring both sides yielding P α T. Which is just the original proportionality relationship. So, no pressure and temperature don't have a square root relationship.

Hope that helps.(6 votes)

- This might be too unrelated, but I have a clear memory of leaving inflated balloons in a room, and after a month or so they are shrivelled up and tiny.

I understand that if there are temperature drops in the room, the balloon's volume will decrease, but surely this would only be a bit.

Assuming the temperature in the room stays roughly the same, why do balloons (or balls, or anything that's inflated for that matter) eventually deflate?

Is it because they are not 100% airtight? Or some other unrelated physics law...

Thanks(1 vote)- Temperature does affect gas volume in a balloon according to Charles’s Law. As temperature decreases, volume decreases too (assuming the pressure and moles of gas remain constant). Likewise, as temperature increases, volume increases too (same assumptions). Mathematically this is represented as: V α T, which means volume of gas is directly proportional to the temperature of the gas.

So temperature would affect a balloon’s volume, but it would be a different pattern then what you’re observing. The temperature throughout the day changes as it transitions from day to night, from hot to cold. So if the volume changes were solely due to temperature, you would observe a periodic change in the balloon’s volume. If would shrink during the night when the temperature is at its minimum, and inflate during the day when the temperature is at its maximum. But if the volume is steadily decreasing regardless of the time of day, then another effect is causing the volume change.

For balloons or any other rubber balls filled with air (tires too), they lose gas particles inside them over time. Technically this is Avogadro’s Law which says that moles of a gas and its volume are directly proportional: n α V. So if there are less moles of gas in the balloon, then the volume will decrease. And unlike the temperature cause, this will be permanent since gas will not reenter the balloon and inflate it again.

The gas particles escaping isn’t so much due to the tied end or the place where you filled the balloon up though. If you make a tight seal, you’ll still see the balloon deflate. Most of the gas particles actually escape by passing through the rubber. Interestingly the gas particles do this primarily by dissolving into the rubber, migrating through the rubber, and ‘evaporating’ into the outside air. Even though the balloon’s rubber is a solid, you can have gas particles dissolve into it and bind to the rubber polymer. It’ll alternatively bind and unbind as it makes its way through the rubber until finally reaching the outside. If you’re interested there’s an old paper called the “Permeability of Rubber to Gases” by Edwards and Pickering published back in 1919 which talks about this and how well different types of gases dissolve in rubber.

So a balloon shrivels up after a while because the gas gradually escapes.

Hope that helps.(5 votes)

- So if velocity is proportional to moles. When V goes up, does n go up as well?(1 vote)
- Yep, that is what Avogadro's Law says. As it mentioned in the video, V=Volume, and n=numbers of moles of the gas. Based on my knowledge of chemistry, the greater the volume, the more molecules it contains. Hence, the more molecules, the greater the molar mass and the more moles it has in the gas. So, if V increases, n increases as well.

Hope my explanation is helpful and understandable.(2 votes)

## Video transcript

- [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.