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## AP®︎/College Chemistry

### Unit 8: Lesson 1

Ideal gas equation- The ideal gas law (PV = nRT)
- Worked example: Using the ideal gas law to calculate number of moles
- Worked example: Using the ideal gas law to calculate a change in volume
- Gas mixtures and partial pressures
- Dalton's law of partial pressure
- Worked example: Calculating partial pressures
- Worked example: Vapor pressure and the ideal gas law
- The Maxwell–Boltzmann distribution
- Non-ideal behavior of gases

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# Gas mixtures and partial pressures

AP.Chem:

SAP‑7 (EU)

, SAP‑7.A (LO)

, SAP‑7.A.1 (EK)

, SAP‑7.A.2 (EK)

For a mixture of ideal gases, the total pressure exerted by the mixture equals the sum of the pressures that each gas would exert on its own. This observation, known as Dalton's law of partial pressures, can be written as follows:

*P*(total) =*P*₁ +*P*₂ +*P*₃ + ... where*P*₁,*P*₂, and*P*₃ are the partial pressures of the different gases in the mixture, and*P*(total) is the total pressure of the mixture.. Created by Sal Khan.## Video transcript

- [Instructor] In this video, we're going to introduce
ourselves to the idea of partial pressure due to ideal gases. And the way to think about it is imagine some type of a container, and you don't just have one
type of gas in that container. You have more than one type of gas. So let's say you have gas one
that is in this white color. And obviously, I'm not
drawing it to scale, and I'm just drawing those
gas molecules moving around. You have gas two in this yellow color. You have gas three in this blue color. It turns out that people
have been able to observe that the total pressure in this system and you could imagine that's being exerted on the inside of the wall, or if you put anything in this container, the pressure, the force per
area that would be exerted on that thing is equal to the sum of the pressures contributed
from each of these gases or the pressure that each
gas would exert on its own. So this is going to be equal to the partial pressure due to gas one plus the partial pressure due to gas two plus the partial pressure
due to gas three. And this makes sense mathematically from the ideal gas law
that we have seen before. Remember, the ideal gas law tells us that pressure times volume is
equal to the number of moles times the ideal gas constant
times the temperature. And so if you were to
solve for pressure here, just divide both sides by volume. You'd get pressure is equal to nR times T over volume. And so we can express both
sides of this equation that way. Our total pressure, that would
be our total number of moles. So let me write it this way, n total times the ideal gas constant times our temperature in kelvin divided by the volume of our container. And that's going to be equal to, so the pressure due to gas one, that's going to be the
number of moles of gas one, times the ideal gas constant
times the temperature, the temperature is not going
to be different for each gas, we're assuming they're all
in the same environment, divided by the volume. And once again, the volume
is going to be the same. They're all in the same
container in this situation. And then we would add that to
the number of moles of gas two times the ideal gas constant,
which once again is going to be the same for all of the gases, times the temperature
divided by the volume. And then to that, we could add the number
of moles of gas three times the ideal gas constant times the temperature
divided by the volume. Now, I just happen to
have three gases here, but you could clearly keep going and keep adding more
gases into this container. But when you look at it
mathematically like this, you can see that the right-hand side, we can factor out the RT over V. And if you do that, you
are going to get n one plus n two plus n three, let me close those parentheses, times RT, RT over V. And this right over here
is the exact same thing as our total number of moles. If you say the number of moles of gas one plus the number of moles of gas two plus the number of moles of gas three, that's going to give you
the total number of moles of gas that you have in the container. So this makes sense
mathematically and logically. And we can use these mathematical ideas to answer other questions or to come up with other
ways of thinking about it. For example, let's say that we knew that the total pressure in our container due to all of the gases is four atmospheres. And let's say we know that
the total number of moles in the container is equal to eight moles. And let's say we know that the number of moles of gas three is equal to two moles. Can we use this information to figure out what is going to be the partial
pressure due to gas three? Pause this video, and
try to think about that. Well, one way you could think about it is the partial pressure due to gas three over the total pressure, over the total pressure
is going to be equal to, if we just look at this
piece right over here, it's going to be this. It's going to be the number
of moles of gas three times the ideal gas constant times the temperature
divided by the volume. And then the total pressure, well, that's just going
to be this expression. So the total number of moles
times the ideal gas constant times that same temperature, 'cause they're all in
the same environment, divided by that same volume. They're in the same container. And you can see very clearly
that the RT over V is in the numerator and the denominator, so they're going to cancel out. And we get this idea that the, I'll write it down here, the partial pressure due to gas three over the total pressure is equal to the number of moles of gas three divided by the total, total number of moles. And this quantity right over here, this is known as the mole fraction. Let me just write that down. It's a useful concept. And you can see the mole
fraction can help you figure out what the partial pressure is going to be. So for this example, if we
just substitute the numbers, we know that the total pressure is four. We know that the total
number of moles is eight. We know that the moles, the number of moles of gas three is two. And then we can just solve. We get, let me just do
it, write it over here, I'll write it in one color, that the partial pressure
due to gas three over four is equal to two over
eight, is equal to 1/4. And so you can just pattern match this, or you can multiply both sides by four to figure out that the partial
pressure due to gas three is going to be one. And since we were dealing
with units of atmosphere for the total pressure, this
is going to be one atmosphere. And we'd be done.