Main content

### 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

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# The ideal gas law (PV = nRT)

The ideal gas law (PV = nRT) relates the macroscopic properties of ideal gases. An ideal gas is a gas in which the particles (a) do not attract or repel one another and (b) take up no space (have no volume). No gas is truly ideal, but the ideal gas law does provide a good approximation of real gas behavior under many conditions. Created by Sal Khan.

## Want to join the conversation?

- Can't you derive the equation from Boyle's, Charles's, and Avogadro's law? Since volume is proportional to temperature, pressure, and moles, can you say that by adding proportionality constant R you can relate these in PV=nRT?(49 votes)
- Yes. The Ideal gas equation is a synthesis of all three laws. Ya science!!(58 votes)

- Why does temperature always have to be converted to Kelvin??(23 votes)
- You have to convert to Kelvin because Celsius is a relative scale. Specifically, 0°C is NOT the point of no heat. Instead, 0°C was set as a convenient point for humans (specifically it is the temperature where water freezes at standard pressure). Likewise, doubling a temperature in Celsius is not doubling the heat (again, because the 0 is not an absolute 0).

But, to do these calculations, you must have an absolute scale. 0 must mean no heat. A doubling of the temperature must mean a doubling of the heat. Thus, you use Kelvin because 0K is the point of no heat. When you double a temperature on the Kelvin scale you really are doubling the heat.

Example: On the Celsius scale, if you double 1°C you would get 2°C -- which is hardly any warmer. The temperature that actually has twice as much heat as 1°C is 275°C (rounded to the nearest degree). On the Kelvin scale, you have this problem corrected.(88 votes)

- If ideal gas laws are applicable only for 'ideal' gases , then why study them at all if there exists no such 'ideal' gas ?(7 votes)
- Two reasons:

1) Lots of gases approximate ideal gases as long as the temperature is high and the pressure is low

2) You have to walk before you can run.(11 votes)

- True or False: If one gas is more dense than another, it's because it has more molecules at 22.4L of the gas at stp.

I don't get it.... where is this explained?(7 votes)- The answer is False.

If you solve the Ideal Gas equation for n (the number of particles expressed as moles) you get:

n = PV/RT

Thus, at STP, the same volume of all gases have the same number of molecules (provided the conditions are suitable for the Ideal Gas Law to apply).

A more dense gas has more MASSIVE molecules, but the same number of particles as compared to a less dense gas under the same temperature, pressure and volume.

So, in summary, the Ideal Gas Law states that under the same temperature, pressure and volume all gases contain the same number of molecules (but not the same mass).

Reminder: The Ideal Gas law does not apply when the temperature and pressure are near the point of transforming into a liquid or solid.(22 votes)

- what is the value of R in the equation??(5 votes)
- R can have multiple values depending on the units of the other variables.

If pressure is in atm, R = 0.0821

If pressure is in kPa, R = 8.314

There are lots of other options, but those are the most common units of R. Hope that helps!(12 votes)

- Why do you need to put in R in the equation?(4 votes)
- R is the constant that makes the equation true, kind of like when you use pi to find the circumference, area of a circle or volume of a sphere.(10 votes)

- My teacher told me that an ideal gas is one that follows Boyle's law, Charles' law and Avogadro's law strictly is an Ideal Gas and it is hypothetical but Sal says something else for describing the ideal gas. Help!! :|(0 votes)
- The Ideal Gas is a model that GREATLY simplifies the math necessary for describing the behavior of real gases. As Andrew M mentioned, if the temperature is high enough and the pressure low enough that the gas is nowhere near liquifying or solidifying, then real gases usually behave close enough to ideal gases that we can use the Ideal Gas law.

If you go through the very difficult math for the equations for real gases, you generally get an answer only a fraction of a percent different from the much easier math of the Ideal Gas Law. However, if the temperature and/or pressure is close to where the gas becomes a liquid or a solid, then the Ideal Gas Law will give you VERY wrong answers. In those cases you have to use the much more difficult mathematics for the real gas equations of state. But, except when we have to, we don't use the other equations of state because they are so hard to work with. Most of the time an error of less than 0.5% is acceptable and we use the Ideal Gas Law.

Just for reference, one of the better equations of state that does much better at predicting the behavior of gases as they begin to condense into liquids is the Peng–Robinson equation. You may learn about it at this link http://kshmakov.org/fluid/note/3/

I think you will quickly see why we avoid that level of complexity whenever we can.(17 votes)

- what is the difference between ideal gas and normal gasses and noble gasses??(0 votes)
- An ideal gas is just that, "ideal", it does not really exist. Instead, it is an approximation that comes very close to the way that most real gases behave near conditions found in Earth's environment. Many times, the approximation is so close to reality that we can just ignore the error because it will be so slight we can barely detect the error at all.

The reason we use such approximations is that calculations closer to reality get EXTREMELY complicated and very hard to understand. So we use the Ideal Gas Law any time we possibly can.

There are, however, situations (mostly when the gas is near condensing into a liquid) that the Ideal Gas Law gives such poor approximations that we cannot use it and must go with something more complex.(14 votes)

- What is the equation PV=nTR?(0 votes)
- This equation is the ideal gas law. It explains the relationship between the different conditions and amounts of the ideal gas.

P = pressure

V = volume

n = moles of gas

T = temperature (in Kelvin)

R = ideal gas constant(17 votes)

- How does the root mean square velocity of an ideal gas at constant pressure varies with density ?

Please help.

Thank you.(3 votes)- The rms velocity of an ideal gas at constant pressure does not vary with density. It varies only with temperature.(3 votes)

## Video transcript

- [Instructor] In this
video we're gonna talk about ideal gasses and how we can describe what's going on with them. So the first question you
might be wondering is, what is an ideal gas? And it really is a bit of
a theoretical construct that helps us describe
a lot of what's going on in the gas world, or at least close to what's going on in the gas world. So in an ideal gas, we imagined that the individual particles
of the gas don't interact. So particles, particles don't interact. And obviously we know
that's not generally true. There's generally some
light intermolecular forces as they get close to each other or as they pass by each other or if they collide into each other. But for the sake of what we're
going to study in this video, we'll assume that they don't interact. And we'll also assume that the particles don't
take up any volume. Don't take up volume. Now, we know that that isn't exactly true, that individual molecules
of course do take up volume. But this is a reasonable assumption, because generally speaking, it might be a very, very infinitesimally
small fraction of the total volume of the space that they are bouncing around in. And so these are the
two assumptions we make when we talk about ideal gasses. That's why we're using the word ideal. In future videos we'll talk
about non-ideal behavior. But it allows us to make
some simplifications that approximate a lot of the world. So let's think about how we
can describe ideal gasses. We can think about the volume of the container that they are in. We could imagine the pressure
that they would exert on say the inside of the container. That's how I visualize it. Although, that pressure would be the same at any point inside of the container. We can think about the temperature. And we wanna do it in absolute scale, so we generally measure
temperature in kelvin. And then we could also think about just how much of that gas we have. And we can measure that in
terms of number of moles. And so that's what this lowercase n is. So let's think about how these four things can relate to each other. So let's just always put
volume on the left-hand side. How does volume relate to pressure? Well, what I imagine is, if
I have a balloon like this and I have some gas in the balloon, if I try to decrease the volume by making it a smaller balloon without letting out any other air or without changing the temperature, so I'm not changing T and n, what's going to happen to the pressure? Well, that gas is going
to, per square inch or per square area, exert
more and more force. It gets harder and harder for
me to squeeze that balloon. So as volume goes down, pressure goes up. Or likewise, if I were to
make the container bigger, not changing, once again, the temperature or the number of moles I
have inside of the container, it's going to lower the pressure. So it looks like volume and pressure move inversely with each other. So what we could say is
that volume is proportional to one over pressure,
the inverse of pressure. Or you could say that
pressure is proportional to the inverse of volume. This just means proportional to. Which means that volume would be equal to some constant divided
by pressure in this case. Now how does volume relate to temperature? Well, if I start with my balloon example, and you could run this example
if you don't believe me, if you take a balloon and you were to blow it up at room temperature, and then if you were to
put it into the fridge, you should see what happens. It's going to shrink. And you might say, "Why is it shrinking?" Well, you could imagine that the particles inside the balloon are a little less vigorous at that point. They have lower individual
kinetic energies. And so in order for them
to exert the same pressure to offset atmospheric
pressure on the outside, you are going to have a lower volume. And so volume you could say is
proportional to temperature. Now how does volume
compare to number of moles? Well, think about it. If you blow air into a balloon, you're putting more
moles into that balloon. And holding pressure and
temperature constant, you are going to increase the volume. So volume is proportional
to the number of moles. If you were to take air out, you're also going to decrease the volume, keeping pressure and temperature constant. So we can use these three relationships, and these are actually known as, this first one is known as Boyle's law, this is Charles' law,
this is Avogadro's law. But you can combine them to
realize that volume is going to be proportional to
the number of moles times the temperature divided by the pressure. Divided by the pressure. Or another way to say it is, you could say that volume is going to
be equal to some constant, that's what proportionality
is just talking about, is gonna be equal to some
constant, let's call it R, times all of this business, RnT over P. Over P. Or another way to think about it is we can multiply both sides by P. And what will you get? We will get P times V,
this might be looking somewhat familiar to some of you, is equal to, and I'll just change
the order right over here, n, which is the number of moles, times some constant times T, our temperature measured in kelvin. And this relationship right
over here, PV is equal to nRT, is one of the most
useful things in chemistry. And it's known as the ideal gas law. And in future videos
we're going to apply it over and over again to
see how useful it is. Now, one question you
might be wondering is, "What is this constant?" It's known as the ideal gas constant. And you can look it up, but
it's going to be dependent on what units you use for a pressure or volume and temperature. And we will see that in future videos.