# Ideal gas law derivation from kinetic theory

Consider the derivation of the ideal gas law from kinetic theory presented here: http://en.wikipedia.org/wiki/Kinetic_theory

I have some questions

1. This derivation assumes the container is a cube, but the law holds for containers of any shape, right?

2. This derivation assumes each particle has the same mass, but the law holds for a mixture of gasses of different masses, like air, right?

3. The derivation assumes that the particle intercepts a give side with frequency $2L/v_x$. Why can we make this assumption?

This derivation assumes the container is a cube, but the law holds for containers of any shape, right?

The resulting law holds for any shape of container. However, the details of the derivation depend on the geometry. A cube is easy, so we use it.

This derivation assumes each particle has the same mass, but the law holds for a mixture of gasses of different masses, like air, right?

Yes, it holds for a mixture of any ideal gases - the key is that only the kinetic energy of the gas molecules matters. Heavier molecules would be moving slower at the same temperature, but they would exert the same force (and therefore pressure) on the walls.

The derivation assumes that the particle intercepts a give side with frequency 2L/vx. Why can we make this assumption?

This assumption works for only one particular case. If you notice the third sentence at the beginning of the pressure derivation (emphasis is mine):

When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed

They are looking at the special case of a collision at a 90 degree angle, which bounces off the opposite wall. In that case, it will take the particle t = L/V time to reach the opposite wall, and t = L/V time to bounce back - so the total is 2L/V. This illustration may help explain it better:

Now this might sound like an unreasonably strict condition, but the reason that it works is that the component velocity distribution for each particle is symmetrical (it's a normal distribution), which means that for every particle that impacts the wall at 95 degrees (for example) there is another that impacts 85 degrees. As a result, the average angle of impact is 90 degrees for each wall, and therefore the average collision time is given by looking at 90 degree impacts for each component of the velocity vector.

• 1.So how do we prove the law holds for any container? 2.If the law holds for a mixture of gasses of different masses (still acting as an ideal gas), how do we prove the law in this case? 3.I believe you are misunderstanding that quote from the article. It should read like: "When a gas molecule collides with [a wall of the container that perpendicular to the x-axis]". 4. How do you prove that the velocity component distribution for each particle is the normal distribution? – Joshua Benabou Nov 24 '14 at 0:20
• Also can someone link me to a rigourous proof of the ideal gas law using kinetic theory? – Joshua Benabou Nov 24 '14 at 0:26
• @JoshuaBenabou If you only want a proof that the ideal gas law holds for any container size it might be easier to simply show general applicability via thermodynamics. But if you want a general proof from kinetic theory I would imagine that this might not be easy and possibly quite lengthy. – Philipp Nov 24 '14 at 1:30
• @JoshuaBenabou - 1. There are lots of ways, one way is to show that it holds empirically for many container shapes, then derive it from first principles for an arbitrary container (a cube in this case). I am not sure if there is way to rigorously show that it holds for all container geometries. 2. You would replace the kinetic energy terms with summations over the different masses. 3. Possibly. But, who is to say the x-axis isn't vertical? 4. You don't, that's empirically known. At any rate, it doesn't have to be normal, just symmetrical about a mean. – thomij Nov 24 '14 at 2:54
• There is a nice general proof using statistical mechanics here: en.wikipedia.org/wiki/Ideal_gas_law. – Joshua Benabou Nov 24 '14 at 3:42