# Kinetic Theory of Gases

1. Jul 12, 2006

### Farsight

Apologies if it's been asked before, but this came up on another thread.

The Kinetic Theory of Gases says that hydrogen molecules at room temperature and pressure are travelling at circa 4000mph. So if I were to suddenly decap a canister of hydrogen in a vacuum chamber, I should see molecules coming out at 4000mph. This seems like an explosive velocity that ought to blast the opposite wall of the chamber in the twinkling of an eye. It doesn't seem to square with my layman's experience.

Can any more knowledgeable posters tell me about any experiments that prove the high velocity of the kinetic gas molecules?

2. Jul 12, 2006

### Hootenanny

Staff Emeritus
I'm afraid I don't know of any experiments which directly prove the kinetic temperature off hand. However, to square this with your layman's experience you may wish to make a number of considerations. Firstly, note that the gas molecules will not all be travelling in the same direction, their motion is random and they are equally likely to be travelling in all directions at any one time; therefore, not all of the molecules will collide with the opposite wall at the same time. Secondly, as we are releasing this gas into a vacuum, the temperature of the gas will rapidly decrease and hence this will reduce the kinetic temperature of the gas. Thirdly, you may wish to consider the moment which these molecules carry and the change of momentum when they collide elastically with the container wall, how big is the resulting impulse? Compare this to a tennis ball rebounding off a wall which will exert an approximate impulse of around 8.55 N.s

3. Jul 12, 2006

### 3trQN

That would be what you expect in a vacuum, but at STP the pressures are about equal and a molecules path is far from a straight line from the container opening as it has i high probabillity of colliding with air molecules. Diffusion of the molecules is then much slower.

Maybe one of the oldest experiments is the smoke particle experiment, while direct observations of the molecules in a gas isnt possible with the naked eye, you can see the effects of the collisions between larger smoke particles and the random-like motion of gas molecules, Brownian motion.

While not proof of the velocity of gas molecules, it was i beleive one of the earliest observations that lead to that line of thinking and the development of the kinetic theory of gasses.

4. Jul 12, 2006

### Staff: Mentor

Why don't you do the math yourself and see for yourself if it makes sense, Farsight.

5. Jul 12, 2006

### Farsight

Because my maths is relatively weak, russ. Besides, I don't have a problem with the mathematics of say pressure in a vessel of volume V containing X moles of gas at temperature T. What I'm asking for is an experiment that underpins the model upon which the mathematics is based. Is there an explosive blast of hydrogen hitting the far wall of the vacuum chamber at 4000mph? And if not, is my expectation misguided?

6. Jul 12, 2006

### PRDan4th

Opening a cantainer of any gas at 14.7 psia in a vacuum will create quite an explosion in all directions! A better question is what would happen in an atmospheric pressure container of air when opening a canistger of atmospheric pressure hydrogen, the H2 would mix very rapidly but you would not experience an explosion!

7. Jul 12, 2006

### Cyrus

Do you realize that you have just created a rocket nozzle?

Provided that it would not fail the material, it would go shooting off in the direction opposite the opening.

Roughly speaking, I would initially guess that the speed that the mass leaves your bottle will be a function of the pressure differential, and the area of the opening.

I do not necessarily agree it will be 4000mph, unless someone can show otherwise.

Last edited: Jul 12, 2006
8. Jul 12, 2006

### Farsight

A couple of queries with this, Hootenay:

If my flask was glass and I shattered it via remote control, can I expect a blast of hydrogen molecules in all directions at an average 4000mph?

If the release is into a vacuum and the gas cools, does this reflect some reduction of molecular velocity, and if so what causes it?

edit: our posts overlapped cyrus, but the above is relevant. As far as I know The Kinetic Theory of Gases says the pressure is because the hydrogen molecules are moving at an average velocity of 4000mph (=1789m/s), so if I remove the walls of the flask, I'm thinking that some hydrogen molecules should hit the walls of the chamber (say 1m away) in under a thousandth of a second.

Last edited: Jul 12, 2006
9. Jul 12, 2006

### PRDan4th

Cause of cooling: see Joules-Thompson effect.

10. Jul 12, 2006

### Farsight

Thanks for that. I looked it up on Wikipedia, and the write-up, well it looks reasonable:

http://en.wikipedia.org/wiki/Joule-Thomson_coefficient

But does anybody know if this is right?

As a gas expands, the average distance between molecules grows. Because of intermolecular attractive forces, expansion causes an increase in the potential energy of the gas. If no external work is extracted in the process ("free expansion") and no heat is transferred, the total energy of the gas remains the same because of the conservation of energy. The increase in potential energy thus means a decrease in kinetic energy and therefore in temperature.

And is this right anybody? Hydrogen warms when expanded at room temperature?

Helium and hydrogen are two gases whose Joule-Thomson inversion temperatures at one atmosphere are very low (e.g., about −222 °C for helium). Thus, helium amd hydrogen will warm when expanded at constant enthalpy at typical room temperatures.

Last edited: Jul 12, 2006
11. Jul 12, 2006

### Q_Goest

The warming is extremly slight for h2. From 14.7 psia, 70 F to vacuum the temp only warms 0.06 F (to 70.06 F)

12. Jul 12, 2006

### jasc15

In the thread you are referring to, i posted a response to a similar question to this. To the best of my understanding, this has to do with mean free path. The molecules do move at around 4000 mph (1/2m*v^2 = 3/2kT, or v = (3kT/m)^1/2 ) but that motion is between the tiny spaces between other molecules (something on the order of 4x10^-10m, anyone know the diameter of a hydrogen molecule?). Im not sure if this is a bad analogy, but it is similar to light slowing down in a dense medium. between the molecules, light is moving at 3x10^8m/s but the path is longer.

13. Jul 12, 2006

### Farsight

14. Jul 12, 2006

### Farsight

Molecular Beams look promising.

15. Jul 12, 2006

### Gokul43201

Staff Emeritus
The math is not very hard to do. Let's quickly go through it.

Start with a 1-liter canister of H2 gas (at STP) in the center of a 1m X 1m X 1m cubic vacuum box. The canister is shattered by some remotely triggered mechanism, and the H2 molecules zip off in different directions. Let's roughly calculate the pressure exerted by these molecules as they hit the walls a fraction of a millisecond later.

The plan is to :
(i) count the number of molecules that hit the wall in some time T,
(ii) multiply by the mass of a molecule to find the total mass impinging on the wall during this time,
(iii) multiply this by the change in velocity to find the total change in momentum,
(iv) divide by the time T, to find the average force exerted on the walls, and
(v) finally divide by the area of the wall to find the pressure exerted.

Now as the gas that once filled a 1L canister is now occupying a 1000L box, its density must go down by a factor of about 1000. In the canister, the density of the gas was about 0.1kg/m3, so after release, the density drops to about 10-4kg/m3.

(i) To find the number of molecules hitting the wall in some time T, we construct a boundary layer of thickness vT (where v=4000mph = 2000m/s). At any instant, only the molecules contained in this layer can reach the wall within a time T. The volume of this boundary layer box is hence V = vt*1*1 = 2000T m3. The number of molecules in this box is the product of V and the number density of molecules.

(ii) Since we are going to multiply by mass in this step, the total mass of molecules in the box, is the volume V=2000T m3 times the mass density d= 10-4kg/m3 (from above).

M = Vd = 0.2T kg

(iii) Let's say the molecules impinge normally upon the wall and bounce back (elastically) at the same speed. The change in velocity is then 2*2000 = 4000 m/s. Multiply the mass by this to get the total change in momentum p, during time T:

p = 4000 m/s * 0.2T kg = 800T N

(iv) Force is the rate of change of momentum. So the average force is simple the change in momentum divided by the time taken,

F = p/T = 800T/T = 800N

(v) The pressure is given by

Pr = F/A = 800N/1m2 = 800N/m2 = 800Pa

This, you must realize, is quite a small pressure. Even the atmosphere exerts as much as 100,000Pa of pressure.

Perhaps your layman intuition failed to factor in the really tiny mass of a H2 molecule?

A simple "measurement" involves recognizing that sound propagates through a gas due to molecular collisions. The speed of sound can hence be no greater than the speed of the molecules in an ideal gas. In air, the speed of sound is about 350m/s and hence you'd expect the mean molecular speed to be higher. It turns out to be around 450 m/s. Hydrogen is about nearly 16 times lighter than air, and so, the speed of sound in hydrogen would be about 4 times higher.

Also, a reason that any hydrogen gas produced on earth doesn't stay very long in the atmosphere is that the fraction of the molecules having speeds in excess of the escape velocity (which is what, 20,000 mph?) is not terribly small. The heavier gases hovewer, being slower have no such luck.

The calculated values agree to a high degree of accuracy with measurements of effusion rates, pressures and other macroscopic properties that depend on the molecular speeds.

Last edited: Jul 12, 2006
16. Jul 12, 2006

### Farsight

Thanks for that Gokul. Much appreciated.

I think "Molecular Beam" could be what I'm looking for in terms of demonstrable high velocity impingement on the walls of the vacuum box.

17. Jul 12, 2006

### DaTario

At this point I have an objection. Not all molecules contained in this layer will hit the wall within the time T. You use the expression "can reach" but mathematically you seem to have used the expression "will certainly reach".

Best Regards.

DaTario

18. Jul 13, 2006

### Gokul43201

Staff Emeritus
Da Tario, it's only an order of magnitude estimate. Besides, that one is not not the only approximation I made (some others include uniform density and normal incidence).