Built on Facts

A mysterious gap in posting? Must be finals time! As of now I’m done. We’ll see how they went. One of them went well for sure, the other sort of depends. My own students are having their own exams as well, and I’ve got my fingers crossed for them.

One of the things that’s going to be on the exam (or at least it in a chapter they covered) is the speeds of molecules in a gas. Like people milling around in a crowd, as the molecules fly around and bump into each other they’ll be exchanging energy and moving at different speeds. Some molecules will have just been hit in just the right way to go very fast until they get hit again and slow down. And some get hit just right to be brought temporarily to a near halt. On average we can calculate how fast the molecules are moving. The equation to do this is called the Maxwell-Boltzmann distribution and it’s a doozy:

That gives you the probability density of finding a gas atom or molecule at a particular speed. A graph will make things more clear. Here’s a graph for nitrogen at 30 degrees Celsius. That’s a fairly warm temperature in the mid 80s for us Americans. Nitrogen molecules (two nitrogen atoms bound together) are the most common component of air.

You can see that most nitrogen molecules are percolating along at around 400 m/s or so. The vast majority (more than 90%) are between about 100 and 800 m/s. Above 1000 m/s or so the odds of a molecule being hit in just the right way to reach such high speeds are slim indeed. Only about 1 in 1000 molecules are above 1200 m/s, and it falls off absurdly fast beyond that. Up in the 3000+ range the fraction is too small to bother with – one in trillions of trillions.

Hydrogen molecules are much lighter in mass. So at the same temperature they’ll be moving much faster. I’ll run the numbers and graph the hydrogen speeds on the same graph as the nitrogen. In the graph below, the hydrogen speeds extend to the right much farther:

The average speed is in the neighborhood of 1500 m/s, and while nearly no nitrogen is moving at 3000+ m/s, almost 7% of the hydrogen is. As high as 10 km/s, hydrogen still has a respectable fraction (small, but more than 10 million per mole) of molecules which are at that speed. This is pretty much the escape velocity of earth, and so a hydrogen molecule at this speed that’s at a high enough altitude not to hit any more molecules on the way out will be gone for good. Essentially no nitrogen molecule ever manages to get fast enough to escape.

There’s a lot of nitrogen in our atmosphere but not much hydrogen. This is the main reason why.

Comments

#1 Uncle Al
December 10, 2008

The Space Scuttle initially re-enters around 7500 m/s. One can appreciate why things are toasty outside (and occasionally elsewhere) even before the kinetic energy burns off. Relative velocity works!

http://www.madsci.org/posts/archives/2001-04/988378293.Ph.r.html
#2 wfr
December 10, 2008

Doesn’t the M-B equation describe the velocity distribution for a single molecular species? Is it valid to claim that the velocities of the two molecules in air are distributed as in the combined graph?
#3 Matt Springer
December 10, 2008

The graph shows the distributions for H2 and N2 separately, which is kosher. The fact that the gases are mixed doesn’t the distributions for each kind of gas, so long as we can approximate the gases as ideal. Most gases are ideal to a good approximation at typical atmospheric temperatures and pressures.
#4 Joshua Zelinsky
December 10, 2008

Isn’t there another issue at work here? Hydrogen is a lot more reactive than nitrogen. In a high oxygen atmosphere like our own some of that hydrogen is forced to become water. If we took all the hydrogen atoms in water on earth and changed that to hydrogen gas would that quantity be on the same order of magnitude as the amount of nitrogen in the atmosphere or not?
#5 The Science Pundit
December 10, 2008

Joshua,

I’m not sure about the hydrogen, but something similar to your explanation is why Earth doesn’t have a predominantly carbon dioxide atmosphere like Venus and Mars do. The presence of oceans (as well as plate tectonics and life) locks up most of our CO2 in the form of carbonates (and carbohydrates too).
#6 wfr
December 10, 2008

OK, I’ll buy that the gases under discussion (H2, N2, air) behave somewhat ideally. But the ideal gas law talks only about bulk quantities. It glosses over the small stuff, like atomic mass and individual velocities.

The M-B equation (about whose derivation I am clueless) does deal with the small stuff, and I would think that the assumptions it makes regarding collisions would be different if collisions between molecules of different masses were taken into account.

If I am right about this, then the resulting equation would be different, and the velocity distributions in a mixed gas would not be the same as in a single element gas.

What am I missing?
#7 Matt Springer
December 10, 2008

Well, the question is definitely an important one. It’s not at all a priori obvious to me that different gases would continue to behave in the same way when mixed. But they do, and the reason why is a subtle and very deep result in statistical mechanics called equipartition of energy. Energy is split evenly among the various degrees of freedom, so the three translation degrees of freedom (corresponding to the three spatial dimensions) have the same average kinetic energy for a given temperature no matter what combinations the gases appear in.
#8 Mark Dow
December 10, 2008

These mean velocity differences also explains, more or less directly, the speed of sound differences between gases, and why your voice is squeaky after breathing in helium.
#9 Bob Sykes
December 11, 2008

Since many of these molecules are supersonic, how does the mean velocity explain the speed of sound?
#10 Mark Dow
December 11, 2008

Sound is a pressure wave, and pressure is determined by how fast and how often molecules collide. The speed of sound ends up being close to 70% the mean speed of molecules — nearer to the mode than the mean.

[Sorry, replace velocity with speed in original post.]

It’s true that many molecules are supersonic, but even a supersonic molecule doesn’t get too far before colliding and being slowed down. The average speed (displacement/time) over time for any one particle is not supersonic. The average displacement for all particles over time, as a sound wave passes, is near zero.
#11 Dante
December 12, 2008

There’s a lot of nitrogen in our atmosphere but not much hydrogen. This is the main reason why.

I find this hard to believe. Even if a very small number of hydrogen molecules achieve escape velocity, enough time has passed that they all leave Earth.
Or not?
#12 magetoo
December 12, 2008

These mean velocity differences also explains, more or less directly, the speed of sound differences between gases, and why your voice is squeaky after breathing in helium.

I’d love to see the squeaky helium voice explained properly some time. Everybody seems to know that it is “because the sound travels faster”, but that really doesn’t explain anything at all about the core question: Why do my vocal cords vibrate at a higher pitch?

I can kind of see how the speed of sound, and how the average speed of molecules in a gas, would affect an instrument such as a flute (intuitively, faster molecules will mediate pressure changes faster); and that obviously a reed instrument would be affected in a similar way, given how the physics of the vibrating air column inside is similar (or exactly the same?). And it is perhaps not too far a stretch to extend the analogy from reed instrument to the human voice, but I think I have only ever seen handwaving with nothing really connecting “higher speed of sound” to the end result “higher pitch”.

Writing this comment seems to have helped in bringing things together a bit, though.
#13 Paul Murray
December 14, 2008

Does this mean that earth will inevitably lose all its atmosphere? That everything volatile will, sooner or later, evaporate off?