Posted by
Bruce Sherwood on
Jun 12, 2012; 8:20pm
URL: http://friam.383.s1.nabble.com/atmospherics-tp7580159p7580174.html
I realize that I didn't address one of the questions (or one of the
possible questions): "Why don't all the air molecules just fall to the
ground and stay there"? In case anyone was wondering about that
question, the answer is that the air molecules DO fall toward the
ground, but they continually run into other air molecules (or
molecules in the ground if they get that far down), all of which share
a nonzero absolute temperature and therefore are in random motion, and
in collisions sometimes a molecule will be knocked upward. When you
work out the statistical mechanics of all this, you get an exponential
falloff of density (in the approximation of a constant-temperature
non-convective atmosphere). This falloff is a bit faster for the
lower-mass nitrogen molecules than for the oxygen molecules, but as I
explained in a previous note, both of these molecular species have
mean heights of around 8000 meters, so you shouldn't expect much
difference in oxygen vs nitrogen between your cellar and your attic.
A picturesque way of looking at this is the following. Imagine there
is no atmosphere, and you're sitting at a table out in the open (in
your spacesuit). Place a cup on the table. The atoms in the bottom of
the cup are in contact with atoms in the top of the table, and all of
these atoms are moving with random thermal motion related to the
absolute temperature. At any given moment, there is a finite (but
exquisitely small) probability that all of the atoms in the table
underneath the cup happen to all be heading upward. In that case the
cup will leap up off the table, knocked upward by the upward-moving
atoms in the table. This would not violate conservation of energy or
conservation of momentum (the Earth would recoil), but it would
violate the Second Law of Thermodynamics, because given the gigantic
number of atoms lying underneath the cup, the probability of all those
atoms simultaneously heading upward is vanishingly small. You might
have to wait for billions of billions of billions of years to observe
the leap.
Suppose instead of placing a cup on the table you place a single
molecule of oxygen. Now it's not so improbable that an atom in the
table might impart a significant upward speed to this single molecule
of oxygen. Statistical mechanics provides the tools for calculating
quantitatively the probabilities of various upward speeds. What you
find is that the average speed imparted to an oxygen molecule by an
atom in a table at room temperature is a speed sufficient for the
oxygen molecule to go up 7920 meters before falling back down!
In other words, statistical mechanics gives the answer (the same
answer) to two different questions:
1) What is the average height attained by one oxygen molecule in
contact with a table at room temperature? (Ans. 7920 m)
2) What is the average height of all the oxygen molecules in a
constant-temperature atmosphere? (Ans. 7920 m)
(I'm deliberately playing rather loose with the word "average" here,
but the basic idea is correct.)
There's yet another source of amusement in this statistical picture.
Suppose you have a box whose sides have an accurately known mass.
Suppose you weigh the box in an airless room (to avoid buoyancy
effects) with and without the box being filled with
atmospheric-density air. You're not astonished that the extra mass
with the air is equal to the mass of the air added to the box. But
maybe you should be astonished, because at any given instant almost
none of the air molecules are touching the inside of the box! The
reason why the scale measures an increase is because of the e to the
(-mgy/kT) density gradient. The air density and pressure are just a
tiny bit higher at the bottom of the air (in contact with the bottom
of the box) than at the top of the air (in contact with the top of the
box). Momentum transfer per second from the bottom of the air to the
bottom of the box is very slightly greater than the momentum transfer
per second from the top of the air to the top of the box. When you
work out the details, you find that this difference provides the
conspiracy that let's you think you're measuring the mass of the air.
The difference is small, but so is the mass of the air.
Sometimes one describes air pressure at sea level as "the weight per
area of the column of air above that area". But almost none of those
air molecules are in contact with your measuring device! However, the
number of molecules per cubic meter, and their average y component of
velocity, is such (conspiratorially) as to hit your area with the same
force as though an object with the mass of the total column of air sat
on this area.
Related amusement: Consider a steel ball bearing dropped from a height
h onto a scale, and rebounding to nearly the same height every time.
If the scale can respond very quickly, you will see sudden sharp
spikes when the ball bearing hits, and zero at other times. Now
suppose that the scale is sluggish, and/or h is small enough that the
ball bearing hits the scale at a high rate (though small speed). What
you can calculate is that the average reading of the scale is exactly
the same as if you simply place the ball bearing at rest on the scale!
Bruce
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