[geeks] Odd (non-computer) question....

[Phil Stracchino]

On 10/12/21 5:41 PM, Mouse wrote:
> Suppose we have a cube of air at STP, one metre on a side.
> 
> Now, consider an ideal plane, passing through the middle of this cube,
> dividing it into two volumes each 1m by 1m by 50cm.
> 
> Question: what is the expected number of molecular bonds that plane
> intersects at any particular moment?
> 
> It's easy enough to work out how many molecules are in that cubic
> metre.  But I'm having trouble figuring out the rest of it.  Given how
> tiny a molecule is as compared to the mean space per molecule, I expect
> the number to be small, but I don't know whether it's small as in the
> expected number is .01, or as in 10, or as in 10000, or what.


Interesting question.  We're going to work in approximations here.  :)


At a first approximation, air is 80% nitrogen, 20% oxygen.  The size of
a nitrogen molecule is about 155 picometers; oxygen is 152 picometers.
Again for working approximation, we'll say they're both 150 picometers.

That cubic meter of air masses about 1.225kg, and the atomic mass of an
"average" air molecule is 28.8 (80% likelihood that it will be nitrogen
with atomic mass 14, 20% chance of oxygen with atomic mass 16, both
diatomic molecules, and ignoring binding energies and trace gases).  So
our 1225 grams of air contains 1225/28.8, or about 42.535, moles of air.
 Multiply by Avogadro's number and we're looking at roughly 256 * 10^25
molecules.

This much, you've already done or know how to do.

Now:  The cube root of that number is roughly 1367980794, so we are
looking at a theoretical cube about 1.37 * 10^9 molecules in each
dimension.  So we can divide the cube into 1.37 * 10^9 theoretical ideal
even slices, each of which contains 1871371452752870436 (roughly 1.87 *
10^18) molecules, assuming all of the molecules in our cubic meter of
air were distributed approximately in a regular cubic lattice, which on
this scale is TOTALLY as good a working approximation as any other.

That "even slice" is roughly .000000000731 (1/1367980794) meters, or 731
picometers, thick.  But we said that our average air molecule has a size
of roughly 150pm.  That's just *slightly* over 20% of the thickness of
the slice.  And that means that an arbitrary plane bisecting that slice
in a plane parallel to its face has *roughly* a 20% probability of
intersecting any given molecule in that slice.  And since we already
calculated that the slice contains on average roughly 1.87 * 10^18
molecules, our ideal plane will intersect approximately 0.374 * 10^18 of
them.


-- 
  Phil Stracchino
  Babylon Communications
  phils at caerllewys.net
  phil at co.ordinate.org
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