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

[Mouse]

>> Suppose we have a cube of air at STP, one metre on a side.

>> Now, consider an ideal plane, [...]

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

Of course.

> 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.

Is that the bond distance from nucleus centre to nucleus centre, or the
size of some sphere's effective radius (molecule collision?), or what?

> [W]e're looking at roughly 256 * 10^25 molecules.

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

Yes, possibly except for the molecule size.  (I hadn't tried to look up
bond lengths in O2 or N2.)

> [...slice the cube up...]

That is a very elegant way of getting the number I wanted.

> 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.

True if we model molecules as 150pm spheres.  If instead we treat them
as 150pm line segments, we need to divide by a factor based on
orientation, something like the integral over all vectors from a
sphere's centre to its surface of the absolute value of the component
of the vector in a particular direction (by symmetry, which direction
doesn't matter to the result).  That too I know how to do, but my
calculus is rusty enough I can't do it in my head while composing an
email.

> 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.

Or 1/(pi^2) of that, or whatever the correction factor turns out to be.

But for my purposes at the moment, this is enough: the number is not
small as in 10 or 100 or even 10000, but not all that large - on the
order of a micromole.  I'm glad I didn't just assume that my initial
guess (that it was a small number, maybe even as high as a million) was
correct!  (I'm writing a piece of fiction and it's relevant in a minor
way at one point.)

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