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I found this on The particle physics of you on Symmetry Magazine:

The size of an atom is governed by the average location of its electrons. Nuclei are around 100,000 times smaller than the atoms they’re housed in. If the nucleus were the size of a peanut, the atom would be about the size of a baseball stadium. If we lost all the dead space inside our atoms, we would each be able to fit into a particle of lead dust.

Could somebody show the math to come to this conclusion?

I don't know a lot about physics but I like to learn. My question aims to asses how much I can trust the analogy in the article: The particle physics of you.

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    $\begingroup$ Possible duplicate of If particles are points, then aren't atoms empty space? $\endgroup$
    – John Rennie
    Commented May 21, 2018 at 15:01
  • $\begingroup$ See also physics.stackexchange.com/q/126512/50583 $\endgroup$
    – ACuriousMind ♦
    Commented May 21, 2018 at 15:04
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    $\begingroup$ Have you attempted to do the calculation? What difficulty are you having? It is a simple proportion of diameters. Basic mathematics, not physics. $\endgroup$
    – sammy gerbil
    Commented May 21, 2018 at 15:30
  • $\begingroup$ Do you mean diameters of the nucleus and electron? $\endgroup$
    – Gabriel F
    Commented May 21, 2018 at 15:36
  • $\begingroup$ I mean the comparison of diameters of atom to nucleus, baseball stadium to peanut, human to dust particle ... all in the same proportion given in the quote. $\endgroup$
    – sammy gerbil
    Commented May 21, 2018 at 16:42

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Well, this is the sort of question that has two answers:

  1. this really makes no sense because this is all quantum-mechanical and what things like the 'radius of the nucleus / atom' is make limited sense;

  2. but you can just get some approximate numbers and do some kind of Fermi estimate and that will be fine.

So, I'll do (2), grabbing values from Wikipedia or wherever I can find them. See the comments for why this is all bogus.

Let's assume we're made of carbon: a carbon atom has a radius of about $7\times 10^{-11}\,\mathrm{m}$, and a carbon nucleus has a radius of about $2.2\times 10^{-15}\,\mathrm{m}$. Atoms, of course, are cubes as are nuclei so everything packs nicely and we don't have any annoying factors of $\pi$ and sphere-packing nonsense: the volume of a carbon atom is therefore about $2.7\times 10^{-30}\,\mathrm{m}^3$ (multiply radius by 2 to get side of the cube) and a nucleus is $8.3\times 10^{-44}\,\mathrm{m}^3$.

OK, so: electrons are pointlike, so they take up no space at all. So if we collapse carbon down to its nuclear size (so there's no space in the atom outside the nucleus) then we can fit

$$\frac{2.7\times 10^{-30}}{8.3\times 10^{-44}} \approx 3.3\times 10^{13}$$

atoms in the space previously occupied by one.

Human beings have a volume of about $66\mathrm{l} = 6.6\times 10^{-2}\,\mathrm{m^3}$.

And now if we take a human and remove all the space in their atoms we compress them by a factor of $3.3\times 10^{13}$: their final volume is thus about $2\times 10^{-15}\,\mathrm{m}^3$. Humans also are perfect cubes (at least all the ones I know are) and so this translates as a side length of $1.3\times 10^{-5}\,\mathrm{m}$.

This is about 10 microns, which is well within the range of things we'd call 'dust'.

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