Before the physics, I want to point out one of the most interesting ScienceBlogs posts I've seen in a long time: Not Exactly Rocket Science discussing the body-swapping illusion. You should read it.

Now, here's a quiz I gave my Physics 201 students. Easy as usual, but I'm a sucker for these order-of-magnitude problems where you get an intuitive feel for what a right answer "feels like". It's taken from one of the homework problems in their textbook:

What is the length of the side of a cube containing a number of molecules equal to the number of people in the world (~6 billion) assuming a gas at standard temperature and pressure?

Air is pretty close to an ideal gas for many purposes, so we can use the ideal gas law:

P is the pressure, V is the volume (what we're looking for), R is the ideal gas constant, T is the Kelvin temperature, and n is the number of moles...

Moles? It's a term of art in chemistry and physics, but really it no more complicated than "a dozen". Except instead of 12 of something, a mole is 6.022 x 10^{23} of something. We need a unit of number that big because atoms are very small and there tends to be a lot of them. So doing the division, we see that six billion molecules is about 9.96 x 10^{-15} moles. Not a lot. So solve the ideal gas equation for V and we'll be set:

Standard temperature and pressure means our gas is at 0 degrees Celsius and a pressure of 1 atmosphere. A little chilly, but convenient for calculations. Plug the numbers in and you get V = 2.23 x 10^{-16} cubic meters. Taking the cube root to get the length of one side and converting units, we see that each side of the cube has length of about 6 microns. That's about the size of a red blood cell.

Atoms are *tiny*.

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A Rydberg hydrogen atom in the n=137 excited state has a radius of about a micron. You would need no more than 216 of them to fill the same volume as 6 billion mediocre gas molecules at STP.

No kidding.

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Excellent 'visualization' that I have not seen before.

Cool.

...tom...

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Spheres pack tighter than that, Uncle Al. Not that treating them as spheres is a good approximation in this case...

This reminds me of a very short notice in Science several years ago about the entropic force and about why large and small particles spontaneously segregate. Could you develop an article along these lines?

STP is 0degC? Since when?

Interesting. The wiki for standard temperature lists several standards. IUPAC is cited as 273K, while NIST is the 293K (20degC) I was taught as an undergrad.