Gas in a (very small) box.

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 10²³ 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.