# 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.

1. #1 Uncle Al
December 3, 2008

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.

2. #2 Kobra
December 3, 2008

Atoms are tiny.

No kidding.

3. #3 ...tom...
December 3, 2008

Excellent ‘visualization’ that I have not seen before.

Cool.

…tom…
.

4. #4 Tercel
December 3, 2008

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

5. #5 Bob Sykes
December 4, 2008

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?

6. #6 Donalbain
December 4, 2008

STP is 0degC? Since when?

7. #7 agm
December 4, 2008

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.

New comments have been temporarily disabled. Please check back soon.