7.1.P6

In our discussion of the ideal gas law, (Kinetic theory: the ideal gas law) we treated our gas as dilute -- the molecules were far apart (compared to their size) and collided rarely (but often enough so that they shared energy). In between collisions we considered them to move in straight lines. This of course means that we were ignoring the effect of gravity on the molecules.

We know this is not entirely reasonable. It is, after all, gravity that is responsible for the increase in pressure in a fluid with depth (see Archimedes' principle) that results in buoyancy. But since molecules don't move very far between collisions, maybe -- at least in air -- we can ignore the effect of gravity. Let's estimate to see.

A. We know some information from chemistry that might be helpful.

• At STP, one mole of gas occupies 22.4 liters.
• One mole of any substance contains Avogadro's number (~6 x 10²³) molecules. 
• An atom has a diameter of about 0.1 nanometer.

Using this information (and anything else you might know), estimate the average separation between the centers of molecules in air at STP. On the average, how far apart are their surfaces? 

B. From our analysis of pressure in kinetic theory (Kinetic theory: the ideal gas law) we have learned that the average kinetic energy of a molecule (½mv²) is equal to 3( ½k_BT), where k_B is Boltzmann's constant. More info from chemistry (and physics) can help here.

• k_B ~ 1.38 x 10^-23 Joules/K
• The molecular weight of air (mostly Nitrogen and Oxygen) is ~30.
• The Kelvin temperature at STP is ~300 K.

Using this information (and anything else you might know), estimate the average speed of an air molecule at STP.

C. To get an idea in the simplest case, consider a molecule moving with the average speed of an air molecule in a horizontal direction. If it travels the average distance between molecules between collisions, estimate how far it will fall in that time. Some useful information from physics:

• g ~ 10 N/kg.

Joe Redish 10/30/14