*The pressure of the air at sea level is approximately 14.7 lbs/in² = 30 inches/Hg, or 760 mm/Hg. While these are useful units for a bicycle mechanic, WWII pilots, and a 1960’s weather person, scientists today prefer the unit “Pascal” defined to be 1 N/m². Air pressure at sea level fluctuates a bit, but it typically is about 10⁵ Pascals, usually written p ~ 100 kPa, where “kPa” means “kiloPascals = 10³ N/m²”.

** The density of the air is not constant but falls off as you go up (as anyone who has ever climbed a high mountain has found out). A better (though still not exact) model is to treat the density of air as a decreasing exponential: What we are then doing is approximating an exponential curve by a constant – like this – so that the area under both curves (equal to the total mass) is the same. The interesting thing is that doing a fairly straightforward integral (the integral of an exponential) shows us that what we have found is not the height of the atmosphere, but it is in fact the “h” in the exponential – the height you need to go to get the atmosphere’s density to fall by a factor of 1/e (=0.37)

To see the math, we'll just slice the column of air into little slivers of area A and thickness dy. Then the mass of that sliver is the density at that height (ρ(y)) times the volume, A dy. Adding them all up (integrating) gives the result.

This tells us that the "h" we get out of our assuming a constant density atmosphere is actually the same as the fall off parameter for the exponentially decreasing atmosphere.