Footballs, Temperature, and Pressure

Have you ever gotten together with friends and family on Thanksgiving Day and gone outside for your annual Turkey Bowl?  You go into a shed and retrieve your football, only to find that it's not quite as robust as it was in the summer.  You may not have a leak in your football.  Air molecules don't bounce around as much in the cold as they do in warm weather.  Inside a football, air molecules bounce around and collide with the interior walls of the ball (bladder, really).  The air's pressure inside a football will go down with temperature.

The deflate-gate controversy has people asking how how a given drop in temperature affects pressure.  Unfortunately, I keep seeing the same mistake over and over again in the various analyses I've read.  Using the ideal gas law is just fine.  Assuming the ball's volume doesn't change is a great approximation.  If there are no leaks -- and no illegal removing of air -- the number of air molecules inside the ball remains constant.  The simple result predicted by the ideal gas law with those assumptions is that pressure is proportional to temperature.

Here is where the problem comes.  The pressure that must be used is the total pressure.  The pressure range that's stated for a legal NFL football is 12.5 psi - 13.5 psi, but those pressures are gauge pressures.  The gauge pressure is what we measure above the normal atmospheric pressure we experience all the time, and never notice.  Atmospheric pressure is about 14.7 psi.  That's right, we all have the weight of a bowling ball pushing on each square inch of our bodies.  Luckily, we evolved in Earth's atmosphere and our cells have interior pressures just above 14.7 psi, so we feel equal forces on each side of our skin.  The legal total pressure inside an NFL football is thus 27.2 psi - 28.2 psi.

Assume that an NFL football is at 13 psi when checked inside a locker room at 70 F (21 C or 294 K).  Now take the ball outside.  Using the ideal gas law, and remembering that temperature must be in Kelvin, the graph below shows what to expect for the ball's interior gauge pressure.  The horizontal axis shows possible outside temperatures and the vertical axis shows the gauge pressure (click on the graph for a larger view).

I put a red, dashed vertical line to find the temperature at which the interior gauge pressure hits 12.5 psi, the bottom of the legal range.  That temperature is 60.4 F (15.8 C or 289 K).  You can see in the above graph how a ball that's legal in the warm locker room can lose pressure in the colder outside.

Now, I wish to make it clear that I do not know how and where referees check balls before games.  I don't know if balls are checked in a warm environment or if they are checked in the outside environment where the game will be played.  I don't know the manner in which the balls were rechecked when 11 of 12 of the balls in the Pats win over the Colts were found to be under-inflated.  I do know, though, that if balls that were checked before the game were rechecked at the same temperature at some later date, and found to be at lower pressure, then air must be missing from the balls.

In what I calculated above, I did not account for changes in humidity or anything else.  If air does not leave the football, temperature change is likely to be the dominant factor in changing interior pressure.