Deflategate: Can science tell us if the Patriots cheated?

Did a difference in air temperature cause the air pressure inside the Patriots' footballs to drop below regulation? Here's the math.

Dear nerds, apparently the jocks need our help. 

Deflategate – or, if you prefer, Ballghazi – has left Patriots fans searching for their high-school chemistry textbooks (maybe propping up the ping-pong table in the attic?) in search of a of perfectly natural, not-in-the-slightest-bit-nefarious scientific explanation that would adequately explain why, during a recent game, the air pressure inside their preferred team's balls were found to have been a couple pounds per square inch short.   

It all began late Sunday night, after the Pats defeated the Indianapolis Colts 45-7 at Gillette Stadium to "win" the AFC Championship. According to ESPN, NFL investigators at the game found that 11 of the 12 balls used by the Pats' offensive team "were inflated 2 pounds per square inch below what's required by NFL regulations," which the network says stipulate that each ball should be inflated to between 12.5 psi and 13.5 psi. The Colts' balls, according to reports, were found to have been inflated properly.

Deflated balls are thought to be easier to handle, prompting critics to accuse the Patriots, a team not widely known for its fanatical devotion to NFL rules, of being a bunch of no-good dirty cheatniks. 

What's the average Patriots fanboy to do? When he's not sitting in traffic on Route 1, that is? He's going to start sciencing!

Those who can't bear the thought of their idols being sullied by accusations of underhandedness have latched on to a peculiar explanation: temperature. It's not completely crazy: All else being equal, cold air takes up less space than warm air. So it stands to reason that, if the Patriots' balls were inflated to regulation pressure inside at room temperature and then taken outside into the 48-degree Foxborough, Mass., air, it's only natural that you'd witness a pressure drop.

But by how much? Turns out there's a formula – Amontons's Law of Pressure-Temperature, after the 17th century French chemist Guillaume Amontons – that can tell us exactly how much.

Amontons's Law states that the pressure of a gas of fixed mass and fixed volume is directly proportional to the gas's absolute temperature, or, to put it mathematically: 

P₁  / T₁   = P₂  / T₂

Let's assume that the balls were initially inflated to the minimum regulation pressure, 12.5 psi with 70-degree air. When the air inside the ball falls to 48 degrees, what pressure is it at? The formula for that would be:

P₂  = (P₁  / T₁ ) × T₂

Amontons's Law deals with absolute pressure, not gauge pressure. To get that, we'll need to add the atmospheric pressure of 14.7 psi to the initial gauge pressure of 12.5 psi. After doing that and then converting our freedom-loving units of pounds per square inch and degrees Fahrenheit into their equivalent socialist units of kilopascals and degrees Kelvin, we get:

P₂   = (187.5 kpa / 294 K) × 282 K.

The resulting absolute pressure is 179.9 kilopascals, or, in American, 26 psi. Subtract one Earth atmosphere's worth of pressure, and you wind up with 11.3 psi.

In other words, the temperature difference accounts for, at most, 1.2 psi. For 70 degree air in a football to drop 2 pounds per square inch, it would have to cool to at least 30 degrees. Or perhaps the temperature in the room where they filled the balls was 88 degrees. 

Of course, none of this necessarily means that the Patriots cheated. For one thing, the gauges used to measure the initial or final pressures could have been faulty. 

And second, take a closer look at the NFL's ball rules:

"The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder enclosed in a pebble grained, leather case(natural tan color) without corrugations of any kind."

Notice how there's no mention of pounds per square inch? It's just pounds. Taken literally, the rules say that the bladder inside the football should weigh between 12.5 and 13.5 lbs.

So there you have it. According to the NFL's own rules, football is meant be played with something approaching the weight of the average bowling ball, not the feathery insubstantial spheroid that these so-called athletes have been prancing around the gridiron with for all these years.