By now you’ve probably read that the Patriots ball pressure was found to be 2 psi (pounds per square inch) below regulation for their AFC Championship game versus the Colts in a new scandal termed #DeflateGate.
Here’s what we do know, and what can safely be assumed:
- The temperature at kickoff was 51 F (284 K).* You can find a gameday temperature vs. time graph for Foxboro, MA (where Gillette Stadium is located) here. Since the balls were outdoors during the game, it’s safe to assume that the air temperature within the balls eventually reached this level, and that this is the internal temperature the NFL found their 2 psi discrepancy at. UPDATE: the low pressure was measured at halftime, which confirms our assumption.
- The regulation pressure for an NFL football is 12.5 to 13.5 psi. We’ll assume a worst case scenario in which the pregame ball pressure measured by the officials was the lowest allowed value of 12.5 psi (86.2 kPa).
- We’ll assume the balls passed pregame inspection by officials, otherwise they would not have been used.
- We’ll assume deflated balls weren’t switched into the game, otherwise the reports would have mentioned that.
- 2 psi below the regulation pressure = 12.5 psi – 2 psi = 10.5 psi (72.4 kPa).
- We’ll assume that the change in ball volume relative to pressure is negligible (engineering term for “small enough to be considered zero”). This is a safe assumption as a smart cheater would want the ball shape to look normal to officials and opposing players but grip different to their passer. The ideal gas law states that for an isovolumetric process (one in which the volume of air doesn’t change), P2/P1 = T2/T1.
- We’ll assume no air leaked from the ball between pregame and postgame inspections.
Based on the above, we can now start filling values into our ideal gas law equation. Because that equation requires absolute (not relative) temperature and pressure, some conversions are necessary. As footnoted, we’ve already made the temperature conversion. The pressure conversion is simply a matter of adding the above temperature values to standard atmospheric pressure (101 kPa). Thus:
P1 = 101 kPa + 86.2 kPa = 187.2 kPa
P2 = 101 kPa + 72.4 kPa = 173.4 kPa
T2 = 284 K
Inserting these values into the ideal gas law equation and simplifying gives the relation:
T1 = T2/0.926
where T1 is the temperature of the air inside the balls when the NFL measured P1. This gives a T1 value of 307 K, which converts to 92.93 F (33.85 C).
This value may sound high, but in fact the NFL has a long tradition of preparing game balls using a variety of heating methods, such as placing them in dryers or ovens. The “Ball” chapter of the latest NFL Rule Book (PDF warning) specifies new, unaltered balls for kicking purposes only:
Ovens are obviously capable of temperatures far exceeding 93 F, and normal dryer inlet temperatures are as high as 175 C (448 K, 347 F). Therefore, it’s theoretically possible for the Patriots balls’ internal air temperature to have been high enough to result in the postgame low pressure reading without the team having broken the rules.
UPDATE: A more rigorous analysis that doesn’t require ovens or dryers and actually corroborates Bill Belichick’s press conference statements has been posted by Dr. Phil Metzger of NASA.
* The ideal gas law requires absolute temperature, hence the use of Kelvin instead of Fahrenheit or Celsius.