For the conditions of the 2015 AFC Championship Game, using Gay-Lussac’s law we can compute that the Patriots’ footballs should have had an on-field pressure of 11.32 psig, or pounds-per-square inch gauge pressure. This is assuming pregame inflation to 12.5 psig in a 71°F locker room, with an on-field temperature of 48°F. (With the Rankine scale, the locker room temperature was 460 + 71 = 531°R and the field temperature was 460 + 48 = 508°R.) The details of the computation are as follows (I encourage you to try it yourself with a calculator):
(12.5+14.7)*(460+48)/(460+71)-14.7 =11.32
where:
12.5 = starting Patriots’ gauge pressure in psig
14.7 = atmospheric pressure (add this to psig to get absolute pressure)
460 = add this to degrees Fahrenheit to get absolute temperature (Rankine scale)
71 = locker room temp, in degrees Fahrenheit
48 = field temperature, in degrees Fahrenheit
The ratio of the two absolute temperatures, (508°R / 531°R = 0.9567) is about 95.7%, meaning that, in absolute terms, the temperature fell by 4.3% from locker room to field. Therefore, the absolute pressure in the footballs should also fall by 4.3%, from 27.2 psi to about 26.02 psi. Subtract 14.7 psi from this last number and you get 11.32 psig, the predicted on field relative pressure for the Patriots' footballs.
Now let’s compare this prediction to the measurements. Using the Logo gauge, the values for the Patriots’ footballs were 11.80, 11.20, 11.50, 11.00, 11.45, 11.95, 12.30, 11.55, 11.35, 10.90 and 11.35 psig. The average of these values is 11.49 psig. Notice that eight of the eleven measured values are greater than the predicted pressure of 11.32 psig.
Accounting for an additional drop in pressure from some balls being wet, the agreement between prediction (11.32 psig) and observation (11.49 psig) is remarkably good. The difference of 0.17 psi is in line with the amount of warming that would be expected to occur during the measurement process, especially when one accounts for the fact that the Patriots’ footballs were kept in a bag before being measured during halftime. The agreement is especially compelling to me in light of the fact that the calibration of the cheap pressure gauges used by the referee can drift substantially over time. For example, Figure 12 in Exponent’s report shows that the calibration of the two gauges shifted in the relevant pressure range by about 0.1 psi over the course of just a few days. Figures 5 and 6 show that even brand new gauges can exhibit a pretty wide range of values when measuring the same pressure.
With such low-cost gauges, an observation of 11.49 psig and a prediction of 11.32 psig, what more could the NFL ask for?
