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Deflate-gate: A simple matter of Boyle's Law

By Dr. Ben Arnold

The great fanfare over the under inflated footballs in the past New England Patriots football game can be explained by physics. The Ideal Gas Law (Boyles Law) gives the relationship among gas pressure, volume, and temperature.
P x V/ T = constant
Lets assume the Patriots' footballs were only inflated to 12.5 psi (the NFL range is 12.5 to 13.5 psi). The News report stated under inflated by 2 psi, we don't know if this was under inflated from 13 or 12.5 psi, lets assume 13, so that the balls had a reported psi of 11 psi as measured by a gauge. If the inside temperature when the balls were prepared was say 75 degrees F and the outside at game time was 50 F, we have a 25 F degree change. One little, big issue with the above equation is that temperature and pressure must be expressed in absolute units i.e. we have to translate the temps into the Kelvin scale and the baseline atmospheric pressure to the gauge pressure. We have to make some more assumptions:

  1. The volume of the balls don't change
  2. The atmospheric pressure is 14.7 psi and
  3. Vapor pressure inside the ball doesn't change.
Then
(12.5+14.7) * V1 / 297 = P2 * V2 / 283.1
Since V1=V2, they cancel out. Solving for P2, we get 25.9 absolute psi, which is 11.2 gauge psi.


This can mostly explain the issue but we need to be more precise by considering the atmospheric pressure. The barometric pressure in Boston on Jan. 18, 2015 at about game start time was 29.9 Hg and dropped to 29.69 Hg (These have to be converted to pascals). I leave it to you to determine this step can explain another about 0.2 psi.

But one of the assumptions is not quite correct. The volumes of the balls do change a little with pressure and temperature. This can be increased by the ball being wet as it was during the rainy game. I have no way to estimate this effect but to be sure the volume would increase as leather expands when wet. These small increases in volume go to lower the ball's psi further.

Finally, the vapor pressure of the water content of the air inside the ball could potentially have another 0.2 psi depending on the conditions when inflating the balls.

These estimates add up to a possible explanation of Inflate-gate, but the news doesn't know or don't want to investigate this because it might shut down the coverage? I suggest the uncertainties of these calculations are less than the verbal accuracies of the news reporters and emotional fans. It appears to me that deflate gate is just old physics newly discovered for the first time and news running ahead too fast to promote controversy.

Play ball!



This story was posted on 2015-01-30 15:46:33
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