Start with the ideal gas law: PV=nRT
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Use a little logic
P1/T1=nR/V=P2/T2 or P1/T1=P2/T2
Rearrange again to get
Now just plug in the numbers, bearing in mind to use the correct units.
For this equation to work, temperature should be measured in Kelvin, which goes to zero at absolute zero, and the pressure is in Pascals.
It's also important to remember that the pressure is not just the pressure you read on a pressure gauge, but the pressure reading PLUS atmospheric pressure, which is about 14.6 psi (100,663 pascals)
P1 = starting pressure in the ball = 12.5 pounds per square inch + atmospheric pressure
=86,184 pascals + 100,663 pascals = 186,847 pascals
T1 = estimate of temperature in the locker room where the balls were inflated
= 70 degrees F = 294.26 Kelvin
T2 = estimate of temperature on the field when the balls were found to be low
= 50 degrees F = 283.15 Kelvin
From one of the equations above
P2 = P1(T2/T1)
= 186,847 pascals(283.15 Kelvin/294.26 Kelvin) = 179,798 Pascals
= 26.07 pounds per square inch = final pressure in the ball + atmospheric pressure
= final pressure in the ball + 14.6
Rearrange the equation again to get
Final pressure in the ball = 26.07 pounds per square inch - atmospheric pressure
= 26.07 pounds per square inch - 14.6 pounds per square inch
= 11.47 pounds per square inch
So, clearly, the Patriots balls could have ended well up below NFL leagues standard pressures, provided they were inflated in a locker room at a reasonable temperature and initially measured by officials to be at 12.5 psi before they cooled to the temperatures on the field.
So who was guilty in DeflateGate? No one, unless you count the people who couldn't understand the math, or were simply were too lazy to do it, but screamed "Cheatriots!" anyway.
And that's the final word on DeflateGate.