### The Final Word on DeflateGate

Start with the ideal gas law: PV=nRT

 Article TL:DR? Try this chart instead.
Rearrange

P/T=nR/V

Use a little logic

P1/T1=nR/V=P2/Tor P1/T1=P2/T2

Rearrange again to get

T2(P1/T1)=P2

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

Which is slightly less than about the pressure supposedly measured in the balls by the officials at halftime.

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.

1. Isn't 70 F = 294.3 K and 50 F = 274.3 K?
This doesn't change much as you then find
P2=(274.3 / 294.3)*(186,847 pascal)= 174kPa = 25.2 psi, and so you get about 10.6 psi, smaller than your estimate.

2. Oops, typos corrected. Thanks, Anonymous!

3. Shouldn't the final word on this be something a bit more public and come from Bill Nye? Say a full episode devoted to making and explaining his apology - which gets picked up and endlessly replayed by all the major news networks.

1. That would be nice, wouldn't it? Except Bill Nye already flubbed the issue worse than I ever would have imagined.

http://physicsbuzz.physicscentral.com/2015/01/bill-nye-drops-ball-on-deflategate.html

Maybe he could redeem himself by making a real attempt at the math and/or an experiment, instead of blowing it off with an answer that didn't even rise to the level of being wrong.

4. If you don't like my last word, try the New York Times article (thanks for pointing it out, Mike).

http://www.nytimes.com/2015/01/30/sports/football/deflation-experiments-show-patriots-may-have-science-on-their-side-after-all.html

The article basically says the same thing I do.

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