Monday, January 26, 2015

Full Frontal Nerdity

Deflategate is if nothing else, a sent-from-heaven excuse for the MSM to quit reporting actual news. That being the case, who am I to pass up the opportunity.

Detailed and necessary information is in short supply, but never let a lack of facts hold up a good story. Firse I've only heard one second-hand report of what the actual pressure in the balls was at the time of the infraction. That was reported to me as 11.5 psi so we'll go with that. Also the minimum required starting pressure which is 12.5-13.5 psi. We'll assume 12.5 just to make it easy for us.

Let's start with a football at 0 psi. That's 0 psig which means the pressure inside is the same as the pressure outside. The ball looks like a ball, but would be easily squashed nearly flat. The actual inside pressure would be 14.6 psia, same as outside. We also assume that the balls were filled indoors at 72F, then taken out to the field where they were played at 45 deg F.

Two formulas apply here: The ideal gas law, PV = nRT, and Gay-Lussec which states that P1/T1 = P2/T2 and describes the change in pressure due to heating or cooling of a fixed quantity of a gas. Run G-L initially assuming 12.5 psig and 27 deg F temp drop (using psia and degrees Kelvin of course, and you get a final pressure of 11.8 psig.

Gay-Lussec is the wrong formula to be using to start as it does not account for adding extra gas. If PV = nRT, and V and R are constants*, then kP = NT. K=12.39 for an initial shot with V=1liter (why not), T = 72F (295 deg K), and n turns out to be .042 moles initially. Change n to .075 moles and P goes up to 1.8 atm (12.5 psia). Solve for temp and it only goes up about 2 degrees C or 3 deg F.
One could tweak this to add the temp rise into the formula which would result in fewer moles added.

NOW use Guy-Lussec and the temp shift is 30 degrees instead of 27 and the pressure drop is from 12.5 to 11.7 instead of 11.8. Not much difference, but closer to 11.5.  Gage calibration could easily account for the rest. Detailed proof is left as an exercise for the student. I've done enough heavy lifting for one day.

Nerds Rule!

I am reminded of the talk radio station I sometimes listen to on the way to work. Any day now I expect to hear: "Nuclear weapon detonated in Washington, Lower Manhatten in flames. This and more after our in-depth interview with Peyton Manning about the Broncos roster shakeup."

*Yes I know the ball swells a bit when inflated. Measure that yourself. R is a constant.

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