Clearing Up the Air on DeflateGate

David Butler II-USA TODAY Sports

Looking at all the sloppiness in the reporting, and in the NFL rule book, I figure it's worth a quick review of everything.

Here's what the NFL rule book has to say:

The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder enclosed in a pebble grained, leather case (natural tan color) without corrugations of any kind. It shall have the form of a prolate spheroid and the size and weight shall be: long axis, 11 to 11 1/4 inches; long circumference, 28 to 28 1/2 inches; short circumference, 21 to 21 1/4 inches; weight, 14 to 15 ounces.

There's a lot of awful science here. First off, pounds are NOT a unit of pressure: pounds per square inch are. Moreover, there is the implicit use of what's known as a gauge pressure: that is, we're measuring relative to whatever atmospheric pressure happens to be. (The defined "standard" atmosphere is about 14.7 psi, and many pressure gauges are calibrated relative to this value.) So that's what we'll work with here.

Now, also note that there's quite a bit of tolerance packed into the size of the football. Even with those quarter inch discrepancies in the long axis and circumference, you can get about a 5 percent change in the volume. So if you put the same amount of air in a football at the lower end of the tolerances and at the higher end of the tolerances (for instance, if you pumped them for the same amount of time), you'd end up with a pressure roughly 5 percent lower for the largest allowed football than for the smallest allowed ball.

The other issue is that pressure and temperature are correlated. Back in high school you might have learned something called the ideal gas law, which tells us how the volume, pressure, and temperature of a gas are related to one another. While it's not completely accurate for engineers working in an industrial plant, it's good enough for the low pressures and temperatures we're dealing with here. Basically, it says:

p * V = n * R * T

where p is your pressure, V is your volume, n is the number of moles of gas, T is the temperature, and R is the universal gas constant. For our purposes, we don't need to worry about anything other than pressure and temperature. If we have two sets of pressures and temperatures for the same volume and amount of gas, then we have:

p1 / p2 = T1 / T2

where the "1's" and "2's" are initial and final state. A little rejuggling gives us:

p1 = p2 * (T1 / T2)

So we can find the initial pressure if we know what the initial and final temperatures were, as well as the final pressure.

Using this equation is a little bit more complicated than just sticking in the numbers that are reported in the press. You can't stick in 50 degrees Fahrenheit, since that's a relative temperature, and you can't stick in 12 psi, since that's a relative pressure. We need to use absolute temperatures. The "offset" for Fahrenheit temperatures is 460 degrees (that puts us in the Rankine scale), and as I mentioned above, the offset for pressure is 14.7 psi. (As an engineer, I have to admit I'm getting dry heaves at the idea of using Rankines and psi in a calculation, but here we are.)

So, if the balls were 2 psi under expected, that means their pressure was

p2 = (12.5 - 2) + 14.7 = 25.2 psi

and the temperatures were (guessing here):

T1 = 75 Fahrenheit = 535 Rankine

T2 = 45 Fahrenheit = 505 Rankine

That means our initial pressure should have been

p1 = 25.2 * (535 / 505) = 26.7 psi

But remember that we have to subtract off the atmospheric pressure to get the gauge pressure, so now we're back to an initial pressure reading of 12 psi.

On the other hand, there are also a bunch of other issues:

  • What is the accuracy of the gauge used to measure the pressure? Most standard gauges that I've seen aren't accurate to more than about ±0.5 psi.
  • What was the temperature indoors in the room where the balls were being inflated? Small temperature differences matter here: each degree of temperature difference adds about a 0.05 psi difference.
  • What size were the footballs relative to the allowed range? As I mentioned above, you can get about a 5 percent discrepancy in pressures just because of the size difference. So that's about another 1.2 psi difference that would possibly need to be accounted for.

So there are a lot of issues that can explain what's going on without invoking nefarious intent on the part of the Patriots. And this doesn't take into the account the fact that the referees are ultimately responsible for maintaining the balls in playable condition, regardless of what the teams provide. (And, as Aaron Rodgers has indicated, teams regularly try to skirt the rules.)

EDIT: The starting temperature was about 52 degrees, and let's say for the sake of argument that the temperature was 50 degrees at halftime when the measurements were made (the data from Weather Underground suggests 50 degrees instead of 45, but that's measured at a slightly different occasion. You get a result of about 11.7 psi. It's not enough to shoot down the argument. And again, it doesn't take away the responsibility from the referees to inflate the balls to the proper pressure (as is their job in the NFL rulebook).

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