The Science of Deflategate

picture of Harry KellerBy Harry Keller
Editor, Science Education

The New England Patriots have muddled up the Super Bowl in a grand fashion. It might as well be an MIT prank but on a national scale. We haven’t see the like since Caltech (MIT’s famous rival in the prank world) jimmied the Rose Bowl’s card stunts half a century ago. (Personal note: I was a member of the Caltech group of twelve that did that. Also see this LA Times article.)

Denials are not going to change any fan’s mind. If you’re a Patriot fan, you probably don’t think it’s important. If you’re not, then you will believe any evil of the New England dynasty.

Before delving into the science, I should note that reporters have said that the ball deflation was discovered during halftime and rectified. As the final score was 45-7, and the second half score was 28-0, even if every Patriot point in the first, seemingly flawed half were rescinded, the score would still be 28-7 in favor of the Patriots. We are not discussing, therefore, who should play in the Super Bowl. We are instead discussing how balls would have become low in pressure.

As any high school physics teacher will tell you, PV=nRT, the ideal gas law. Okay, that’s gibberish to many, but it’s really very simple. So simple in fact that you can do the calculations yourself with calculator or even readily with paper and pencil. Let’s deconstruct this equation. 

P is pressure. That’s the factor under discussion. One unit of pressure is PSI or pounds per square inch. According to a rule buried in two hundred pages of NFL rules, it says that the balls must be inflated between 12.5 and 13.5 PSI. It does not specify the conditions under which this pressure is to be tested. In our tax code, we call this a loophole.

Next is V, the volume of the gas. In a football, increasing or decreasing the pressure a bit will make very little change in volume. For the purposes of this discussion, we can consider the volume to be constant.

The factor, n, is the amount of gas present, measured in moles. We don’t have to think about what a mole is. The crucial element here is that it is the amount of air put into the ball with that inflation needle. More air means more pressure and/or volume if R and T are constant. Less air reduces pressure. If you put the needle into the ball without a pump attached, air will escape, and the pressure will drop.

R is the gas constant. That last word is all that matters. This is a constant.

T is the temperature measuring on an absolute scale. Usually, this scale is Kelvin. In the Kelvin scale, absolute zero is zero degrees or 0K. In the Fahrenheit scale, the value is about -459°F. In the Celsius scale, it’s about -273°C. Increasing the temperature of a gas increases its pressure and/or volume as you can see from the ideal gas law.

We’ll take the volume as constant along with the amount of air and the constant R. Then, we get that P = kT, where k is some constant depending on the other values. If you double the temperature on the Kelvin scale, then you double the pressure.

After all of that nerdy discussion, you come to the nub of the issue. Usually, people will begin with 12.5 PSI and room temperature to calculate the pressure on the field at playing temperature. I will do this backward instead.

The NFL says that the footballs measured at halftime on the field were at 10.5 PSI. According to reports, they really said that the pressure was two pounds low. Scientists don’t use such sloppy descriptions. We cannot know if two pounds means 2.0 pounds or 1.7 pounds. Both numbers, when reported with a single digit of precision are the same.

Let’s give the Patriots the benefit of any doubt here and take 1.7 as the difference and 10.8 PSI as the actual measured pressure. Eleven balls measured low and certainly were not at the exact same pressures. One ball did not measure low, a strange occurrence that no one has explained.

We know the field temperature in Foxborough that day was in the high forties (Fahrenheit). Let’s call it 48°F. The absolute temperature scale using Fahrenheit as the unit is called Rankine. Add 459 to 48 and get 507°R. Let the pressure be 10.8 PSI to calculate the proportionality constant (that’s what scientists call it) as 10.8 / 507 PSI/°R = 0.021 PSI/°R. We now know that each degree Fahrenheit changes the pressure by 0.021 PSI.

Getting the pressure back up to 12.5 from 10.8 means raising the temperature by

(12.5 – 10.8) PSI / 0.021 PSI/°F

The units in °F are the same size as in °R allowing the substitution you see above. The calculation gives you 81°F as the required temperature rise from our posited 48°F field temperature to a rather high 129°F.

Locker rooms are notoriously high-temperature places, but not that high. It might be 80°F in the locker room. A temperature rise of between 40°F and 50°F must have taken place if the balls were untouched after preparation. Otherwise, there are other factors. Could the volume have changed? How about a leak in eleven footballs? Neither is likely.

There is another factor not included here: water vapor. If the air in the ball is saturated in water vapor as in most locker rooms, then it may constitute 3% or more of the air. At the low temperatures of the game, it would be between 0% and 1% of the air with much of the water having condensed out. The quantity of “air” would decrease by 2% or more and so would the pressure. That factor alone would bring the ball down by around 0.2 PSI and require only about 119°F as the air temperature in the ball.

Bill Belichick told the world that he was able to prepare footballs in the usual way and observe as much as a 1.5 PSI change in pressure after preparation. This information helps us a great deal. These footballs are banged up and rubbed to make them behave as much older footballs would. Brady likes his footballs “old” and soft. Rubbing vigorously, as Lord Kelvin (he of the Kelvin scale) told us, generates heat and lots of it if you rub especially hard as with some machine. You do not have to rub if you consider another factor: compression.

Air is warmed when it is compressed. How much? That depends on the temperature and pressures. The equation is a bit daunting.

T2 = T1*(P2/P1)(k-1)/k

T2 is the absolute temperature after compression; T1 is before. P2 is the pressure after; P1 is before. And, k is a factor that depends on the gas and is about 1.4 for air. Using 540°R as T1, 14.7 psi as P1 and 27.2 psi as P2, you can use a calculator to find T2 = 644°R or 184°F. That’s rather hot. Of course, the football will heat up and take some of that heat away. In any event, you can readily have a temperature of 119°F inside the football during inflation with a pressure of 12.5 psi with respect to ambient air pressure (14.7 psi).

You can impute nefarious motives to Belichick and company if you will. Or, you can assume that once again, they are following the rules to the letter and not a whit further. They are allowed to prepare the footballs by rubbing them. They are not told by the rules the environmental conditions under which pressure measurements are supposed to be made. The measurement certainly was made while inflating the footballs. Everyone knows that Tom Brady likes his footballs to be at the soft end of the spectrum and to be roughed up as an old football would be. The Patriots staff might be doing all of this deliberately because they understand the ideal gas law very well. They might just have a preparation procedure that leaves the air inside the footballs warm enough to make the pressure change seen.

So, it all comes down to two things: intent and the obscure NFL rules, 200 pages of them. The intent may have been to make Tom Brady happy without him knowing a thing about the details. And why would he? The intent may have been to follow the letter of the rules only. Or, it may have been accidental that it all worked out this way. I doubt that we’ll ever know given the tight organization that the Patriots team is.

The only answer is to note that the Patriots were the victors of the AFC championship no matter how you slice it and to consider ways to change the rules (again!) so that this silliness never happens again.

I’d also remark that Bill Nye the Science Guy got it wrong this time. Look it up. He said that you cannot change the pressure without a needle. He also said, “Go Seahawks!”

Another personal note: I coached a Science Bowl team to several regional championships over several years. On two successive National Science Bowls, my team did something that was within the rules but that the judges did not like. In each following year, the rules had been changed to prevent a recurrence. Rules are rules. If you follow them and someone complains, then they have no recourse other than changing the rules in the future.

2 Responses

  1. Despite my article, just sit back and enjoy the Super Bowl. It should be a close and exciting game.

  2. Someone suggested that leather stretches when wet. How much? Did the rain add an additional factor to the pressure reduction?

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