This Deflate-Gate controversy had me thinking about 9th grade chemistry and wondering about the effect of temperature on the balls. I saw that CAR kicker Graham Gano recently tweeted that he is often frustrated by how the balls naturally lose their air pressure when going from an indoor setting, where the inflation pressure is set and checked, to a cold outdoor setting. Could NE's balls have lost their pressure due to the temperature?

We'll need to dust off the Combined Gas Law to figure this one out. That's the physical law that says gas temperature, pressure, and volume are all related. Pressure and temperature are directly proportional--As temperature goes up, pressure goes up proportionally and vice versa. Pressure and volume are inversely proportional--as volume expands, pressure decreases and vice versa. And so it follows that volume and temperature must also be directly proportional. Put mathematically, we can say that for a given mass of air, the ratio of product of the pressure and volume and the temperature must remain constant.

PV/T = k (where k is a fixed constant)

If we change any one of those variables, we can solve for the others. Because k=k, we can write:

P1*V1/T1 = P2*V2/T2

It's all coming back now, isn't it? (Aww man, I've got a quiz in English next period and I didn't do the reading. What happened in chapter 7?)

Back to New England. The NFL rules are that the balls must be inflated to between 12.5 and 13.5 psi. The report I just read said that 11 of the 12 balls were under-inflated by 2 psi each. I assume that means they must be about 10.5 psi. Let's say it was 70 deg F indoors. The temperature at kickoff was 51 deg F according to the official NFL record, fairly warm for Foxboro this time of year. One gotcha is that temperature and pressure must be expressed in absolute terms, so we need to translate the temps into the Kelvin scale and the baseline atmospheric pressure to the gauge pressure.

Assuming V1=V2 (the ball does not significantly grow between 10 and 13 psi) and that no deflation occurred, the mass of air remained the same and we can write

(12.5+14.7) * V1 / 294.3 = P2 * V2 / 283.7

Since V1=V2, they cancel out. Solving for P2, we get 26.2 absolute psi, which is 11.5 gauge psi. That only explains a half of the discrepancy, and there's still 1.0 psi left to account for. But the volume of the ball does change when inflating, which is the whole point of deflating it to make it easier to hold, throw, and catch. But it only does so slightly in that range of pressure, probably by a few percent. And a shrinking ball would have the effect of mitigating the loss of pressure.

Any warming that may have occurred during halftime when the balls were re-examined would reduce the effect of temperature, and leave more of the discrepancy unaccounted for. Certainly, a full half of play might deflate the balls, but 11 of the 12 were under-pressurized, so I doubt that would explain it. Some balls would be affected much more than others depending on what kind of abuse they get.

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• ### Guest - Jim Welsh

Dear Mr. Burke,
It is not necessary to assume that rules were broken. And Bill Nye the Nonsense guy is all wet. Bill Belichick is a genius, and the NFL owes him a big apology. All Bill Nye knows is PV=nRT. Belichick knows how to gain every possible advantage within the confines of the NFL rule book.
So Bill Nye, the Nonsense guy, you should stick with dancing shoes. You are sadly misinformed, you do not understand thermodynamics, and you owe Bill Belichick an apology also. The NFL has just stepped on a hand grenade, and this is so funny, if only to people who understand the joke "Knock, knock." "Who's there?" "Well, it could be Heisenberg." Need proof? Put on your thinking caps.
So Mr. Burke, what would you say if I told you that it is within the realm of reason and scientifically provable that the likely hood is that Bill B did not cheat, followed the rules to the letter, and still knows little or nothing of the complex subject called thermodynamics?
I will spare you the complexities of how a quick pressurization can be approximated as an adiabatic-reversible process. I will leave that as an assignment for you. Links are provided below. Please show all work. Ha!
Instead I will give you an example. If you want to clean some electronics you can buy a pressurized can of air. If you use it, you will notice that the can gets cold. You set it down and it warms back up.
The same thing happens (in reverse) with a football, if you use a hand pump. The mere act of injecting more air into a partially inflated ball raises the air temperature. I will save you the math, but if the Pats were to have submitted balls to the officials that were inflated to 10 psig and 70 deg F, then these balls were hand pumped up, BY THE OFFICIALS to the 12.5 psig as Bill B. requested them to do, the mere act of pumping those balls up to 12.5 psig would have also raised the air temp on the interior of the ball to 85 degrees F. There is nothing in the rule book that states what the temperature of the air is inside the balls when the balls are examined. The rule's just states that the balls be presented to the refs, and that the refs will set them to the proper pressure. It also explains how the Colts' balls were not similarly deflated by halftime. They were at equilibrium (70 deg F) and near or in the specification when delivered to the refs.
An 85 deg. ball will begin losing pressure immediately if the ambient temperature is 70, and will lose exactly the measured pressure of 10.5 psig if the temp fell to 45 degrees by halftime.
Sorry Bill B. I guess every team will know this trick now, as if many others don't already know.
The funny thing is that Bill B. used the ref's elbow grease to pump up the balls that gave his team a perfectly legal advantage. He can deny knowing the particulars of the thermodynamics involved, because the details are complex, and probably figured out by some smart guy maybe like an engineer who deals with these types of things on a daily basis, who Bill B asked to help with the problem, but leave him in the dark about the details. Or maybe a smart ball boy or equipment manager figured out that if you start with a 10 lb pressure at 70 deg., pump it up to 12.5psig, then take it out in the cold the ball feels more "comfy" that way, and the pressure drops. I can understand Belichick not wanting to come out with this "trade secret".
But the secret is out, as can be seen in this physics researchers, academics and student’s question and answer site that’s been kicking around the issue.
http://physics.stackexchange.com/questions/161035/gay-lussacs-law-the-patriots

Either way, no rules were broken, advantage gained, headed to the big game, and Roger Godell once again has egg on his face. Genius.
The sad part is I think they are gonna try to pin this on the ball boy? Evil genius, perhaps.

Where y of air = 1.4
T2 = T1 (P2/P1)^n, where n = (y-1)/y;

• ### Guest - Mark

If the balls were inflated with a pressure tank, the resulting air temp inside the ball would be less than room temperature. Do we know how the balls were inflated?

• ### Guest - Mark Malonson

"Guest - Mark": It's not necessarily true that the air temperature inside the football goes lower than room temperature if it is filled from a pressure tank. It would depend on the relative volume of the pressure tank and football. I suppose if the pressure tank were small enough, the adiabatic cooling effect on the source tank during the fill process would outweigh the adiabatic heating of the air in the football, and you could end up with lower than room temp air in the football. Without going through the calculations, I would think the source tank would have to be so small as to be useless in that case. But from what I've seen and heard, its more likely they use electric or manual pumps.

Here's the good/bad news for the Patriots: the latest "leaks" I'm hearing from the NFL is that the referees didn't actually write any numbers down, they just checked to see if the footballs "passed" or "failed". 11 of the 12 balls "failed" (and as we all should realize by now if they started at the low end of the spec when warm, and then were measure when cold, they ALL would have failed). Further, the 2 psi number may have only been the worst ball. I've heard new leaked info that most of the balls were more like 1 psi low. But again, no one wrote the numbers down.

So I'm guessing the NFL will eventually make a statement saying that they have no evidence that Patriots tampered with the footballs. But because of the sloppy way the whole fiasco was handled, they also cannot say for sure that the footballs WEREN'T tampered with (if they had just confiscated and set aside the low pressure balls, even just one or two of the worst ones, for analyzing later, instead of pumping the cold balls back up to spec and sending them back onto the field, all these questions could have been avoided!) So the Patriots won't get penalized. But all football fans outside of New England will continue to believe the Patriots cheated.

• ### Guest - JERALD ZANDMAN

1. DONT YOU HAVE TO USE THE TEMPERATURE OF THE AIR INSIDE THE BALL, WHICH, DUE TO THE HEAT OF COMPRESSION WHEN THE BALL WAS INFLATED, MAKES THE INTERNAL TEMPREATURE HIGHER THAN THE 70 DEG INDOOR TEMPERATURE? THUS WHEN THE AIR INSIDE THE BALL FALLS TO THE 51 DEG OUTSIDE TEMPERATURE, GTHE PRESSURE DROP IS GREATER THAN YOUR CALCULATION.
2. SINCE KICKERS LIKE THE BALL INFLATED TO THE MAXIMUM ALLOWABLE PRESSURE, PERHAPS ONE OF THE PATRIOTS FOOTBALLS STARTED OUT INFLATED TO THE UPPER END OF THE ALLOWABLE RANGE.

• ### Guest - Richard

All the science is interesting but there is but one truly fair solution, which for everyone to play with the same balls not selected by the quarterback, but the game judge. Hell go to Wall Mart buy 24 balls and game on!

• ### Guest - Mark Malonson

Here is the latest information from NFL.com on Deflategate. This was reported on Feb 1, in http://www.nfl.com/news/story/0ap3000000466783/article/more-details-on-the-investigation-of-patriots-deflated-footballs:

>> Eleven of the 12 footballs used in the first half were judged by the officials to be under the minimum of 12.5 PSI, but just one was two pounds under. Many of them were just a few ticks under the minimum.

If they are using the type of pressure gauge that I think they are, then a "tick" is 0.2 psi. So many of the balls were about a half a psi below the minimum. Less than the loss that Brian Burke's calculation predicted.

This is sounding more and more like a ...ahem.... normal distribution. Like what you would expect from the natural variation in conditions. The NFL still hasn't confirmed whether the numbers were actually written down at halftime. But if they were, I'd like to go on record and make a prediction of what the measurements were. This is based on educated guesses of the average and variation in ball temperature for the two measurements, the likely initial measurements, and the likely precision of the gauge. No deflation required:

Ball 1 10.8
Ball 2 11.0
Ball 3 11.2
Ball 4 11.3
Ball 5 11.4
Ball 6 11.5
Ball 7 11.6
Ball 8 11.7
Ball 9 11.9
Ball 10 12.0
Ball 11 12.1
Ball 12 12.4

And, by the way, If this story was first reported not as "11 of 12 balls were 2 psi low", but instead, "one ball was about 1 psi lower than expected, and the rest were in the normal expected range, given the temperature conditions" would there even be a story? Would Brian have done his analysis? Would anyone be questioning the Patriots' fumble rates?

Unfortunately, no matter what the Wells reports ultimately concludes, you can't get the toothpaste back in the tube.