What is DeflateGate?

I don't follow football, but anyone living in eastern Massachusetts as I do can hardly fail to know something about it. Apparently the Patriot football team was accused of deliberately deflating footballs in a game, supposedly to give the players who handle the ball, principally the quarterback, a better grip. The Patriots denied deliberately deflating balls and suggested that temperature changes were responsible for the undeniable fact that at least one ball was in fact not properly inflated. Can high-school physics shed any light on the affair?

According to Wikipedia, the rules specify that balls must be inflated to between 12.5 and 13.5 psi. The temperature at which this pressure should be maintained is not specified (which should tell you that football rulemakers don't understand high-school physics). The Patriots claimed that a temperature decrease was sufficient to reduce the pressure to below the minimum, without anyone needing to remove air. Let's see if this defense is credible.

Assumption 1: Air is an ideal gas

This means that the ideal gas law specifies the relationship between temperature and pressure. This law states:

PV = nRT, where

P is pressure,
V is volume,
n is the amount of gas (in moles, but we don't need to know that),
R is the ideal gas constant, that is, a constant of proportionality, and
T is temperature.

I will examine whether this assumption is warranted, but for the moment it is a very convenient idealization. And we can simplify it further: Since the contention of the Patriots is that nobody let any air out of the footballs, the factor n in the equation, the amount of air, is constant. I'll roll it into the other constant factor, the gas constant R, to get:

PV = (constant)*T

Assumption 2: The volume of the footballs is constant

That is, the volume didn't change so much as to make the ball flabby, just to change its firmness. This assumption too will be examined, but assuming it to be true for the moment, we can roll that constant into the others to get

P = (constant)*T

And that says that the pressure and temperature are proportional. If one increases or decreases by x%, the other will increase or decrease by x% also.

Just the facts

The contention is that the officials inflated all the footballs to 13psi, midway between the 12.5 to 13.5 specification, before the game. During the game, one of the balls was found to be deflated by 2psi. I don't know whether that means it was at 10.5 psi (2psi below the minimum specification) or at 11psi (2psi below the pre-game pressure). 10.5 would be 81% of the pre-game pressure, a drop of 19%, and 11psi would be 85% of the pre-game pressure, a drop of 15%. Is a temperature change of 15 or 19 percent believable?

Beware of the temperature's units!

Temperature in the gas law has to be absolute temperature. That is, it must be relative to absolute zero, not to the freezing point of water as in Celsius, or to whatever the heck Fahrenheit degrees are relative to. This makes intuitive sense: If it were otherwise, when the temperature reached zero degrees the law would predict the pressure would also be zero. We know this cannot be the case. Footballs at zero Celsius or zero Fahrenheit do not lose all pressure.

Since this is an American scandal in an American sport we'll use American degrees. To convert Fahrenheit degrees relative to absolute zero we add 459; that is, absolute zero is -459 degrees Fahrenheit.

Not credible!

What was the temperature at which the balls were initially inflated? It was a winter game in New England, but to give maximum leeway to the Patriots, suppose it was in a warm room in the stadium, say 80F, which is 539 absolute. A pressure drop of 15% would require a temperature drop of 15%, to about 458 absolute or -1F; a drop of 19% shows a temperature of -22F. The outside temperature during the game was about 20F.

I think someone cheated, by letting air out of at least the one football.

The ideal gas law is derived assuming that the gas consists of point particles. Heat makes the points move, which bounce off the walls of the container, producing the pressure on the walls. The key is that the only effect of adding or subtracting heat is to make the points move faster or slower. But air isn't composed of point particles. Being mostly N2 and O2, the molecules are like sticks. Additional heat can do more than just make them move faster; it can make them whirl end over end, or rotate along the long axis, or (perhaps) the individual atoms in the molecules might move independently of one another. None of those motions will make the molecules hit the walls any harder, and so they will not contribute to an increase in pressure. Thus temperature and pressure will not be proportional; reducing the temperature by some percentage will reduce the pressure by less than that percentage. This further strengthens our conclusion: No credible temperature change will produce the required pressure change.

Is it reasonable to assume the volume will remain constant? Footballs are elastic, after all; decreasing the pressure will reduce the size to some degree. This still further strengthens the conclusion: If the temperature decreases and the volume does also, then the pressure will have to decrease less than it would if the volume remained constant. That underinflated football lost some air.

Any other mitigating factors?

There are always other possibilities. How accurate were the pressure gauges? Were the same gauges used to inflate the balls and to check their inflation? Did the suspect football have a leak? Could the temperature of the air out of the pump be much greater than the ambient temperature? How many times was the underinflated ball checked? How much air is lost each time the pressure is checked? Were the balls really inflated properly before the game?

Every conclusion of science is tentative. Here's the tentative conclusion: