Scientific Explanation of Why Patriots are Innocent

The New England Patriots have been accused of cheating for deflating footballs before the start of the AFC Championship game against the Indianapolis Colts. Let's take a scientific approach to determine whether the Patriots could have cheated or if elements outside of their control have led to a false positive for cheating.  Let's also accept that I went overboard on this analysis, but, hey, it's not often I get to combine Chemistry and sports.

Qualification: I graduated with a Chemical & Biomolecular Engineering degree from Rice University and work as a senior research engineer for a large oil and gas company, so there is a basis for these arguments.  I am an avid Houston sports fan and despise Boston for all the success its sports teams have achieved this millennium.  That being said I cannot stand idly by while someone is falsely accused of cheating.  It is virtually impossible for the Patriots to have deflated the footballs intentionally, given the information available to date.

By rule, NFL footballs must be inflated to between 12.5-13.5 psig by the officials before game time.  12 footballs are given to each team and another set kept for the kicking game.   The Patriots have stated they prefer their footballs at 12.5 psig, so I will use that as a starting point. Reports are mixed, but let's assume the footballs were deflated by 2 psi, to 10.5 psig, when measured by the officials after the game.

Before we get started, I just have to point out that in the scientific calculations, temperature and pressure must be converted to absolute values, as in amounts above zero.  Inflated pressure is typically reported in gauge pressure, which is psi above atmospheric pressure of 14.7 psi. Temperature must be converted from Fahrenheit (F) to Kelvin (K), which is the degrees above absolute zero.

12.5 psig = 27.2 psia
10.5 psig = 25.2 psia
70 F = 293.15 K
50 F = 283.15 K

1. Magnitude of Change - 7%
The press is overreacting to the 2 psi change in pressure.  Viewed on an absolute scale, this is only a 7% reduction in pressure, so it is not going to be hard to build a credible scientific argument to explain why a variable would change by 7%.

((27.2-25.2)/27.2)*100 = 7%

2. Measurement Error
Measurement error in controlled lab environments is typically +/- 3%, due to a combination of inherent inaccuracy in the measurement device and human error.  I gather the NFL was not using the high quality measurement devices used in laboratories, was not standardizing the conditions between measurements, and was just sticking a needle in the football without accounting for air escaping. Also, I am assuming that different people took the pressure measurements before and after the game, which, in and of itself, adds an incredible amount of uncertainty.  Do we even know if the same gauge was used?  Let's just say the quality of measurement being between +/- 7% would not hold up in court.  

3. Temperature Effects
An equation of state can be used to relate  how changes in one state variable affect another.  For this simple problem, let's use the simplest equation of state: The Ideal Gas Law:

PV=nRT
n is the number of molecules of air (otherwise known as moles)
R is a constant

The temperature dropped by 3.4% (293.15 F locker room -> 283.15 F outdoors), which explains a 0.9 psi reduction in pressure to 11.6 psia.

I have read in several articles that "it wasn't even that cold outside."  This is faulty reporting without understanding the facts.  The 3.4% reduction in temperature accounts for HALF of the reduction in pressure.  That's huge.

4. Air Escaping
I have not read a single article discussing the impact of air escaping the  football.  In fact this article (http://www.wcsh6.com/story/weather/2015/01/20/inflate-gate-weather-roll/22065861/)  which does a good job applying the Ideal Gas Law, completely omits the loss of air.  Remember, in PV=nRT, "n" is the number of molecules.  If air escapes the football it will have a direct impact on the pressure.

Footballs are not perfectly sealed.  They will lose air over time.  When world-class athletes  squeeze the football on every single play, they apply an external pressure to the ball, which forces some of the air to escape.  Thus, there will be less molecules in the ball by the end of the game, reducing the internal air pressure of the football.

Per our equation above, only 4% of the molecules would have to escape in order to account for the rest of the 7% reduction in pressure.  I cannot estimate how much air would escape, but it is not hard to believe 4% of the initial air inside the footballs would escape over a 3 hour game.

5. Football Preparation
Belichick stated that, during their trial run, the air pressure increased by 1 psi after "roughing up the footballs."  This is legal, and every team does it.  Equipment managers rough up the footballs to give it a better grip, right before the footballs are handed to the officials.  Bill Nye the Science Guy went onto Good Morning America (https://www.youtube.com/watch?v=JZ-onepOA4g) and stated that Belichick's claim is unfounded.  I disagree.

Both Belichick and Brady have stated the equipment managers rough up the footballs right before the officials inflate the footballs.  Roughing up the footballs involves rubbing the footballs aggressively for several minutes I would imagine.  This creates friction on the football, which causes the football to heat up.  Do this experiment: Rub your hands together.  What happens? They get hot really quickly.  Imagine doing that for several minutes how hot the football must get.  In fact, we have evidence to estimate this:

Belichick stated that, during their trial run, the air pressure increased by 1 psi after roughing up the footballs.  That correlates to a temperature rise of 20 F.  (Note: this assumes the air pressure is 10 psig when roughing up the footballs.)  Why is that so important?  Because the balls are inflated IMMEDIATELY AFTER roughing up the balls.  If the balls are inflated to 12.5 psig when the temperature internal to the football is 90 F (20 F above ambient), then the initial pressure measurement is artificially high.  That really means there is a 40F drop in temperature between the initial point of pressurization and the temperature outside.  A 40 F drop in temperature would account for 1.9 psi drop in pressure, and that is without taking into account measurement error or air escaping the football.

5. Equilibrium
Bill Belichick covered this beautifully.  He states it takes 1-2 hours for each football to reach its original pressure after roughing up the football, thus reaching equilibrium.  Equilibrium is the point at which the process has reached steady state and is no longer changing.  All of these measurements are worthless if they are not taken at equilibrium at the exact same temperatures and pressures.

6. Why were the Colts footballs still within the acceptable air pressure limits?
I have not heard anything about the Colts footballs, but I assume they were tested and found to be within the 12.5-13.5 psig limits.  This is all speculation, but maybe the Colts inflated their footballs to the 13.5 psig limit (or more).  As I wrote above, the preparation activities and time between preparation and pressurizing the footballs is crucial.  Maybe there was some difference between the two teams' processes.  The Colts certainly ran fewer offensive plays, so maybe less air escaped from the footballs.

7. Conclusion
There is overwhelming evidence that the natural causes of the temperature dropping, air escaping, and friction on the football are the reasons the measured pressure on the football dropped by 2 psi, or 7%, over a 3 hour period in which the measurement conditions were never the same, introducing significant amount of uncertainty.  In the end, this is just a minuscule change in pressure that could be a result of a myriad factors.  It is highly unlikely that the Patriots did this intentionally, given that the officials during the game could not tell the difference between the "inflated" and "deflated" footballs.