The Science Behind Deflate-gate

A few inches of snow in Maryland allowed me to have a little extra time with my family yesterday.  After my two year old son Crosby’s afternoon nap, my wife and I thought he might like to go outside to play in the snow.  After getting him all bundled up, I took him outside planning on pulling him around the yard in our sled.

The sledding lasted all of about 30 seconds.  We spent a couple minutes making and throwing snowballs, but then he got down to business.  He made it very clear he wanted me to get the footballs out and kick them.  For some reason he thinks it’s pretty awesome watching his 42 year old dad trying to place kick and punt.  Or maybe I’m just kidding myself and he thinks it’s hilarious watching me trying to relive my football glory days!

Anyway, I couldn’t help but notice that the footballs all seemed pretty deflated when I was trying to kick them.  The balls were really hard to kick, but I could pick them up pretty easily.  Hmmm…  Crosby kept dropping balls he normally would hold on to, but I think that was because of his mittens.  I’ll need to buy him some Stickum for future play sessions.

It got me thinking about Tom Brady, Bill Belichick, the Patriots and all those deflated footballs.  I decided to do some quick calculations to see if their explanation that changing temperatures might have caused the balls to have pressures below the range required by NFL rules.

The whole controversy relates to the behavior of gases.  In order to perform calculations like the ones needed to be able to tell if the Pats claims of innocence hold any water you will need to know about the volume, pressure, temperature and number of gas particles inside of a legally inflated ball.

I made some assumptions to make the calculations easier.  The first is that the volume of the ball will not change significantly in a ball inflated to 10.5 pounds per square inch (psi) or 13.5 psi.  The second assumption in the calculations is that the number of gas particles in the balls did not change from when they were checked in with the officials before the game and when they were checked again at halftime which means no one let any air out of the balls.  This leaves just temperature as a factor which could cause the pressure of inflation to change.

Let’s imagine that the footballs were inflated and tested indoors where the temperature is 70 degrees and measured 12.5 pounds per square inch or psi.  The pressure of the balls would drop as the air in the balls cooled when the balls were taken outside where the game-time temperature was 50 degrees (51 actually, but I rounded off).

Things get a bit science-y now.  In chemistry, we use a law called Gay-Lussac to calculate the change in volume caused by temperature changes.  Yes, I know, I am being serious…

Gay-Lussac states:  P1/T1 = P2/T2

We also need to convert temperatures from Fahrenheit to the Kelvin scale.  You can do this with the calculator feature in Google, just type “70F to Kelvin” and “50F to Kelvin” as your search and you will find the following conversions.

70F = 294K  and 50F = 283K

So let’s figure out what the pressure of the balls would have been at halftime (P2) given the drop in temperature from 70F (T1 = 294K) to 50F (T2 = 283K)

A little math gives us the following:

P2 = (P1*T2)/T1

P2 = (12.5psi*283K)/294K

P2 = 12psi

So this alone cannot account for the drop in pressure from 12.5psi all the way down to 11.5 psi.

If the balls had been inflated with an internal temperature of 90F (305K) at half-time had dropped to 50F (283K), that could explain the balls pressure dropping to 11.5-11.6 psi.  This is within the range that Mike Florio (profootballtalk.com) stated the balls had after they were re-checked during the AFC title game.

P2 = (P1*T2)/T1

P2 = (12.5psi*283K)/305K

P2 = 11.6 psi

So I think that it’s most likely the case that no one on the Patriots intentionally let air out of those footballs.  Do I think the Pats might have done everything they could to manipulate the system to create a competitive advantage for themselves?  If you thought inflating footballs at 90F would help you make it to the Super Bowl, would you do it?

In the interest of full disclosure, I was born in Washington state.  Curt Warner (not Kurt Warner, although I really liked him too) was one of my earliest football heroes.  I did root for the Seahawks when I was growing up and I’ll be rooting for them on Sunday.   But I… am a Vikings fan… a long-suffering Vikings fan, and I’ll leave it at that.