Footballs and Boyle’s : On Cavitation and Deflate-gate

OK!  Full disclosure: I am a New England Patriots fan through and through, having grown up and lived much of my life in Massachusetts.  Leading up to Superbowl  XLIX, there  was incessant media coverage discussing potential “cheating” in the AFC title game through the use of deflated footballs.

Cavitation as Results From a Propeller*

Whether or not someone intentionally deflated game balls cannot be disproven by science, but … science can at least offer a plausible alternative, using the same equation that explains how cavitation occurs during peak rarefactional pressure (and not the peak compressional phase of the pressure – as is often mistakenly assumed).  The equation derives from the ideal gas law.

The term cavitation comes from the Latin term “cavus” which mean a cavity, pocket, or hole.  Remember that cavitation is a rapid phase transition to a gaseous state, creating a cavity within the medium.  When discussing why cavitation occurs during rarefaction, I usually refer to Boyle’s law.  In reality, Boyle’s law is  just a special case of the ideal gas law.

PV = nRT

This equation states that there is a relationship between pressure (P), Volume (V), and Temperature (T).  The variable n stands for the “amount of the gas, specified in moles”, and R is known as the gas constant.  Cavitation typically occurs in an extremely short time duration, implying that the number of moles (n) and the temperature (T) remain constant.  This means that the ideal gas law can simplify down to an equation that states pressure times volume equals a constant, also known as Boyle’s Law.

How does Boyle’s law explain cavitation occurring during rarefaction and not compression?  During compression, the exterior pressure on the gas bubble increases, making the bubble shrink.  In other words, as the external pressure goes up, the internal volume goes down.  This is similar to what happens when a balloon is brought deep under water; the balloon collapses but never pops.  Conversely, during rarefaction, the pressure exterior to the gas pocket decreases allowing the internal pressure to expand the bubble.  Again, since the pressure times the volume is a constant, as the pressure decreases, the volume increases.  If the bubble (volume) expands enough, the membrane is stretched too thin and the bubble implodes.  This is similar to taking a balloon up in an airplane.  As the plane gains altitude, the cabin air pressure decreases and the balloon gets larger. Eventually, the balloon will expand to the point where its latex/rubber film cannot maintain the stretch, and the balloon bursts inwardly.

Returning to football and “Deflate-gate” … How does this same equation potentially exonerate the Patriots from wrongdoing?  In this case, we cannot simplify the equation in quite the same way as in our prior discussion since the temperature cannot be assumed to be constant.  In fact, the whole argument hinges on the temperature NOT remaining constant.  In essence, the equation tells us that for a fixed volume (i.e. no conspiracy to remove air from the footballs) as the temperature changes the pressure changes.  Since Tom Brady of the Patriots seemingly likes the football less inflated than some other quarterbacks, the Patriots would naturally inflate the ball to the lowest allowed limit.  If the footballs were inflated in a warm room, when they are taken outdoors on a winter afternoon in Foxboro, Massachusetts, the temperature decrease could result in a pressure decrease below the allowed limit.

But why would this not also be true for the Colts?  The answer is simple, although the stadium temperature was clearly the same for the Colts, the starting condition could easily be different.  Let’s say that the Colts generally inflate their footballs to a higher allowed pressure (perhaps because the ball flies farther when kicked).  Let’s also assume that Indianapolis inflates the footballs in a room that is colder than the room in which the Patriots inflate their footballs.  Now when that ball is taken outside to colder temperature, the pressure drops less than for the Patriots, still remaining within the allowed limits.  Hence, science can provide a reasonable explanation for what otherwise requires assuming that everyone is trying to cheat in a way that makes little to no sense.  Besides, the Patriots scored only 17 points in the first half of the game when the footballs had lower pressure, and scored 28 points after the footballs were re-inflated to NFL specifications.   It sounds like the argument for advantage just fell into a cavus.

Frank Miele, MSEE  President, Pegasus Lectures, Inc. , graduated cum laude from Dartmouth College with a triple major in physics, mathematics, and engineering. While at Dartmouth, he was a Proctor Scholar and received citations for academic excellence in comparative literature, atomic physics and quantum mechanics, and real analysis.

* [Cavitation photo] Reprinted with permission from Ultrasound Physics and Instrumentation, 5e. Photo courtesy of Professor S.A. Kinnas, Ocean Engineering Group, University of Texas at Austin.