Football Pressure Drop by Temperature
Footballs have three components: a rubber bladder, a leather skin, and the string to tie the skin together. The leather skin gets harder as temperature drops and can thus hide pressure changes to the user. Same effect is seen with car and bicycle tires. The rubber bladder is very pliable and thus expands quickly. Only about 3 psi of added pressure are needed to cause the bladder to fill totally the volume of the skin that holds it in. Certainly, in the range of 10 to 15 psi the volume is constant. Thus the gas laws called Charles’ Law and Boyle’s Law do not apply.
On a similar note, the amount of molecules inside the bladder is important. The use of the ball in such a violent sports does cause the number of molecules to decrease inside the bladder. This effect can be analyzed by measuring the pressures immediately before and an hour after a game. This needs to be done inside a room within the stadium to control for temperature. The effect probably is not more than a 2% loss of molecules. This translates by Avogadro’s Law into a 2% loss of pressure.
Overall, pressure is directly related to number of molecules and temperature and inversely related to volume. By the above analysis, we can assume that the number of molecules and the volume do not change much. Thus pressure varies with temperature for the air in a football. There are many gas laws to cover all these variables. The necessary law for footballs is called Gay-Lussac’s Law and is fundamental to the study of gases. It was derived in 1808 by Joseph Louis Gay-Lussac.
The Kelvin temperature needs to be used to study changes since it is directly tied to the motion of molecules. The Kelvin temperature of a 75-degree Fahrenheit room is 297 K. The Kelvin temperature of a 35-degree Fahrenheit field is 275 K. This is a normal room temperature. Halftime was at about 8:25 in Foxboro and the temperature was about 35 F.
The basic gas law will use Pf to mean the final pressure as read on the field. Pi means the pressure as read indoors. Tf mean the temperature as read on the field. Ti means the temperature as read inside. P means pressure, T temperature, f means final state, and i means initial state. The mathematical form of the gas law is as follows:
Pf = Pi x (Tf/Ti)
The ratio of the temperatures is 0.926. That is easy but the pressure to plug in is more difficult. The NFL stipulates an internal pressure of at minimum 12.5 psi (pounds per square inch). That however is the gauge pressure reading. There is normal atmospheric pressure both inside and outside a ball before inflation. This normal pressure is 14.7 psi. Inflation merely increases the pressure inside the ball beyond the normal. At the start of inflation the pressure difference is 0 between inside and outside. To inflate to the NFL minimum of 12.5 psi means the gauge might read 12.5 psi but the true pressure inside is this plus atmospheric pressure of 14.7. Thus at minimum there is 27.2 psi inside. Since there is 14.7 outside the ball due to the atmosphere, the pressure gauge reads the difference of the two so 12.5. Physics is based upon what is really there so the Pi to use is 27.2 psi.
Use the gas law equation now and you will find that the Pf on the field is 25.2 psi. Now stick a pressure gauge inside the ball on the field. The gauge reads the difference between ball pressure and atmospheric pressure. So 25.2 minus 14.7 which gives 10.5 psi of internal pressure. This is what the NFL is reporting as the difference. They stated a pressure drop of 2.0 psi. The calculations show a drop from 12.5 to 10.5 psi due to the temperature drop. Thus there is no scandal but rather basic physics at work. At the half, the footballs were given more air while cold. This will bring the pressure up to the 12.5 to 13.5 range. Since the temperature did not change during the second half, there was no pressure change inside the footballs. When measured directly after the game when cold, they would maintain that pressure.
This analysis is for the New England team. The Indianapolis team stated that they liked working with footballs at a pressure that gauged at 13.5 psi. This is an internal pressure of 28.2 psi. At the same temperature change, the pressure would decrease to 26.1 psi. The pressure as read by the gauge would be thus 11.4 psi. This is also under the pressure range as suggested by the NFL for its footballs. Both decreased due to the temperature in the stadium. Colts’ footballs would have decreased too with the amount dependant on the location and time at which measurements were taken. But when were they measured? Much also depends on the gauge being used. The table is the probable pressure variation of the footballs with time. First column is the time, second the Patriots’ pressure, third the Colts’ pressure, fifth the temperature, and sixth the environment.
5 12.5 13.5 75 indoors
6 12.5 13.5 75 indoors
7 11 12 45 on field
8 10.5 11.5 35 on field
9 12.5 11.5 35 air added at 8:30 halftime
10 12.5 11.5 35 on field
11 14.5 13.5 75 indoors
The pressure reading on the Colts’ footballs would have gauged at 11.5 at halftime. The referees apparently checked all 24 footballs in 10 minutes. They would have missed on a rough gauge that the Colts’ were at 11.5 and thought it to be 12 psi and thus good enough. Hopefully, more information will be available soon.