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Deflate-gate and the Ideal Gas Law


Howdy MHR!

Long time lurker/commenter/used-to-be-poster here (hopefully recognized by at least a few of you)

Many theories have been thrown around regarding deflate-gate, temperature drops, pressure-temperature relationships and I thought I'd try to delve into the math and physics involved in a somewhat simpified manner. While physics seems to scare a large portion of the population, the necessary science here is no more than high school level stuff.

The Ideal Gas Law

Most everyone who has mentioned the science related to dropping pressure in a football does so by stating a relationship between pressure and temperature. This relationship is known as the Ideal Gas Law. It's not new, it's not controversial, and it's very accurate at relatively low pressures and high temperatures (say less than 1000 psi and above -100 to -200 deg. C) At atmospheric conditions the ideal gas law is very accurate if all the parameters are known and/or controlled. In equation form, the ideal gas law states:

PV=nRT or PV=mRT

  1. P = absolute pressure (psia rather than psi)
  2. V = Volume
  3. R = Gas constant
  4. T = absolute temperature (Kelvin or Rankine rather than C or F)

The difference is the two equations given is that n represents the number (moles) of gas molecules in the system while m represents the mass of the gas. Two different numbers but the relationships don't change. With some slight algebraic manipulation we can re-organize the equation to

P/T=mR/V

Since it is a closed system (no mass is moving across the boundary) we can assume m is constant for all temperatures and pressures. Similarly the gas constant, as implied by it's name, remains constant. Here, we will assume that the volume change in the ball is negligible across the pressure range of interest. This all comes together such that the right side of our equation (mR/V) is constant for each pressure and volume. Since mR/V is the same at state 1 (when the balls were tested prior to the game) and state 2 (when they failed at halftime) we can set P/T at state 1 equal to P/T at state 2. This is where the relationship between pressure and volume that everyone likes to reference comes from.

P1/T1=P2/T2

Re-organize this to solve for T2 results in

T2=P2*T1/P1

Calculations

Assuming the balls were inflated and tested in a room at about 70 deg. F, here are the values used for calculations

- 70 deg F converted to absolute temperature is 529.67 deg Rankine = T1
- 12.5 psi gauge pressure is converted to an absolute pressure of 27.2 psia = P1
- 10.5 psi gauge pressure is converted to an absolute pressure of 25.2 psia = P2

That's all that's needed. Using the above values and plugging them into the equation found in the previous section and we find:

T2 = (25.2psia)*(529.67R)/(27.2psia)
T2 = 490.72 R = 31.05 F

490.72 deg Rankine converts back to 31.05 deg. F. This is the temperature at which you would expect a 2 psi drop from an initial temperature of 70 F. This gives us an absolute maximum temperature which must also be understood in light of our assumptions. Remember where we assumed V to be constant? It likely is pretty close. You wouldn't notice much difference in the size of a ball inflated to 10.5 psi compared to one at 12.5 psi. What is important to note, is that if we use V as a variable the equation from above becomes PV/T=mR and when combining the equations for state 1 and 2 we get T2=P2*T1/P2*(V2/V1). Notice it only changes by multiplying by a new factor (V2/V1). Since the ball is not an ideal rigid body, the volume will decrease by some amount when pressure is decreased. This means that V2

Anyway, hope that helps some of the numbers being thrown around. I'm not here to give a hard number (one of the first things we learn as engineers is that there is no such thing as an exact measurement) but to shed some light on the relationship between temperature and pressure, and hopefully put to rest some questions on where the numbers being thrown around are coming from. Hopefully this makes some sense and you can see that the science in this case is very simple.

This is a Fan-Created Comment on MileHighReport.com. The opinion here is not necessarily shared by the editorial staff of MHR.