# Relationship between Pressure and volume with Air

So the below question might sound like a homework question but in reality it is an oversimplified explanation of what I'm trying to measure.

Let's say we have a perfect machine that pushes out 600cfm into a box. On the other end of the box is another machine that only consumes 300cfm from the box.

What is the pressure in the box and how do I calculate that? I've looked into Boyle's Law and something just doesn't seem right about the results I'm getting so I'm trying to make sure that this is the right law I should be looking into.

If it's relevant, assume a volume of 360in3 and temperature of 310 K.

• You're talking about cfm, which is a rate, so the pressure will be time-dependent. What work have you done to try to solve the problem? Have you looked at the ideal gas law? Aug 11, 2016 at 18:42
• I've tried to use Boyles law as well as the ideal gas law but nothing that would take into account the "hole in the box". Unless I'm missing something, the only formula that took into account another set of volume variables was Boyle's law which does not appear to give me a correct solution. Aug 11, 2016 at 18:50
• The hole in the box doesn't really matter if you assume reversible compression; you then just look at the net mass flow. Aug 11, 2016 at 19:07

You can use the ideal gas law at first for a rough estimate. Assuming we start at atmospheric pressure, and the flow rate is in standard cubic feet per minute: $$P = 14.7 psia$$ $$V = 360 in^3 = 0.2083 ft^3$$ $$R = 10.731 \frac{ft^3*psi}{R*lbmol}$$ $$T = 310K = 558^\circ R$$ Starting point in the box at time = 0: $$n=\frac{PV}{RT}=0.0005115 mol$$ Flow rate for 300 cfm converted to moles per second: $$\frac{14.7 psi * (\frac{300 cfm}{60s/min})}{10.731}*558^\circ R = 0.01227 mol/s$$ After one second of pushing 0.01227 moles of air into a box that already contains 0.0005 moles of air: $$n = 0.0128 mol$$ $$P = \frac{0.0128 mol * 10.731 * 558R}{0.208 ft^3} = 367.5 psi$$

In this situation, the box fills up so fast that the outlet flow would increase drastically to a certain point until you either deadhead the blower or get a box failure. Or as @Ben Welborn suggested, you may create nuclear fusion resulting in a wormhole between the engineering SE and physics SE

• Can you explain to me how you got R? Aug 12, 2016 at 13:57
• It's the universal gas constant. Quoting from wikipedia, it is "the constant of proportionality that happens to relate the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered." Aug 12, 2016 at 14:01

600 - 300 = 300 cfm.

You are pushing 300 cubic feet per minute into a 0.20833 cubic foot (or 5.89925 liter) box.

What's the resulting pressure of 300 cubic feet per minute into a 0.20833 cubic foot box? Assuming that the pressure is an unstoppable force, going into an indestructible box, the pressure will increase variably and periodically until you have a solid.

But, at first, generally, gasses will compress into liquids, once the liquid state is reached it becomes "incompressible"... so the pressure will then escalate quickly. For simplicity, lets just use nitrogen as a guinnea pig.

Supposing that you can perfectly remove heat from the system to maintain 310 kelvin (98.33 °F?), as nitrogen becomes liquid the pressure will be maintained about 45,000 psi (give or take a few hundred psi, until the box is full). For understanding, consider how the boiling point of water is maintained until it is all gone- the same thing is true of the boiling point of nitrogen- the pressure is also maintained (until the box is full, since you are reversing the boiling process).

A mole of gas is 22.4L, so you are compressing about 13.4 moles per minute into 5.9 liters. The density of liquid nitrogen is 0.807 kg/L... and one mole (of N2) weighs about 28 grams. So, 5.9 L x 0.807 kg tells us the net weight of the full box of liquid nitrogen... about 4.76 kgs. 4.76kg / 28 g tells us how many moles of liquid N2 will fit in the box.

So, 170 moles of liquid N2 will fit in the box... how long do we have before it's full? At 13.4 moles per minute, you have about 12 minutes and 41.2 seconds before the pressure really escalates. Then, liquid nitrogen will compress into solid nitrogen. Making all the same assumptions, at 98 °F, the pressure will rise sharply to about 75,000 psi, and there it will stay pretty steady until the box is full of solid N2.

Solid nitrogen weighs 1.027 kg per liter. Simply stated, the proportional difference in density will tell us the proportional volume left in the vessel. So 100 x (1.027 - 0.807) / 1.027 = the percentage of room left in the vessel (before it becomes full of solid N2). 21.42% of 5.9 L is 1.264 liters. 1.264 L x 1.027 kg per liter tells us how many kilograms of solid nitrogen we can add before the box is full of solid nitrogen. 1.3 kg of solid nitrogen / 28 g per mol is 46.43 moles (neglecting rounding errors). At 13.4 moles per minute being added to the box, you will have about 3 minutes and 28 seconds before essentially the box is full of solid N2.

At this point it is even more incompressible and the pressure will be applied toward the nuclear fusion of nitrogen. Now I don't know the pressure at which nitrogen will fuse, but for what it's worth you are talking about binding energies... and there's talk of it on the physics SE. In a practical way, the binding energy is measurably the difference in mass between the atomic number and the representative isotope. Remember, mass is energy. For example, the energy (by weight) of N14 is actually 14.0030740052 g/mol.

Anyway, the binding energy is something like 7.7 MeV or about 1.23368e-12 joules per nitrogen atom and you have about 206.5 moles of solid N2; that's about 2.5x10^26 atoms. And that's about 3x10^14 joules - holding a box of nitrogen together. For silicon, the binding energy is 8.45 MeV (1.3534777464e-12 joules/atom). Now, since 2 nitrogen atoms will fuse (in my mind) to become 1 silicon atom, that's half the atoms. So, 14.4 MeV - 8.45 MeV is how much energy is absorbed turning one turning 2 atoms of N into 1 atom of Si; and you now have 1.25x10^26 atoms- that's 9.53295e-13 joules x 1.25x10^26. So, you need 1.2x10^14 joules to turn 5.9L of solid N2 into silicon.

Also, for convenience let's totally bend physics to our whim and say one joule equals 0.003245 psi. So, 1.2x10^14 / 0.003245 equals the total psi before nitrogen "alchemizes" into silicon.

At 3.67x10^16 psi, you will have silicon. How long will this take? Well I'm not sure, but we can be sure that the pressure going from a liquid to solid went up by about 30,000 psi in probably well under a minute. So, assuming that it's pretty close to the "incompression rate/limit" let's just pick a nice easy figure like 100,000 psi per minute. Now, 3.67x10^16 / 1x10^5 = 3.67x10^11 minutes... or about 700,000 years (give or take maybe 300,000 years). But this is really messy because some of it will become silicon but some of it won't.

Assuming nitrogen is only becoming silicon, the density of silicon is 2.3296 kg/L. Now, you have to fill about half of the box with silicon and don't forget that after another say 700,000 years nitrogen is once again being reduced to about half it's volume. Eventually all of the silicon will fuse to become nickel. I think it's safe to say the amount of energy or psi for this has never actually been measured... so I will just leave the rest of these calculations alone. The density of nickel is 8.912 kg/L.

Now, I just want to make mention of this because it seems interesting to note that nickel should fuse to become barium, but as the density of barium is only 3.51 kg/L (which is only slightly greater than than 1/2 of 8.912)- at this point you will convert nickel to barium without much leftover volume in the box! You may proceed relatively quickly (remember the void turns to silicon then nickel and then barium) at this step and move right on to Copernicium (aka ununbium). The density of Copernicum is unknown, but it should be about 23.7 kg/L.

Beyond that, I just don't know. Maybe it becomes a black hole or something. But to really answer the question, the limit of the pressure might be 100,000 psi per minute (when compressing a solid). Otherwise it will be the vapor pressure of the liquid or the melting pressure of the solid (perhaps for a few minutes, I guess that depends on the substance).