Gases and Heat

We divers all learn our gas laws and, if we are lucky, we remember them. However, although we might remember the universal gas law (PV/T=constant), we always use Boyle's law that PV = constant at constant temperature.
Then we go for a fill and feel the tank getting warmer and we know that the warmth means that the pressure is higher for the quantity of gas so when we get it home and it has cooled down it will have less pressure than we hoped.
Typical dive shop. Short filled again. Why can't they use cool air when they know I'm in a hurry?

This short page is to discuss the effects of compressing a gas on its temperature so we have some feel for the numbers involved and maybe excuse the dive shop a bit.

The whole problem with the universal gas law is that it has three variables in it. If we know the first state exactly, ie: we know pressure, volume and temperature, and we know two of the final values we can calculate the third. Just so you can check you're with it consider that we have two tanks: the first 200bar in 12L at 20°C and a second tank with 100bar at 10L also at 20°C and we want to even them up so we can both do a second dive together so we connect them together with a whip and then we will wait till it all cools back to 20°C:
200*12/(273+20) + 100*10/(273+20) = P*22/(273+20) so P = 154.5bar.

But what if we only know the volume and we don't know the temperature?

We really can't solve problems that can flop about in lots of variables but if we are prepared to accept certain constraints we can work things out and then allow for those constraints. So...

Adiabatic gas laws

Adiabatic is a big word that just says that rather than keeping the temperature constant we just will not consider heat entering or leaving the gas. This is quite realistic if we are considering a sudden change and when we do the sums we see why we don't want to do sudden changes
The law is PV^γ = const where "γ" (the Greek letter gamma) represents a constant that is about 1.4 for air. Actually for monatomic gases (like helium) it gets up to 1.66 but for big multi-atom (hydrocarbons) it is almost down to 1.

OK let's take 10L of air at 100bar and instantly halve its volume to 5L. What happens?

so	P1V1γ = P2V2γ
as in	P2/P1 = V1γ/ V2γ
otherwize	P2/P1 = (V1/ V2)γ
well	V1/ V2 = 10/5 = 2
hence	(V1/ V2)γ = 21.4 = 2.64 = P2/P1

So the pressure didn't go from 100 to 200bar but to 264bar and the ideal gas law tells us that the temperature did PV/T=const
100*10/(273+20)=264*5/(273+T) so T=115°C ie: above the boiling point of water

And that was just halve the volume. Nothing like compressing gas from 1 to 200 bar.

You want to do 200 times compression? Well the gas laws are falling apart by then but to give you the flavour....
OK we need a cylinder with 200x12 ie: 2400L (2.4 cubic meters of air) in it. We want a piston that will squeeze this down to a little 12L tank and we need something to push the piston. I'll supply the gas. You get the hardware. OK?

well	V1/ V2 = 2400/12 = 200
hence	P2/P1 = 2001.4 = 1665

Yes you read that right! 1665bar. The ideal gas law is a joke at those pressures but for the record that works out at 2175°C. Your aluminium tank melted at about 650°C and your steel one at 1515°C.
Incidentally that is about 7M joules, about 2Kw - hours to buy from the electricity supply people and the equivalent of leaving a 2 bar fire on for an hour so no wonder the compressor room of the dive shop is warm.

So why don't tanks melt? Because the compressor delivers hot compressed air to its cooling coils and then to the bank. Even if it supplies it directly to your tank it has cooled a lot before it reaches the valves in the filling panel and then as it passes from the high pressure panel into you still relatively empty tank it expands as the pressure drops so the dive centre is filling your tank with very cold gas which then compresses warming up a bit.