rather than thinking about this as a pressure conversion, would it make more sense to think of it as an energy conversion?
converting that stored potential energy to kinetic energy.
in essence isnt a 1st stage just a valve?....and isnt lowering the IP in essentially just slowing the speed of the gas leaving the valve?
well if we are slowing the speed of the gas leaving the 1st stage, that gas is going to have a lower kinetic energy..
as we know the higher the kinetic energy, the higher the temperature imparted to that gas......well that energy has to come from somewhere, mainly the stored potential energy of the rest of the gas in the cylinder, and the metal of the first stage, and subsequently the surrounding water...
.....which in theory is why a higher IP results in a higher temperature drop over the 1st stage.
Ide imagine it should be much like taking 2 full cylinders and opening the valve only a crack on one of them....and fully open on the other.
The wide open cylinder is going to output that air at a higher pressure....but will subsequently get much much colder than the partially open cylinder
I'm pretty sure the gas laws, like charles' and gay lussac's law, only applies to closed systems.....and I think that's why they seems to fall apart when applied to an open system like a scuba tank
that was my understanding of it, but admittedly i havent really sat down to look at the math.