Question

Is there a formula that details the amount a gas will heat up and cool down when the pressure is changed?

Answer 1

PV=nRT

Answer 2

P = Pressure
V = Volume
n = number of moles of gas
R = gas constant
T = temperature in kelvin

Answer 3

In real world situations (such as pumping up a bicycle tyre) the temperature will not be constant, but will increase as the pressure increases and volume decreases. Is there a formula that details a variable pressure and temperature and neglects volume rather than assuming it to be constant?
Or is there a particular way to re-arrange the ideal gas law so that it fits this purpose?

Answer 4

Yes, if you have all variables constant but one and want to compare it to another condition like temp you can use \[\frac{P1}{T1}=\frac{P2}{T2}\] So if you know the pressure at one temp and would like to know it at another temp, fill and solve for P2.

Answer 5

Explaining it out has lead to a eureka moment. The pressure in the system will be constant because as the pressure increases the piston will be forced back and a pressure equilibrium will be maintained. So by leaving the pressure and molar content as constants and adjusting the other two variables I should get the answer I want.
Essentially: \[{{V _{1}} \over {T_{1}}} = {{V_{2}} \over {T_{2}}}\]
I have left this "eureka paragraph" below for all to see.

The eureka paragraph:
The reason for this is that I'm designing a Stirling engine and I would like to know <roughly> what its ideal output would be. I understand that when the air inside the engine is heated, it will expand, but as it expands, I know the temperature will drop. Due to this temperature drop over volume expansion it will cause the pressure to drop also, which I theorise will cause the temperature to drop further, in accordance to the ideal gas law, but the amount this gas will cool is what I want to know so I can figure out what would be the maximum toque I could output on this device.