# Air pressure/temperature relationship

When you compress air it heats, and when you decompress it it cools, but how much?
Working from http://en.wikipedia.org/wiki/Gay-Lussac's_law I find the equation
pressure/temperature=constant
so I derived for example
if the air is 10°C (283.16°K) and the pressure is 14PSI so...
14/283.16 = 0.04944
increase the pressure by 90 PSI
104/2103.5 = 0.04944
maintian pressure, but decrease temperature to almost ambient (16.84°C)
104/290 = 0.3586
decrease pressure to ambient
14/39.04 = 0.3586
i.e. I compress a tank of air to 90psi above ambient air pressure of 14psi at 10 degrees celcius, I let it cool to almost ambient (just for easier math) and when I shoot the air out it decompresses and cools to 39 Kelvin. It doesn't sound right.
Then I realised that equation required mass and volume to remain constant.
http://en.wikipedia.org/wiki/Combined_gas_law gives me a more complete equation
(pressure x volume)/temperature = constant
but that would mean when I increassed the pressure by compacting the air I would, for example, decrease the volume by ten times thus increasing the pressure ten times and the temperature would remain constant.

Andrew Mason
Homework Helper
When you compress air it heats, and when you decompress it it cools, but how much?
Working from http://en.wikipedia.org/wiki/Gay-Lussac's_law I find the equation
pressure/temperature=constant
so I derived for example
if the air is 10°C (283.16°K) and the pressure is 14PSI so...
14/283.16 = 0.04944
increase the pressure by 90 PSI
104/2103.5 = 0.04944
maintian pressure, but decrease temperature to almost ambient (16.84°C)
104/290 = 0.3586
decrease pressure to ambient
14/39.04 = 0.3586
i.e. I compress a tank of air to 90psi above ambient air pressure of 14psi at 10 degrees celcius, I let it cool to almost ambient (just for easier math) and when I shoot the air out it decompresses and cools to 39 Kelvin. It doesn't sound right.
Then I realised that equation required mass and volume to remain constant.
http://en.wikipedia.org/wiki/Combined_gas_law gives me a more complete equation
(pressure x volume)/temperature = constant
but that would mean when I increassed the pressure by compacting the air I would, for example, decrease the volume by ten times thus increasing the pressure ten times and the temperature would remain constant.

Because the gas is contained in a smaller volume the pressure is still much greater than the original pressure (Pf/Pi = Vi/Vf if T is the same). To find the compressed volume, you have to use the adiabatic condition: $PV^\gamma = \text{Constant}$ where $\gamma = C_p/C_v$.