# Air pressure/temperature relationship

Tags: relationship
 P: 2 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. Please, what am I missing???
HW Helper
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 $\gamma = C_p/C_v$.