A rigid, well-insulated gas cylinder contains N0 moles of an ideal gas at a temp

Question

A rigid, well-insulated gas cylinder contains N0 moles of an ideal gas at a temperature To  and pressure Po. A venting valve is opened and the gas is pumped out of the tank. Derive an expression for the instantaneous temperature in the cylinder as a function of the number of moles of gas remaining in the cylinder. What is the final temperature Tf, if the tank is evacuated to Nf = 0? Is the prediction physically realistic? The heat  capacity at constant volume of the gas is independent of temperature.

Explanation / Answer

PV = nRT

nT = PV/R = constant

nT = K

Tdn + ndT = 0

Tdn = - ndT

dT/dn = - T/n

dT/T = -dn/n

Integrating both sides

ln T = - ln n + C

at n=No ; T =To

so

ln To = - ln No + C

C = ln To + ln No = ln To*No

so

ln T = ln To*No - ln n

ln T = ln (To*No/n)

T = (To*No)/n

as n = Nf = 0

Twill tend to infinity...this is not practically possible

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