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. Solution 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 .

1. 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.
Solution
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

2. 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