# A rigid- well-insulated gas cylinder contains N0 moles of an ideal gas.docx

A rigid, well-insulated gas cylinder contains N0 moles of an ideal gas at a temperature ToÂ Â 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 heatÂ Â 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 .

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### A rigid- well-insulated gas cylinder contains N0 moles of an ideal gas.docx

• 1. A rigid, well-insulated gas cylinder contains N0 moles of an ideal gas at a temperature ToÂ Â 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 heatÂ Â 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
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