An insulted vertical cylindrical vessel of crosssecional are

An insulted vertical cylindrical vessel of crosssecional area A=.12 m² contains 20 mol of He gas at T₀ =300 K and P₀=.5 bar. It is capped by a moveable piston of mass m= 150 kg supported from the ceiling by a wire. The top of the vessel is open to the atmosphere (P_atm = 1 bar, T_atm=300 K). The wire is cut and the weight falls and compresses the gas.

a) Obtain the pressure and temperature of the gas a shrot time after the wire has been cut, when negligible amount of heat tranfer has taken place. Comment on whether your solution takes into account the inevitable oscillations of the piston around its equilibrium position before it comes to rest.

b)After a long time, the He gas thermally equilibrates with the environment by heat transfer through the piston. How much heat is exchanged with the environment during this process?

Solution

a) Pressure due to piston + atmosphere = final pressure, Pf = 101 kpa + 150*9.81/0.12 pa = 113.25 kpa
Now, no energy exchange, so dU = pdV = nRdT
(113.25 - 50)kpa * dv = 20*8.31*dT
dV = 0.98dT * 10^-3
now, pV = nRT
113.25kpa*(Vo - dV) = 20 * 8.31 * (To + dT)
Vo = 20*8.31*300/50*1000 = 0.9972
113.25kpa(0.9972 - 0.98dT * 10^-3) = 20*8.31(300 + dT)
dT = 227.65K, Tf = 527.65K
The osscilations of the piston would make it necessary for the heat to be exchanged with the surroundings which is not assumed in this case.

b) dU = dH - pdV
nCvdT = nCpdT - pdV
20*8.31dT/[gamma + 1] = 20*8.31*[gamma]dT/[gamma+1] - [113.25 - 50]*10^3dV
gamma for He = 5/3
so, dV = -6.5*1-^-4 dT

pV = nRT
113.25*10^3 *(0.9972 - 6.5*10^-4dT) = 20*8.31*(300 + dT)
dT = 262.25
dH = nCpdT = 2.72*10^4 J