A weightless piston divides a thermally insulated cylinder into two parts of volumes `V` and `3V. 2` moles of an ideal gas at pressure `P = 2` atmosphere are confined to the part with volume `V =1` litre. The remainder of the cylinder is evacuated. The piston is now released and the gas expands to fill the entire space of the cylinder. The piston is then pressed back to the initial position. Find the increase of internal energy in the process and final temperature of the gas. The ratio of the specific heats of the gas, `gamma =1.5`.

Explanation

In free expansion, no work is done, temperature remains same. Since number of moles remains same
`n=(PV)/(RT)=(P_1.4V)/(RT)`
`P_1=(P)/(4)`
In the case compression is adiabatic
`P_1(4V)^(gamma)=P_2V^(gamma)impliesP_2=4^(gamma)P_1=(4)^((3//2)((P)/(4))=2P`
`DeltaU=-DeltaW_(adi)=(P_2V_2-P_1V_1)/(gamma-1)=(2P.V-(P)/(4).4V)/((3)/(2)-1)`
`=2PV`
`=2xx2xx10^5xx1xx10^-3=400J`
`DeltaU=nC_VDeltaT=(nRDeltaT)/(gamma-1)=(2xx8.3DeltaT)/((3)/(2)-1)=33.2DeltaT`
`DeltaT=(DeltaU)/(33.2)=(400)/(33.2)=12K`
Final temperature`=T_1+DeltaT+(27+273)+12=312K`
(Room temperature `27^@C`)