One mole of an ideal gas `(y=1.4)` at `500K`, is filled in an adiabatic cylinder. The piston is free to move against atmospheric pressure. A non-conducting fan is inserted into the gas space and rotated vigrously, so that the gas expands slowly, till its volume is double. Find work done by the fan in `KJ`.
(Take `R=8 j//`mole. `k,` neglect heat capacity of the cylinder, piston and the fan).

Explanation

Using `1st` law of `TD`
`Q=W+Uuparrow`
`0=((-W_(tan))+PDeltav)+n(f)/(2)RDeltaT`
`W_(tan)=nRDeltaT+n(f)/(2)RDeltaT`
`W_(tan)=nC_(P)DeltaT`
`W_(tan)=(1)(R+(f)/(2)R)(k)` (as the gas is expending slowly so `p=` constant, so `Tpropv`)
`W_(tan)=14 kj`