A sample of dry air has an initial pressure p1 = 1000 hPa and temperature T1 = 300K. It undergoes a process that takes it to a new pressure p2 = 500 hPa with unchanged temperature T2=T1. Calculate the mechanical work per unit mass performed with the sample under the following scenarios:
1. Isothermal expansion to final state.
2. Isochoric pressure reduction to p2 followed with isobaric expansion to final state.
3. Isobaric expansion to final specific volume \alpha _{2} followed with isochoric pressure reduction to final state.