Q$1.$ A gas at a rate of $1\mathrm{.}25k\frac{g}{s}$ is reversibly compressed from a pressure $(P_1)$ of $0\mathrm{.}3$ MPa to a final pressure $(P_2)$ of $750$kPa. You may assume the gas to behave ideally and is supplied to the compressor at a temperature of $25C.$

$($i$)$ Determine the power requirement if compression is performed isothermally.

$($ii$)$ Determine the power requirement if compression is performed adiabatically.

$($i$)$ Instead of compressing directly from $P_1$ to $P_2$ as in $($ii$)$ above, the gas is withdrawn at an intermediate pressure given by $P_{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}}=(P_1{P}_2{)}^{\frac{1}{2}}$ and cooled to $40C.$ The gas is then returned to the compressor and compressed to $P_2.$ Calculate the power requirement for each compression stage; and hence, the total power for this process. Take for the gas, $R=0\mathrm{.}518k\frac{J}{k}g.K$ and $=1\mathrm{.}229.$