$2-2-21$ A piston$-$cylinder device is used to compress a gas by pushing the piston with an external force. During the compression process, heat is transferred out of the gas in such a manner that the stored energy in the gas remains unchanged. Also, the pressure is found to be inversely proportional to the volume of the gas. $($a$)$ Determine an expression for the heat transfer rate in terms of the instantaneous volume and the rate of change of pressure of the gas. $($b$)$ At a given instant, when the gas occupies a volume of $0\mathrm{.}15m^3,$ the rate of change of pressure is found to be $2kP\frac{a}{s}.$ Determine the rate of heat transfer.