$1.$ The motion actuator for a robotic manipulator consists of a simple piston$-$cylinder system where the pressure of the gas contained in the cylinder can be controlled by heating or cooling. The moving piston provides the actuation. In a particular application, the actuator starts from state $1($T$1,$ P$1,$ V$1,...)$ and moves to state $2($T$2,$ P$2,$ V$2,...).$ During this change in position, the heater is regulated such that the pressure increases linearly with the volume, P$=($P$1/$V$1)$V$.$ Derive an expression for the amount of energy transfer to the gas as heat, Q$,$ for this process. Using the ideal gas property relations, express your results in terms of the pressure P and volume V at state $1$ and $2$ and constants for the gas.