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 (T₁, P₁, V₁,. . .) and moves to state 2 (T₂, P₂ , V₂,...). During this change in position, the heater is regulated such that the pressure increases linearly with the volume, P = P₁ / V₁ 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 result in terms of the pressure P and volume V at states 1 and 2 and constants for the gas.