explain this solution and the theory behind each step

A cylinder with a piston restrained by a linear spring contains 2 kg of carbon dioxide at 500 kPa, 400 C. It is cooled to 40 C, at which point the pressure is 300 kPa. Calculate the heat transfer for the process. Solution: C.V. The carbon dioxide, which is a control mass. Continuity Eq.: m2 - m1 = 0 Energy Eq.: m (u2 - uj) = 1Q2 - 1W2 Process Eq.: P =A + BV (linear spring) 1W2 = SPdV = =(P, + P2)(V2 - V1) Equation of state: PV = mRT (ideal gas) State 1: V1 = mRT /P1 = (2 x 0.18892 x 673.15) kJ /500 kPa = 0.5087 m3 State 2: V2 = mRT2/P2 = (2 x 0.18892 x 313.15) kJ /300 kPa = 0.3944 m3 1 W2 = =(500 + 300) kPa (0.3944 - 0.5087) m' =-45.72 k.J From Table A5, Cvo = 0.653 kJ/kg K Q = m Cvo (T2 - T1) + W= 2 kg x 0.653 kJ/kg K (40 - 400) K - 45.72 kJ = -515.88 kJ