A vertical frictionless piston cylinder arrangement is made up of a freely moving piston of mass M and thickness t and a cylinder of height and cross-sectional area A. The cylinder is open at the top to the atmosphere which is at a pressure P and closed at the bottom. The space below the piston is occupied by a gas with an initial volume of V.. The space above the piston is filled with sand of density p. Heat is now added to the gas causing the piston to move up slowly until all the sand has been spilled. Derive expressions for the following in terms of the given quantities: (a) initial pressure, final pressure of the gas and the pressure at any intermediate instant (b)work interaction for the gas. Sketch the process undergone by the gas on a PV diagram and show through appropriate hatching the areas corresponding to the work interactions for the atmosphere, sand and the piston.