AP Physics 2 FRQ: Heating a gas under a piston

Mathematical Routines, 10 pointsAbout 22 minutes · 10 points

A vertical cylinder contains a sample of monatomic ideal gas sealed under a piston of mass M and cross-sectional area A. The piston can slide up and down without friction, and its top is open to the atmosphere, which is at pressure P_atm. Initially the gas is at a temperature of 300 K and the piston is at rest a height h = 0.15 m above the base of the cylinder, as shown. A heater under the cylinder then slowly transfers energy to the gas, and the piston rises slowly until it is 0.21 m above the base. The piston has a mass of 20 kg, A = 0.010 m², and P_atm = 1.0 × 10⁵ Pa.

(A) (i) Starting from Newton's second law applied to the piston, derive an expression for the absolute pressure P of the gas in terms of M, A, P_atm and physical constants, as appropriate. (ii) Calculate P. 3 points

(B) (i) Consider a PV diagram of the process, with pressure on the vertical axis and volume on the horizontal axis. Describe the path the gas follows on the diagram: its shape, the volumes at which it starts and ends, and the direction in which the gas moves along it. (ii) Calculate the work W done on the gas during the process. 3 points

(D) The gas is returned to its initial state. The piston is now clamped so that it cannot move, and the heater transfers the same amount of energy Q to the gas as in part C. Will the final temperature of the gas be greater than, less than, or equal to the final temperature in the original process? Justify your answer. 2 points