Thermodynamic Cycle with a Spring-Loaded and Stopper-Controlled Piston

Problem Statement

A cylindrical container with a frictionless, massless piston is fixed with its axis pointing vertically upward. The container holds a monatomic ideal gas. The atmospheric pressure outside the cylinder is constant at $P_0$P0​.

Above the piston, a spring with spring constant $k$k is attached vertically. The upper end of the spring is fixed, and the lower end just touches the top face of the piston (at natural length) when the piston is at its reference position. The spring constant satisfies $k = \frac{2P_0S^2}{3V_0}$k=3V0​2P0​S2​.

The gas is taken through the following thermodynamic cycle (State A $\to$→ State B $\to$→ State C $\to$→ State A):

• State A: The gas volume is $V_0$V0​ and pressure is $P_0$P0​; the piston is at rest. The spring is at natural length and just touching the piston.
• Process A $\to$→ B: Heat is slowly added to the gas. The gas expands, the piston rises, and the spring compresses. When the volume reaches $4V_0$4V0​, heating stops. This is State B.
• Process B $\to$→ C: At State B, a stopper is inserted to hold the piston fixed. Heat is then slowly removed from the gas until the pressure drops to $P_0$P0​. This is State C.
• Process C $\to$→ A: At State C, while the piston remains fixed, the spring is gently detached from the piston. The stopper is then removed, and heat is slowly removed from the gas while keeping the pressure equal to $P_0$P0​ throughout, until the volume returns to $V_0$V0​ (State A).

Find the thermal efficiency $e$e of this one-cycle process.

Constraints

• Atmospheric pressure: $P_0 = 1.0 \times 10^5\,\mathrm{Pa}$P0​=1.0×105Pa
• Gas volume at State A: $V_0 = 1.0 \times 10^{-3}\,\mathrm{m^3}$V0​=1.0×10−3m3
• Cross-sectional area of cylinder: $S = 1.0 \times 10^{-2}\,\mathrm{m^2}$S=1.0×10−2m2
• Spring constant: $k = \frac{2P_0S^2}{3V_0} = \frac{20000}{3}\,\mathrm{N/m}$k=3V0​2P0​S2​=320000​N/m
• Gas constant: $R = 8.31\,\mathrm{J/(mol \cdot K)}$R=8.31J/(mol⋅K)

Input Format

The thermal efficiency $e$e is expressed as an irreducible fraction $\frac{a}{b}$ba​ ($a, b$a,b are coprime positive integers).

Compute $a + b$a+b and give the answer as a positive integer.