Thermal Efficiency of a Special Heat Cycle

Problem Statement

A vertical cylinder with a smooth-moving piston contains $n$n moles of a monatomic ideal gas. The gas undergoes the following cycle $A \to C \to B \to A$A→C→B→A:

• State A: Gas volume is $V_0$V0​, pressure is $P_0$P0​.
• Process A $\to$→ C: While the piston is fixed, heat is supplied until the pressure reaches $2P_0$2P0​, reaching state C.
• Process C $\to$→ B: The piston is released, and heat is supplied while the pressure is kept at $2P_0$2P0​ until the volume reaches $2V_0$2V0​, reaching state B.
• Process B $\to$→ A: Heat is slowly removed while the piston is pushed down, causing pressure $P$P and volume $V$V to change along a straight line on the $P$P-$V$V diagram, returning to state A.

Find the thermal efficiency $e$e of this cycle (the net work done per cycle divided by the total heat absorbed per cycle).

Constraints

• The gas is a monatomic ideal gas.
• $V_0 = 1.0 \times 10^{-3}\ \mathrm{m^3}$V0​=1.0×10−3 m3
• $P_0 = 1.0 \times 10^5\ \mathrm{Pa}$P0​=1.0×105 Pa

Input Format

The thermal efficiency $e$e is expressed as an irreducible fraction $\frac{X}{Y}$YX​. Give the value of positive integer $X+Y$X+Y.