Two-Stage Process and Heat Cycle for a Monatomic Ideal Gas

Problem Statement

A cylinder with a frictionless piston contains $n$n moles of a monatomic ideal gas. Initially, the gas has pressure $P_0$P0​, volume $V_0$V0​, and temperature $T_0$T0​ (state A). The following two processes are performed in sequence:

1. Process 1 (Isochoric heating): The gas is heated at constant volume $V_0$V0​ until the pressure reaches $2P_0$2P0​ (state B).
2. Process 2 (Isobaric expansion): The gas is heated at constant pressure $2P_0$2P0​ until the volume reaches $2V_0$2V0​ (state C).

Let $R$R be the gas constant and the molar heat capacity at constant volume of the monatomic ideal gas be $C_V = \frac{3}{2}R$CV​=23​R. Find the total heat $Q$Q absorbed by the gas from outside during the combined process (processes 1 and 2).

Constraints

• Initial pressure: $P_0 = 2.0 \times 10^3 \text{ Pa}$P0​=2.0×103 Pa
• Initial volume: $V_0 = 0.50 \text{ m}^3$V0​=0.50 m3
• Gas constant: $R = 8.3 \text{ J/(mol\cdot K)}$R=8.3 J/(mol\cdotK) (you may use the relation $P_0 V_0 = nRT_0$P0​V0​=nRT0​ in calculations)

Input Format

Find the value of $Q$Q and give the integer part of the answer.