An ideal monoatomic gas is contained inside a cylinder-piston assembly connected to a Hookean spring, as shown in the figure. The piston is frictionless and massless, with cross-sectional area 100 cm² and a stopper 5 cm away. Ambient pressure is 1 bar. Initial gas conditions: V_initial = 2 L, T_initial = 300 K. The spring constant is 10 kN/m and is initially unstretched. The gas is expanded reversibly by adding 362.5 J of heat, until the piston presses against the stoppers at the final equilibrium state. Neglecting heat loss to the surroundings, the final equilibrium temperature of the gas is ____ K (rounded off to the nearest integer). Given: for a monoatomic ideal gas, C_v = (3/2)R, where R = 8.314 J/(mol·K).

Question

Accepted Answer

The correct answer is 595 to 605. GATE 2025 official key: 595 to 605 K. ΔV = Area×stroke = 0.01 m²×0.05 m = 0.0005 m³. Force balance gives P(x) = P_amb + kx/A, so work done by gas W = P_amb·ΔV + ½k·x² = 100000×0.0005 + 0.5×10000×0.05² = 50 + 12.5 = 62.5 J. Energy balance: ΔU = Q − W = 362.5 − 62.5 = 300 J. Since n = P₁V₁/(RT₁) and nR = P₁V₁/T₁ = 100000×0.002/300 = 0.667 J/K, ΔU = n·(3/2)R·ΔT = (3/2)×0.667×ΔT = 300 ⇒ ΔT = 300 K, so T_final = 300+300 = 600 K.