An ideal gas undergoes a cyclic process 1 2 3 1 described as follows: Step 1 2: Isobaric expansion from volume V to 2V . Step 2 3: Isochoric cooling until the pressure is halved. Step 3 1: Isothermal compression at 400 K back to the initial state 1. If the magnitude of the net work done in the entire cycle is 2988 J, the number of moles of the gas is _______. [Given: R = 8.3 J K ⁻¹ mol ⁻¹ , 2 = 0.7 ]

Step-by-step solution

Let the number of moles of the gas be n . In state 1, the gas is at temperature T₁ = 400 K (since step 3 1 is isothermal at 400 K and returns to state 1). Step 1 2 is an isobaric expansion from V to 2V . Since V T at constant pressure, T₂ = 2T₁ = 800 K. Work done W_ 1 2 = -nR(T₂ - T₁) = -n 8.3 (800 - 400) = -3320n J. Step 2 3 is an isochoric process. Since volume is constant, work done W_ 2 3 = 0 . Step 3 1 is an isothermal compression at 400 K from 2V back to V . Work done W_ 3 1 = -nRT₁ ( V₁ V₃ ) W_ 3 1 = -n 8.3