An ideal gas ( 0.5 mol), initially at 2 bar pressure, is compressed at a constant temperature of 600 K in two steps: first, against a constant external pressure of P bar ( 2 < P < 8 ), and then against constant external pressure of 8 bar. At each step, the compression is stopped only when the pressure of the gas becomes equal to the external pressure. The total work done on the gas in these steps is W . Considering a

Step-by-step solution

Let the initial, intermediate, and final volumes of the gas be V₁ , V₂ , and V₃ respectively. Using the ideal gas equation V = nRT P , we have: V₁ = nRT 2 V₂ = nRT P V₃ = nRT 8 The work done on the gas during the first step against a constant external pressure P is: W₁ = -P_ ext,1 (V₂ - V₁) = -P ( nRT P - nRT 2 ) = nRT ( P 2 - 1 ) The work done on the gas during the second step against a constant external pressure of 8 bar is: W₂ = -P_ ext,2 (V₃ - V₂) = -8 ( nRT 8 - nRT P ) = nRT ( 8 P - 1 ) The total work done on