An ideal gas ( = 1.5 ) at 300 K and 100 kPa with an initial volume of 1 litre undergoes a cycle: it is suddenly compressed to half its volume, cooled at constant pressure to 300 K , and finally expanded isothermally to its original volume of 1 litre. Calculate the total work done by the gas in this cycle.

Step-by-step solution

The total work done in the cycle is the sum of the work done in the adiabatic, isobaric, and isothermal processes. From the adiabatic compression phase, the initial state is P₁ = 10^5 Pa , V₁ = 10⁻³ m ^3 and the final volume is V₂ = 0.5 10⁻³ m ^3 . The work done is W₁₂ = P₁ V₁ - P₂ V₂ - 1 = 100 - 100 2 0.5 -82.8 J . During the isobaric cooling at pressure P₂ = 2 2 10^5 Pa , the gas cools from T₂ = 300 2 K to T₃ = 300 K . The work done is W₂₃ = nR(T₃ - T₂) = 1 3 (300 - 300 2 ) = 100(1 - 2 ) -41.4 J . During the isot