A sample of an ideal gas ( C_p / C_V = ) has an initial pressure p₀ and volume V₀ . The gas is isothermally taken to a pressure p₀ / 2 and from there adiabatically to a pressure p₀ / 4 . Determine the final volume.

Step-by-step solution

For the isothermal process, the temperature remains constant, so p V = constant . Let the volume after the isothermal process be V₁ . p₀ V₀ = ( p₀ 2 ) V₁ V₁ = 2 V₀ For the subsequent adiabatic process, p V^ = constant . Let the final volume be V₂ . p₁ V₁^ = p₂ V₂^ ( p₀ 2 ) (2 V₀)^ = ( p₀ 4 ) V₂^ 1 2 2^ V₀^ = 1 4 V₂^ V₂^ = 4 1 2 2^ V₀^ V₂^ = 2 2^ V₀^ V₂^ = 2^ + 1 V₀^ V₂ = 2^ ( + 1)/ V₀