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

Step-by-step solution

For the initial adiabatic process, the equation of state is p₁ V₁^ = p₂ V₂^ . Substituting the given values p₁ = p₀ , V₁ = V₀ , and p₂ = p₀ 2 : p₀ V₀^ = p₀ 2 V₂^ V₂^ = 2 V₀^ V₂ = 2^ 1/ V₀ For the subsequent isothermal process, the temperature remains constant, so p₂ V₂ = p₃ V₃ . Substituting p₂ = p₀ 2 and p₃ = p₀ 4 : p₀ 2 V₂ = p₀ 4 V₃ V₃ = 2 V₂ Substituting the value of V₂ obtained from the adiabatic process: V₃ = 2 2^ 1/ V₀ V₃ = 2^ ( + 1)/ V₀