Consider a given sample of an ideal gas (Cp/Cv = γ) having initial pressure p0 and volume V0. (a) The gas is isothermally taken to a pressure p0/2 and from there adiabatically to a pressure p0/4. Find the final volume. (b) The gas is brought back to its initial state. It is adiabatically taken to a pressure p0/2 and from there isothermally to a pressure p0/4. Find the final volume.

Sol. Initial pressure = P base 0 Initial Volume = V base 0 γ = C base P/C base V (a) Isothermally to pressure P base 0/2 P base 0V base 0 = P base 0/2 V base 1 = 2 V base 0 Adiabatically to pressure = P base 0/4 P base 0/2 (V base 1)^γ = P base 0/4 (V base 2)^γ ⇒ P base 0/2 (2V base 0)^γ = P base 0/4 (V base 2)^γ ⇒ 2^γ+1 V base 0^γ = V base 2^γ ⇒ V base 2 = 2^(γ+1)/γ V base 0 ∴ Final Volume = 2^(γ+1)/γ V base 0 (b) Adiabetically to pressure P base 0/2 * (V’)γ P base 0 * (2^γ+1 V base 0^γ) = P base 0/2 * (V’)γ Isothermal to pressure P base 0/4 P base 0/2 * 2^1/γ V base 0 = P base 0/4 * V” ⇒ V” = 2^(γ+1)/γ V base 0 ∴ Final Volume = 2^(γ+1)/γ V base 0