The initial pressure and volume of a given mass of a gas (Cp/Cv = γ) are p0 and V0. The gas can exchange heat with the surrounding. (a) It is slowly compressed to V0/2 and then suddenly compressed to V0/4. Find the final pressure. (b) If the gas is suddenly compressed from the volume V0 to V0/2 and then slowly compressed to V0/4, what will be the final pressure?

Sol. Given C base P/C base v = γ P base 0 (Initial Pressure), V base 0 (Initial Volume) (a) (i) Isothermal compression, P base 1V base 1 = P base 2V base 2 or, P base 0V base 0 = P base 2V base 0/2 ⇒ P base 2 = 2P abse 0 (ii) Adiabatic Compression P base 1V base 1^γ = P base 2V base 2^γ or 2P base 0 (V base 0/2)^γ = P^1 (V base 0/4)^γ ⇒ P’ = V base o^γ/2^γ * 2P abse 0 * 4^γ/V base 0^γ = 2^γ * 2 P base 0 ⇒ P base 02^γ+1 (b) (i) Adiabatic compression P base 1V base 1^γ = P base 2V base 2^γ or P base 0V base 0^γ = P’(V base 0/2) ^γ ⇒ P’ = P base 2^γ (ii) Isothermal compression P base 1V base 1 = P base 2V base 2 or 2^γ P base 0 * V base 0)/2 = P base 2 * V base 0/4 ⇒ P base 2 = P base 02^γ+1