During the reversible adiabatic expansion, we have d S = 0. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as: T = T i 2 − − R / C V = T i 2 − − 2 / 3 for the monoatomic ideal gas. Heating the gas up to the initial temperature T i increases the entropy by the amount Δ Δ S = n ∫ ∫ T T i C V d T ′ T ′ = n R ln ⁡ ⁡ 2.