A gaseous mixture consisting of 1 mole of helium and 1 mole of oxygen is initially at a temperature T₀ and volume V₀ . The mixture undergoes an isothermal expansion to a new volume V₁ = V₀ e^ 1/3 , doing work W . Alternatively, if the same initial mixture undergoes an adiabatic expansion to a volume V₂ , it does the identical amount of work W . The ratio of the final to initial volume in the adiabatic process, V₂ V₀

Step-by-step solution

First, determine the properties of the gas mixture. Helium is monoatomic ( C_ V1 = 1.5R ) and oxygen is diatomic ( C_ V2 = 2.5R ). The effective molar heat capacity at constant volume for the 2 -mole mixture is: C_ V,mix = n₁ C_ V1 + n₂ C_ V2 n₁ + n₂ = 1(1.5R) + 1(2.5R) 2 = 2R The effective ratio of specific heats for the mixture is: = C_ P,mix C_ V,mix = C_ V,mix + R C_ V,mix = 2R + R 2R = 1.5 The work done during the isothermal expansion of the 2 -mole mixture is: W_ iso = nRT₀ ( V₁ V₀ ) = 2RT₀ (e^ 1/3 ) = 2 3 RT