A rigid container holds Helium gas at an initial temperature of 127^ C . The total translational kinetic energy of the gas is 50 ~kJ . A valve is opened, allowing some gas to escape, and the remaining gas is cooled to 27^ C . If the new total translational kinetic energy of the gas is 15 ~kJ , what is the mass of the Helium gas that escaped? [Given: R = 25 3 ~J ~mol ⁻¹ ~K ⁻¹ and molar mass of He = 4 ~g/mol ]

Step-by-step solution

The total translational kinetic energy of an ideal gas is given by E = 3 2 nRT . For the initial state ( T₁ = 127^ C = 400 ~K ): n₁ = 2E₁ 3RT₁ = 2 50000 3 ( 25 3 ) 400 = 100000 25 400 = 10 ~mol For the final state ( T₂ = 27^ C = 300 ~K ): n₂ = 2E₂ 3RT₂ = 2 15000 3 ( 25 3 ) 300 = 30000 25 300 = 4 ~mol Number of moles of Helium that escaped = n₁ - n₂ = 10 - 4 = 6 ~mol Mass of escaped Helium = moles molar mass = 6 ~mol 4 ~g/mol = 24 ~g Answer: 24 ~g