Consider three identical flasks filled with different gases. Flask \(\mathrm{A} : \mathrm{CO}\) at 760 torr and \(0^{\circ} \mathrm{C}\) Flask \(\mathrm{B} : \mathrm{N}_{2}\) at 250 torr and \(0^{\circ} \mathrm{C}\) Flask \(\mathrm{C} : \mathrm{H}_{2}\) at 100 torr and \(0^{\circ} \mathrm{C}\) a. In which flask will the molecules have the greatest average kinetic energy? b. In which flask will the molecules have the greatest average velocity?

According to the Kinetic Theory of Gases, the average kinetic energy of an ideal gas depends only on its temperature. All the flasks are at \(0^{\circ}\mathrm{C}\), which means they all have the same temperature. Therefore, the average kinetic energy of the gas molecules in all three flasks will be the same. So, the answer to question (a) is that the molecules will have the greatest average kinetic energy in all the flasks.

The average velocity of the gas molecules can be determined using the following formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) Where \(v_{rms}\) is the root mean square velocity of the gas molecules, \(R\) is the ideal gas constant (8.314 J K⁻¹ mol⁻¹), \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kg/mol. Since the temperature in all three flasks is \(0^{\circ}\mathrm{C}\) or 273.15 K, we can calculate the average velocity of the molecules in each flask by plugging in the values for each gas's molar mass (in kg/mol) and solving for \(v_{rms}\). For Flask A (carbon monoxide, CO): Molar mass of CO is 28.01 g/mol, which is 0.02801 kg/mol. \(v_{rms,CO} = \sqrt{\frac{3(8.314)(273.15)}{0.02801}}\) For Flask B (nitrogen, N₂): Molar mass of N₂ is 28.02 g/mol, which is 0.02802 kg/mol. \(v_{rms,N_2} = \sqrt{\frac{3(8.314)(273.15)}{0.02802}}\) For Flask C (hydrogen, H₂): Molar mass of H₂ is 2.02 g/mol, which is 0.00202 kg/mol. \(v_{rms,H_2} = \sqrt{\frac{3(8.314)(273.15)}{0.00202}}\) Solving the equations, we get: \(v_{rms,CO} ≈ 510.41 \,\text{m/s}\) \(v_{rms,N_2} ≈ 510.55 \,\text{m/s}\) \(v_{rms,H_2} ≈ 1930.23 \,\text{m/s}\) As we can see, the molecules in Flask C (hydrogen gas) have the greatest average velocity. So, the answer to question (b) is that the molecules will have the greatest average velocity in Flask C.