Which of the following gases has the highest average speed at 400 K?

• A. O_2
• B. CO_2
• C. He
• D. N_2

Answer: (C)

Verified step by step guidance

1. Recall that the average speed of gas molecules is related to the root-mean-square speed, which depends on temperature and molar mass. The formula for the root-mean-square speed \(v_{rms}\) is given by: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] where \(R\) is the gas constant, \(T\) is the temperature in kelvin, and \(M\) is the molar mass of the gas in kilograms per mole.
2. Note that the temperature \(T\) is the same for all gases (400 K), so the variable that affects the average speed is the molar mass \(M\). The lighter the gas (smaller \(M\)), the higher the average speed.
3. Identify the molar masses of the gases given: - \(O_2\) (oxygen) has a molar mass of approximately 32 g/mol, - \(CO_2\) (carbon dioxide) has a molar mass of approximately 44 g/mol, - \(He\) (helium) has a molar mass of approximately 4 g/mol, - \(N_2\) (nitrogen) has a molar mass of approximately 28 g/mol.
4. Since helium has the smallest molar mass among the options, it will have the highest average speed at the same temperature according to the formula.
5. Therefore, by comparing molar masses and applying the root-mean-square speed formula, you can conclude that helium (\(He\)) has the highest average speed at 400 K.