Which of the following gas samples has the greatest average molecular speed at 25 °C?

• A. CO_2
• B. O_2
• C. He
• D. N_2

Answer: (C)

Verified step by step guidance

1. Recall that the average molecular speed of a gas is related to the root-mean-square speed, which depends on the temperature and the molar mass of the gas. The formula for the root-mean-square speed is given by: \[ u_{rms} = \sqrt{\frac{3RT}{M}} \] 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 since the problem states 25 °C (which should be converted to Kelvin by adding 273.15). This means the variable that affects the average molecular speed is the molar mass \(M\).
3. Identify the molar masses of each gas: - \(CO_2\) (carbon dioxide) has a molar mass of approximately 44 g/mol, - \(O_2\) (oxygen) has a molar mass of approximately 32 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 the average molecular speed is inversely proportional to the square root of the molar mass, the gas with the smallest molar mass will have the greatest average molecular speed at the same temperature.
5. Compare the molar masses and conclude that helium (\(He\)), having the smallest molar mass, will have the greatest average molecular speed at 25 °C.