Which of the following gases has a root mean square speed of 412 m/s at 191 K?

• A. Oxygen (O_2)
• B. Nitrogen (N_2)
• C. Helium (He)
• D. Chlorine (Cl_2)

Answer: (D)

Verified step by step guidance

1. Recall the formula for the root mean square (rms) speed of a gas molecule: \[ u_{rms} = \sqrt{\frac{3RT}{M}} \] where \(u_{rms}\) is the root mean square speed, \(R\) is the ideal gas constant, \(T\) is the temperature in kelvin, and \(M\) is the molar mass of the gas in kilograms per mole.
2. Identify the given values: the rms speed \(u_{rms} = 412\ \text{m/s}\) and the temperature \(T = 191\ \text{K}\). The gas constant \(R\) is typically \$8.314\ \text{J/(mol\cdot K)}$.
3. Rearrange the rms speed formula to solve for the molar mass \(M\): \[ M = \frac{3RT}{u_{rms}^2} \] This will allow you to calculate the molar mass of the gas that has the given rms speed at the specified temperature.
4. Calculate the molar mass \(M\) using the values for \(R\), \(T\), and \(u_{rms}\). Remember to keep units consistent, especially converting molar mass to kilograms per mole (1 g/mol = 0.001 kg/mol).
5. Compare the calculated molar mass to the molar masses of the given gases (Oxygen \(\approx 32\ \text{g/mol}\), Nitrogen \(\approx 28\ \text{g/mol}\), Helium \(\approx 4\ \text{g/mol}\), Chlorine \(\approx 70.9\ \text{g/mol}\)) to identify which gas matches the calculated molar mass and thus the rms speed.