At what temperature (in K) does hydrogen gas (H_2) have a root mean square speed of 745 m/s?

• A. 273 K
• B. 500 K
• C. 600 K
• D. 400 K

Answer: (B)

Verified step by step guidance

1. Recall the formula for the root mean square (rms) speed of a gas: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(v_{rms}\) 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: \(v_{rms} = 745\) m/s, and the gas is hydrogen (\(H_2\)). The molar mass of \(H_2\) is approximately 2.016 g/mol, which must be converted to kilograms per mole by dividing by 1000.
3. Rearrange the rms speed formula to solve for temperature \(T\): \(T = \frac{M v_{rms}^2}{3R}\).
4. Substitute the known values into the rearranged formula: use \(M\) in kg/mol, \(v_{rms} = 745\) m/s, and \(R = 8.314\) J/(mol·K).
5. Calculate the temperature \(T\) using the substituted values to find the temperature at which hydrogen gas has the given rms speed.