Question

The rms speed of oxygen at room temperature is about \[ 500\,\text{m s}^{-1}. \] The rms speed of hydrogen at the same temperature is about:

• A. \(125\,\text{m s}^{-1}\)
• B. \(2000\,\text{m s}^{-1}\)
• C. \(8000\,\text{m s}^{-1}\)
• D. \(500\,\text{m s}^{-1}\)

Hint:

At the same temperature, \[ v_{\text{rms}}\propto \frac{1}{\sqrt{M}}. \] Lighter gases move faster than heavier gases.

Answer 1

The Correct Option is B

Step 1: Write the formula for rms speed.
The root mean square speed of gas molecules at temperature $T$ is:
\[ v_\text{rms} = \sqrt{\frac{3RT}{M}} \] where $R$ is the gas constant, $T$ is the absolute temperature, and $M$ is the molar mass.

Step 2: Write the ratio of rms speeds for two gases at the same temperature.
For hydrogen and oxygen at the same temperature $T$:
\[ \frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{3RT/M_{H_2}}{3RT/M_{O_2}}} = \sqrt{\frac{M_{O_2}}{M_{H_2}}} \]

Step 3: Insert the molar masses.
$M_{O_2} = 32\,\text{g/mol}$ and $M_{H_2} = 2\,\text{g/mol}$:
\[ \frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]

Step 4: Solve for the rms speed of hydrogen.
\[ v_{H_2} = 4 \times v_{O_2} = 4 \times 500 = 2000\,\text{m\,s}^{-1} \]

Step 5: Interpret the result.
Hydrogen is 16 times lighter than oxygen per molecule, so it moves 4 times faster at the same temperature. This is why hydrogen gas escapes from containers more easily than heavier gases.

Step 6: State the answer.
\[ \boxed{2000\,\text{m\,s}^{-1}} \]