Question

According to the kinetic theory of gases, the ratio of the root mean square velocity of molecular oxygen and molecular hydrogen at 300 K is

• A. 1 : 1
• B. 1 : \( 2\sqrt{2} \)
• C. 1 : 4
• D. 1 : 16

Hint:

The root mean square velocity is inversely proportional to the square root of the molar mass of the gas. For two gases at the same temperature, the lighter gas will have a higher velocity.

Answer 1

The Correct Option is C

Step 1: Understanding the root mean square velocity.
The root mean square velocity \( v_{rms} \) of a gas is given by the equation:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \] where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas. Step 2: Analyzing the ratio.
The ratio of root mean square velocities for oxygen (O$_2$) and hydrogen (H$_2$) at the same temperature can be written as:
\[ \frac{v_{rms, \text{O}_2}}{v_{rms, \text{H}_2}} = \sqrt{\frac{M_{\text{H}_2}}{M_{\text{O}_2}}} \] The molar masses are \( M_{\text{H}_2} = 2 \) g/mol and \( M_{\text{O}_2} = 32 \) g/mol. Thus, the ratio is:
\[ \frac{v_{rms, \text{O}_2}}{v_{rms, \text{H}_2}} = \sqrt{\frac{2}{32}} = \frac{1}{4} \] Thus, the ratio of the root mean square velocities is 1 : \( 2\sqrt{2} \).
Step 3: Conclusion.
The correct answer is (B) 1 : \( 2\sqrt{2} \).