Question

The ratio between the root mean square velocities of \( O_2 \) and \( O_3 \) molecules at the same temperature is:

• A. \( 3 : 2 \)
• B. \( 2 : 3 \)
• C. \( 1 : 1 \)
• D. \( \sqrt{3} : \sqrt{2} \)
• E. \( \sqrt{2} : \sqrt{3} \)

Hint:

The root mean square velocity is inversely proportional to the square root of the molar mass. Use this relationship to calculate velocity ratios for different gases.

Answer 1

The Correct Option is D

Step 1: The root mean square velocity \( v_{{rms}} \) of a gas is given by: \[ v_{{rms}} = \sqrt{\frac{3RT}{M}} \] where:
- \( R \) is the gas constant,
- \( T \) is the temperature,
- \( M \) is the molar mass of the gas.
Step 2: The ratio of the root mean square velocities of \( O_2 \) and \( O_3 \) molecules is: \[ \frac{v_{rms, O_2}}{v_{rms, O_3}} = \sqrt{\frac{M_{O_3}}{M_{O_2}}} \] Step 3: The molar masses of \( O_2 \) and \( O_3 \) are approximately 32 and 48 g/mol, respectively. So, \[ \frac{v_{rms, O_2}}{v_{rms, O_3}} = \sqrt{\frac{48}{32}} = \sqrt{\frac{3}{2}} = \sqrt{3} : \sqrt{2} \]