Question

If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is :

• A. 2.0
• B. 0.5
• C. 1.5
• D. 1.0

Answer 1

The Correct Option is B

Given: - Root mean square (rms) velocity of hydrogen (\(v_{H_2}\)) = 2 km/s - Molecular mass of hydrogen (\(M_{H_2}\)) = 2 g/mol - Molecular mass of oxygen (\(M_{O_2}\)) = 32 g/mol

Step 1: Relationship for Root Mean Square Velocity

The root mean square velocity of a gas is given by:

\[ v_{\text{rms}} \propto \frac{1}{\sqrt{M}} \]

where \(M\) is the molar mass of the gas.

Step 2: Calculating the Ratio of Velocities

Using the inverse square root relationship for hydrogen and oxygen:

\[ \frac{v_{O_2}}{v_{H_2}} = \sqrt{\frac{M_{H_2}}{M_{O_2}}} \]

Substituting the given values:

\[ \frac{v_{O_2}}{2} = \sqrt{\frac{2}{32}} \] \[ \frac{v_{O_2}}{2} = \sqrt{\frac{1}{16}} \] \[ \frac{v_{O_2}}{2} = \frac{1}{4} \]

Multiplying both sides by 2:

\[ v_{O_2} = \frac{1}{4} \times 2 = 0.5 \, \text{km/s} \]

Conclusion:

The root mean square velocity of oxygen at the same conditions is 0.5 km/s.