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

The root mean square velocity of oxygen at the same temperature and pressure is determined using the formula \(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\), where \(k\) is Boltzmann's constant, \(T\) is the absolute temperature, and \(m\) is the molecular mass.

Given that the root mean square velocity of hydrogen is 2 km/s (\(v_{\text{rms, H}_2}\)), we can find the root mean square velocity of oxygen (\(v_{\text{rms, O}_2}\)) using the relationship between velocities and molar masses:

\(\frac{v_{\text{rms, H}_2}}{v_{\text{rms, O}_2}} = \sqrt{\frac{M_{\text{O}_2}}{M_{\text{H}_2}}}\)

The molar masses are:

• \(M_{\text{H}_2} = 2 \text{ g/mol}\)
• \(M_{\text{O}_2} = 32 \text{ g/mol}\)

Substituting these values yields:

\(\frac{2}{v_{\text{rms, O}_2}} = \sqrt{\frac{32}{2}}\)

\(\frac{2}{v_{\text{rms, O}_2}} = \sqrt{16}\)

\(\frac{2}{v_{\text{rms, O}_2}} = 4\)

Solving for \(v_{\text{rms, O}_2}\) gives:

\(v_{\text{rms, O}_2} = \frac{2}{4} = 0.5 \text{ km/s}\)

Therefore, the root mean square velocity of oxygen under the same conditions is 0.5 km/s.