Question

The root mean square (rms) speeds of Hydrogen atoms at 500 K, \(V_{H}\), and Helium atoms at 2000 K, \(V_{He}\), are related as:

• A. \(V_H > V_{He}\)
• B. \(V_H < V_{He}\)
• C. \(V_H = V_{He}\)
• D. \(V_H \gg V_{He}\)

Hint:

RMS speed of a gas is proportional to \(\sqrt{\frac{T}{M}}\). Equal ratios of \(T/M\) give equal rms speeds.

Answer 1

The Correct Option is C

Step 1: RMS speed formula.
The root mean square (rms) speed of gas molecules is given by \[ V_{rms} = \sqrt{\frac{3kT}{m}} \] where \(T\) is temperature and \(m\) is molecular mass.

Step 2: Ratio of rms speeds.
\[ \frac{V_H}{V_{He}} = \sqrt{\frac{T_H / m_H}{T_{He} / m_{He}}} = \sqrt{\frac{T_H \, m_{He}}{T_{He} \, m_H}} \] Given \(T_H = 500\text{ K}, T_{He} = 2000\text{ K}\), and \(m_{He} = 4m_H\): \[ \frac{V_H}{V_{He}} = \sqrt{\frac{500 \times 4m_H}{2000 \times m_H}} = \sqrt{1} = 1 \]

Step 3: Conclusion.
Hence, \(V_H = V_{He}\).