Question

Let \( \bar{V} \), \( V_{\text{rms}} \), \( V_p \) denote the mean speed, root mean square speed, and most probable speed of the molecules of mass \( m \) in an ideal monoatomic gas at absolute temperature \( T \) Kelvin. Which statement(s) is/are correct?

• A. No molecules can have speed greater than \( \sqrt{2} V_{\text{rms}} \).
• B. No molecules can have speed less than \( \frac{V_p}{\sqrt{2}} \).
• C. \( V_p<\bar{V}<V_{\text{rms}} \).
• D. Average kinetic energy of a molecule is \( \frac{3}{4} m V_p^2 \).

Hint:

In an ideal monoatomic gas, the most probable speed, the mean speed, and the root mean square speed are related by the formulas \( V_p<\bar{V}<V_{\text{rms}} \). Always remember that the distribution of molecular speeds is not uniform, and different speeds correspond to different probabilities.

Answer 1

The Correct Option is C

We consider \( \bar{V} \), \( V_{\text{rms}} \), and \( V_p \), representing the mean, root mean square, and most probable speeds of molecules in an ideal monoatomic gas. These speeds depend on the temperature \( T \) and mass \( m \) of the molecules. The speed formulas are: 1. Most probable speed \( V_p \): \[ V_p = \sqrt{\frac{2kT}{m}} \] 2. Root mean square speed \( V_{\text{rms}} \): \[ V_{\text{rms}} = \sqrt{\frac{3kT}{m}} \] 3. Mean speed \( \bar{V} \): \[ \bar{V} = \sqrt{\frac{8kT}{\pi m}} \] where: - \( k \) is the Boltzmann constant, - \( T \) is the absolute temperature, - \( m \) is the molecular mass. Step 1: Analyzing Option (A) - Speed Greater than \( \sqrt{2} V_{\text{rms}} \) The Maxwell-Boltzmann distribution describes molecular speeds. Although some molecules can have high speeds, the probability decreases rapidly. Therefore, no molecule can exceed \( \sqrt{2} V_{\text{rms}} \), making this statement true. Thus, Option (A) is correct. Step 2: Analyzing Option (B) - Speed Less than \( \frac{V_p}{\sqrt{2}} \) \( V_p \) is the most probable speed. The distribution is broad, and some molecules will move slower than \( \frac{V_p}{\sqrt{2}} \), so this statement is incorrect. Thus, Option (B) is incorrect. Step 3: Analyzing Option (C) - \( V_p<\bar{V}<V_{\text{rms}} \) From the speed equations: \[ V_p = \sqrt{\frac{2kT}{m}}, \quad \bar{V} = \sqrt{\frac{8kT}{\pi m}}, \quad V_{\text{rms}} = \sqrt{\frac{3kT}{m}} \] we find that \( V_p<\bar{V}<V_{\text{rms}} \), since: \[ V_p = \sqrt{\frac{2}{3}} V_{\text{rms}} \quad \text{and} \quad \bar{V} = \sqrt{\frac{8}{3\pi}} V_{\text{rms}} \] Therefore, the relationship \( V_p<\bar{V}<V_{\text{rms}} \) is correct. Hence, Option (C) is correct. Step 4: Analyzing Option (D) - Kinetic Energy Formula The average kinetic energy per molecule is: \[ E_{\text{kinetic}} = \frac{3}{2} kT \] Option (D) incorrectly suggests kinetic energy is proportional to \( \frac{3}{4} m V_p^2 \). The correct formula depends on temperature, not directly on speed. Therefore, this statement is incorrect. Thus, Option (D) is incorrect. Conclusion The correct statement is: \[ \boxed{(C) \, V_p<\bar{V}<V_{\text{rms}}} \]