Which of the following is true ?

Options

A. `u_("rms") gt bar(v) gt alpha`
B. `u_("rms") lt bar(v) lt alpha`
C. `u_("rms") gt bar(v) lt alpha`
D. `u_("rms") lt bar(v) gt alpha`

Explanation

To determine which of the following statements is true regarding the velocities of gas molecules, we need to analyze the relationships between the root mean square velocity (VRMS), average velocity (V̄), and most probable velocity (α).

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Root Mean Square Velocity (VRMS)**: This is defined as the square root of the average of the squares of the velocities of the gas molecules. The formula is given by:
\[ V_{RMS} = \sqrt{\frac{8RT}{\pi m}} \]
- **Average Velocity (V̄)**: This is the average of the velocities of the gas molecules. The formula is:
\[ V_{avg} = \sqrt{\frac{3RT}{m}} \]
- **Most Probable Velocity (α)**: This is the velocity at which the maximum number of molecules are moving. The formula is:
\[ V_{mp} = \sqrt{\frac{2RT}{m}} \]

2. **Calculate the Ratios**:
- We need to compare these three velocities. To do this, we can express them in terms of ratios:
\[ \frac{V_{RMS}}{V_{mp}} = \frac{\sqrt{\frac{8RT}{\pi m}}}{\sqrt{\frac{2RT}{m}}} = \sqrt{\frac{8}{2\pi}} = \sqrt{\frac{4}{\pi}} \]
\[ \frac{V_{RMS}}{V_{avg}} = \frac{\sqrt{\frac{8RT}{\pi m}}}{\sqrt{\frac{3RT}{m}}} = \sqrt{\frac{8}{3\pi}} \]
\[ \frac{V_{avg}}{V_{mp}} = \frac{\sqrt{\frac{3RT}{m}}}{\sqrt{\frac{2RT}{m}}} = \sqrt{\frac{3}{2}} \]

3. **Establish the Order of Velocities**:
- From the calculations, we can see that:
\[ V_{RMS} > V_{avg} > V_{mp} \]
- This indicates that the root mean square velocity is the greatest, followed by the average velocity, and the most probable velocity is the least.

4. **Select the Correct Answer**:
- Based on the established order of velocities, we can conclude that the correct statement is that VRMS is the greatest, followed by V̄, and then α is the least.

### Conclusion:
The correct answer is that the order of velocities is:
\[ V_{RMS} > V_{avg} > V_{mp} \]