Most probable velocity, average velocity and root mean square velocity are related as

Options

A. `1 : 1.128 : 1.224`
B. `1 : 1.128 : 1.424`
C. `1 : 2.128 : 1.224`
D. `1 : 1.428 : 1.442`

Explanation

To solve the problem of relating most probable velocity, average velocity, and root mean square velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Formulas**:
- Most Probable Velocity (\(v_{mp}\)):
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
- Average Velocity (\(v_{avg}\)):
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
- Root Mean Square Velocity (\(v_{rms}\)):
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Here, \(R\) is the universal gas constant, \(T\) is the temperature, and \(M\) is the molar mass.

2. **Set Up the Ratios**:
- We need to find the ratios of \(v_{mp}\), \(v_{avg}\), and \(v_{rms}\):
\[ \frac{v_{mp}}{v_{avg}} \quad \text{and} \quad \frac{v_{avg}}{v_{rms}} \]

3. **Calculate the Ratios**:
- For \(v_{mp}\) to \(v_{avg}\):
\[ \frac{v_{mp}}{v_{avg}} = \frac{\sqrt{\frac{2RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} = \sqrt{\frac{2}{\frac{8}{\pi}}} = \sqrt{\frac{2\pi}{8}} = \sqrt{\frac{\pi}{4}} = \frac{\sqrt{\pi}}{2} \]

- For \(v_{avg}\) to \(v_{rms}\):
\[ \frac{v_{avg}}{v_{rms}} = \frac{\sqrt{\frac{8RT}{\pi M}}}{\sqrt{\frac{3RT}{M}}} = \sqrt{\frac{8}{3\pi}} \]

4. **Express the Ratios in a Common Format**:
- Now, we can express the ratios in terms of numerical values:
- \(v_{mp} : v_{avg} : v_{rms} = 1 : \sqrt{\frac{\pi}{4}} : \sqrt{\frac{8}{3\pi}}\)

5. **Calculate Numerical Values**:
- Calculate the numerical values for the ratios:
- \(v_{mp} \approx 1\)
- \(v_{avg} \approx 1.128\)
- \(v_{rms} \approx 1.224\)

6. **Final Ratio**:
- Therefore, the final ratio of most probable velocity, average velocity, and root mean square velocity is:
\[ v_{mp} : v_{avg} : v_{rms} = 1 : 1.128 : 1.224 \]

### Summary of the Ratios:
- Most Probable Velocity (\(v_{mp}\)) = 1
- Average Velocity (\(v_{avg}\)) = 1.128
- Root Mean Square Velocity (\(v_{rms}\)) = 1.224