For gaseous state, if most probable speed is denoted by `C^(**)` average speed by `barC` and root square speed by `C`, then for a large number of molecules, the ratios of these speeds are

Options

A. `C^(**) : barC : C = 1 : 1.225 : 1.128`
B. `C^(**) : barC : C = 1.225 : 1.128 : 1`
C. `C^(**) : barC : C = 1.128 : 1 : 1.225`
D. `C^(**) : barC : C = 1 : 1.128 : 1.225`

Explanation

To solve the problem, we need to find the ratios of the most probable speed (C*), average speed (C̅), and root mean square speed (C) for a large number of gas molecules.

### Step-by-Step Solution:

1. **Identify the formulas for the speeds:**
- Most Probable Speed (C*):
\[ C^* = \sqrt{\frac{2RT}{M}} \]
- Average Speed (C̅):
\[ C̅ = \sqrt{\frac{8RT}{\pi M}} \]
- Root Mean Square Speed (C):
\[ C = \sqrt{\frac{3RT}{M}} \]

2. **Set up the ratios:**
We want to find the ratios of these speeds:
\[ \text{Ratio} = C^* : C̅ : C \]

3. **Substitute the formulas into the ratio:**
\[ C^* : C̅ : C = \sqrt{\frac{2RT}{M}} : \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{3RT}{M}} \]

4. **Cancel common factors:**
Since \(RT/M\) is common in all three terms, we can cancel it out:
\[ = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

5. **Calculate the numerical values:**
- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{3} \approx 1.732\)
- \(\sqrt{\frac{8}{\pi}} \approx \sqrt{\frac{8}{3.14}} \approx \sqrt{2.546} \approx 1.597\)

6. **Express the ratios in a simplified form:**
Now we can express the ratios:
\[ C^* : C̅ : C \approx 1.414 : 1.597 : 1.732 \]

7. **Convert to a simpler ratio:**
To simplify, we can divide each term by the smallest value (1.414):
\[ \approx 1 : 1.128 : 1.225 \]

8. **Final Ratio:**
Thus, the final ratio of the speeds is:
\[ C^* : C̅ : C \approx 1 : 1.128 : 1.225 \]

### Conclusion:
The ratios of the most probable speed, average speed, and root mean square speed for a large number of molecules are approximately:
\[ 1 : 1.128 : 1.225 \]