The ratio among most probable velocity, mean velocity and root mean velocity is given by

Options

A. `1:2:3`
B. `1:sqrt(2):sqrt(3)`
C. `sqrt(2):sqrt(3):sqrt(8//pi)`
D. `sqrt(2):sqrt(8//pi):sqrt(3)`

Explanation

To find the ratio among the most probable velocity (V_mp), mean velocity (V_mean), and root mean velocity (V_rms) of gas molecules, we can use the following formulas:

1. **Most Probable Velocity (V_mp)**:
\[ V_{mp} = \sqrt{\frac{2RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

2. **Mean Velocity (V_mean)**:
\[ V_{mean} = \sqrt{\frac{8RT}{\pi M}} \]

3. **Root Mean Square Velocity (V_rms)**:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

Now, we will find the ratio \( V_{mp} : V_{mean} : V_{rms} \).

### Step 1: Write down the formulas
- \( V_{mp} = \sqrt{\frac{2RT}{M}} \)
- \( V_{mean} = \sqrt{\frac{8RT}{\pi M}} \)
- \( V_{rms} = \sqrt{\frac{3RT}{M}} \)

### Step 2: Set up the ratio
We want to find the ratio \( V_{mp} : V_{mean} : V_{rms} \).

### Step 3: Express each velocity in terms of a common factor
Let's express each term in the ratio using the same denominator for clarity:

- \( V_{mp} = \sqrt{\frac{2RT}{M}} \)
- \( V_{mean} = \sqrt{\frac{8RT}{\pi M}} \)
- \( V_{rms} = \sqrt{\frac{3RT}{M}} \)

### Step 4: Simplify the ratio
Now we can write the ratio as:
\[ \frac{V_{mp}}{V_{mean}} = \frac{\sqrt{\frac{2RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} = \sqrt{\frac{2RT}{M}} \cdot \sqrt{\frac{\pi M}{8RT}} = \sqrt{\frac{2\pi}{8}} = \sqrt{\frac{\pi}{4}} = \frac{\sqrt{\pi}}{2} \]

Now for \( V_{mp} : V_{mean} \):
\[ V_{mp} : V_{mean} = \frac{\sqrt{2}}{1} : \frac{\sqrt{8/\pi}}{1} = \sqrt{2} : \sqrt{\frac{8}{\pi}} = \sqrt{2\pi} : 2 \]

Next, we can find the ratio \( V_{mp} : V_{rms} \):
\[ \frac{V_{mp}}{V_{rms}} = \frac{\sqrt{\frac{2RT}{M}}}{\sqrt{\frac{3RT}{M}}} = \sqrt{\frac{2}{3}} \]

### Step 5: Combine the ratios
Now we have:
- \( V_{mp} : V_{mean} = \sqrt{2\pi} : 2 \)
- \( V_{mp} : V_{rms} = \sqrt{2} : \sqrt{3} \)

To find a common ratio, we can express everything in terms of \( V_{mp} \):
\[ V_{mp} : V_{mean} : V_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

### Final Step: Simplify the ratio
This gives us the final ratio:
\[ V_{mp} : V_{mean} : V_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]

### Conclusion
The ratio among the most probable velocity, mean velocity, and root mean velocity is:
\[ V_{mp} : V_{mean} : V_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \]