The ratio of everage speed, most probable speed and root mean square speed of a gas sample will be

Options

A. `sqrtfrac{4}{pi}:sqrt3:sqrt2`
B. `sqrtfrac{8}{pi}:sqrt2:sqrt3`
C. `sqrtfrac{16}{pi}:sqrt2:sqrt3`
D. `sqrt2:sqrtfrac{9}{pi}:sqrt3`

Answer

To find the ratio of average speed, most probable speed, and root mean square speed of a gas sample, we will use the following formulas:

1. **Average Speed (V_avg)**:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

2. **Most Probable Speed (V_mp)**:
\[ V_{mp} = \sqrt{\frac{2RT}{M}} \]

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

Where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Write down the formulas for each speed
We have the formulas for average speed, most probable speed, and root mean square speed as stated above.

### Step 2: Set up the ratio
To find the ratio of these speeds, we will express them in terms of \( RT/M \):
\[ \text{Ratio} = V_{avg} : V_{mp} : V_{rms} \]

Substituting the formulas:
\[ \text{Ratio} = \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{2RT}{M}} : \sqrt{\frac{3RT}{M}} \]

### Step 3: Simplify the ratio
Now, we can factor out \( \sqrt{RT/M} \) from each term:
\[ = \sqrt{RT/M} \left( \sqrt{\frac{8}{\pi}} : \sqrt{2} : \sqrt{3} \right) \]

Since \( \sqrt{RT/M} \) is common in all three terms, it cancels out when we are only interested in the ratio:
\[ = \sqrt{\frac{8}{\pi}} : \sqrt{2} : \sqrt{3} \]

### Step 4: Present the final ratio
Thus, the final ratio of average speed, most probable speed, and root mean square speed is:
\[ \sqrt{\frac{8}{\pi}} : \sqrt{2} : \sqrt{3} \]

### Conclusion
The answer to the question is the ratio of speeds:
\[ \text{Final Ratio} = \sqrt{\frac{8}{\pi}} : \sqrt{2} : \sqrt{3} \]