The ratio of average spreed to the rms spreed of the gas is

Options

A. `sqrt8:sqrt3pi`
B. `sqrt3pi:sqrt8`
C. `sqrt6:sqrt5pi`
D. `sqrt8:sqrt5pi`

Answer

To solve the problem of finding the ratio of average speed to the RMS speed of a gas, we can follow these steps:

### Step 1: Write the formula for average speed
The average speed (\(V_{avg}\)) of gas molecules is given by the formula:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi m}} \]
where:
- \(R\) is the gas constant,
- \(T\) is the absolute temperature in Kelvin,
- \(m\) is the molar mass of the gas.

### Step 2: Write the formula for RMS speed
The root mean square (RMS) speed (\(V_{rms}\)) of gas molecules is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{m}} \]

### Step 3: Set up the ratio of average speed to RMS speed
We need to find the ratio of average speed to RMS speed:
\[ \frac{V_{avg}}{V_{rms}} = \frac{\sqrt{\frac{8RT}{\pi m}}}{\sqrt{\frac{3RT}{m}}} \]

### Step 4: Simplify the ratio
To simplify the ratio, we can rewrite it as:
\[ \frac{V_{avg}}{V_{rms}} = \frac{\sqrt{\frac{8RT}{\pi m}}}{\sqrt{\frac{3RT}{m}}} = \frac{\sqrt{8RT}}{\sqrt{3RT}} \cdot \frac{1}{\sqrt{\pi}} \cdot \sqrt{m} \]
The \(RT\) terms cancel out:
\[ = \frac{\sqrt{8}}{\sqrt{3}} \cdot \frac{1}{\sqrt{\pi}} = \frac{\sqrt{8}}{\sqrt{3\pi}} \]

### Step 5: Final expression
Thus, the ratio of average speed to RMS speed is:
\[ \frac{V_{avg}}{V_{rms}} = \sqrt{\frac{8}{3\pi}} \]

### Conclusion
The final answer is:
\[ \frac{V_{avg}}{V_{rms}} = \sqrt{\frac{8}{3\pi}} \]

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