The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is

Options

A. `1.085 : 1`
B. `1:1.086`
C. `2 : 1.086`
D. `1.086 : 2`

Explanation

To find the ratio of the root mean square velocity (\( U_{rms} \)) to the average velocity (\( U_{avg} \)) of a gas molecule at a particular temperature, we can follow these steps:

### Step 1: Write down the formulas for \( U_{rms} \) and \( U_{avg} \)

The formulas for the root mean square velocity and average velocity of a gas molecule are:

\[ U_{rms} = \sqrt{\frac{3RT}{M}} \]

\[ U_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

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

### Step 2: Set up the ratio \( \frac{U_{rms}}{U_{avg}} \)

We need to calculate the ratio of \( U_{rms} \) to \( U_{avg} \):

\[ \frac{U_{rms}}{U_{avg}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} \]

### Step 3: Simplify the ratio

This can be simplified as follows:

\[ \frac{U_{rms}}{U_{avg}} = \frac{\sqrt{3RT}}{\sqrt{8RT/\pi M}} = \frac{\sqrt{3RT} \cdot \sqrt{\pi M}}{\sqrt{8RT}} \]

Now, we can cancel out \( \sqrt{RT} \) from the numerator and denominator:

\[ = \frac{\sqrt{3\pi M}}{\sqrt{8M}} = \frac{\sqrt{3\pi}}{\sqrt{8}} \]

### Step 4: Further simplify the expression

Now, we can express this as:

\[ \frac{U_{rms}}{U_{avg}} = \sqrt{\frac{3\pi}{8}} \]

### Step 5: Calculate the numerical value

Calculating the numerical value:

\[ \sqrt{\frac{3\pi}{8}} \approx \sqrt{\frac{3 \times 3.14}{8}} \approx \sqrt{\frac{9.42}{8}} \approx \sqrt{1.1775} \approx 1.085 \]

### Conclusion

Thus, the ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is approximately:

\[ 1.085 : 1 \]

### Final Answer

The correct option is \( 1.085 : 1 \).

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