At a given temperature the ratio of RMS and average velocities is 

Options

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

Answer

To find the ratio of the root mean square (RMS) velocity to the average velocity at a given temperature, we can follow these steps:

### Step 1: Understand the formulas for RMS and average velocity
The RMS velocity (\(v_{rms}\)) and average velocity (\(v_{avg}\)) for a gas can be expressed as:
- RMS velocity:
\[ v_{rms} = \sqrt{\frac{3RT}{M_w}} \]
- Average velocity:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M_w}} \]
Where:
- \(R\) = universal gas constant
- \(T\) = temperature in Kelvin
- \(M_w\) = molecular weight of the gas

### Step 2: Set up the ratio of RMS velocity to average velocity
We need to find the ratio:
\[ \frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3RT}{M_w}}}{\sqrt{\frac{8RT}{\pi M_w}}} \]

### Step 3: Simplify the ratio
Since both expressions have common terms, we can simplify:
\[ \frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3RT}{M_w}} \cdot \sqrt{\frac{\pi M_w}{8RT}} = \sqrt{\frac{3\pi}{8}} \]

### Step 4: Calculate the numerical value
Now we calculate the value of \(\sqrt{\frac{3\pi}{8}}\):
1. Calculate \(\pi \approx 3.14\).
2. Substitute \(\pi\) into the equation:
\[ \sqrt{\frac{3 \cdot 3.14}{8}} = \sqrt{\frac{9.42}{8}} = \sqrt{1.1775} \approx 1.086 \]

### Step 5: Conclusion
The ratio of RMS velocity to average velocity at a given temperature is approximately \(1.086\). Thus, the correct answer is:
\[ \text{Ratio of } v_{rms} \text{ to } v_{avg} = 1.086 : 1 \]

### Final Answer
The correct option is \(1.086 : 1\).

---