if average velocity becomes 4 times, then what will be the effect on rms velocity at the temperature?

Options

A. 1.4 times
B. 4 times
C. 2 times
D. `(1)/(4)` times

Explanation

To solve the question, we need to analyze the relationship between average velocity and root mean square (rms) velocity in terms of temperature and mass. Here’s the step-by-step solution:

### Step 1: Understand the formulas for average velocity and rms velocity.
- The formula for average velocity (V_avg) is given by:
\[ V_{\text{avg}} = \sqrt{\frac{8RT}{\pi m}} \]
- The formula for root mean square velocity (V_rms) is given by:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{m}} \]

### Step 2: Set up the relationship between V_rms and V_avg.
- To find the relationship between V_rms and V_avg, we can take the ratio of the two velocities:
\[ \frac{V_{\text{rms}}}{V_{\text{avg}}} = \frac{\sqrt{\frac{3RT}{m}}}{\sqrt{\frac{8RT}{\pi m}}} \]

### Step 3: Simplify the ratio.
- Simplifying the above expression:
\[ \frac{V_{\text{rms}}}{V_{\text{avg}}} = \sqrt{\frac{3RT}{m}} \cdot \sqrt{\frac{\pi m}{8RT}} = \sqrt{\frac{3\pi}{8}} \]
- This shows that the ratio of V_rms to V_avg is a constant value, specifically \(\sqrt{\frac{3\pi}{8}}\).

### Step 4: Analyze the effect of increasing average velocity.
- If the average velocity becomes 4 times its original value, we can express this as:
\[ V_{\text{avg}}' = 4V_{\text{avg}} \]
- Since the ratio of V_rms to V_avg remains constant, we can express the new rms velocity (V_rms') as:
\[ V_{\text{rms}}' = \left(\frac{V_{\text{rms}}}{V_{\text{avg}}}\right) \cdot V_{\text{avg}}' = \sqrt{\frac{3\pi}{8}} \cdot (4V_{\text{avg}}) \]
- Therefore, we can conclude:
\[ V_{\text{rms}}' = 4 \cdot V_{\text{rms}} \]

### Step 5: Conclusion
- If the average velocity becomes 4 times its original value, the root mean square velocity will also become 4 times its original value.

### Final Answer
- The root mean square velocity will be 4 times its original value.

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