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

Answer

To solve the problem, we will analyze the relationship between average velocity and root mean square (RMS) velocity in the context of kinetic theory.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The average velocity (\(v_{avg}\)) of gas molecules is given by the formula:
\[ v_{avg} = \sqrt{\frac{8RT}{M}} \]
- The root mean square (RMS) velocity (\(v_{rms}\)) is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
Where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(M\) is the molecular weight of the gas.

2. **Identify the Relationship**:
- Both \(v_{avg}\) and \(v_{rms}\) are proportional to the square root of the temperature \(T\). This means:
\[ v_{avg} \propto \sqrt{T} \quad \text{and} \quad v_{rms} \propto \sqrt{T} \]

3. **Given Condition**:
- The problem states that the average velocity becomes 4 times its original value:
\[ v_{avg}' = 4v_{avg} \]

4. **Effect on Temperature**:
- Since \(v_{avg}\) is proportional to \(\sqrt{T}\), if \(v_{avg}\) increases by a factor of 4, we can express this as:
\[ 4v_{avg} \propto \sqrt{T'} \]
- Therefore, we can write:
\[ 4\sqrt{T} \propto \sqrt{T'} \]

5. **Squaring Both Sides**:
- Squaring both sides gives:
\[ (4\sqrt{T})^2 = 16T \propto T' \]
- This implies that the new temperature \(T'\) is 16 times the original temperature \(T\):
\[ T' = 16T \]

6. **Effect on RMS Velocity**:
- Now, substituting \(T' = 16T\) into the RMS velocity formula:
\[ v_{rms}' = \sqrt{\frac{3R(16T)}{M}} = \sqrt{16} \cdot \sqrt{\frac{3RT}{M}} = 4v_{rms} \]
- Thus, the RMS velocity also becomes 4 times its original value.

### Conclusion:
If the average velocity becomes 4 times its original value, the RMS velocity will also become 4 times its original value.

### Final Answer:
RMS velocity will also become 4 times.

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