In the temperature changes from `27^(@)C` to `127^(@)C`, the relative percentage change in RMS velocity is

Options

A. 1.56
B. 2.56
C. 15.6
D. 82.4

Answer

To solve the problem of finding the relative percentage change in the RMS (Root Mean Square) velocity when the temperature changes from 27°C to 127°C, we can follow these steps:

### Step-by-Step Solution:

1. **Convert Temperatures to Kelvin**:
- The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]
- For 27°C:
\[ T_1 = 27 + 273 = 300 \, K \]
- For 127°C:
\[ T_2 = 127 + 273 = 400 \, K \]

2. **Understand the Formula for RMS Velocity**:
- The RMS velocity (\(v_{rms}\)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- Here, \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.

3. **Calculate the RMS Velocities**:
- For \(T_1\) (300 K):
\[ v_{rms1} = \sqrt{\frac{3R \cdot 300}{M}} \]
- For \(T_2\) (400 K):
\[ v_{rms2} = \sqrt{\frac{3R \cdot 400}{M}} \]

4. **Find the Ratio of RMS Velocities**:
- The ratio of the two RMS velocities can be expressed as:
\[ \frac{v_{rms2}}{v_{rms1}} = \frac{\sqrt{400}}{\sqrt{300}} = \sqrt{\frac{400}{300}} = \sqrt{\frac{4}{3}} \approx 1.1547 \]

5. **Calculate the Change in RMS Velocity**:
- The change in RMS velocity can be expressed as:
\[ \frac{v_{rms2} - v_{rms1}}{v_{rms1}} = \frac{v_{rms2}/v_{rms1} - 1}{1} \]
- Substituting the ratio:
\[ \frac{v_{rms2} - v_{rms1}}{v_{rms1}} = 1.1547 - 1 = 0.1547 \]

6. **Calculate the Percentage Change**:
- The percentage change in RMS velocity is:
\[ \text{Percentage Change} = \left(0.1547\right) \times 100 \approx 15.47\% \]

### Final Answer:
The relative percentage change in RMS velocity when the temperature changes from 27°C to 127°C is approximately **15.4%**.