The temperature of an ideal gas is increased from `27^(@)C to 127^(@)C`, the percentage increase in `V_(rms)` is [2013]

Options

A. `37%`
B. `11%`
C. `33%`
D. `15.5%`

Explanation

To solve the problem of finding the percentage increase in the root mean square velocity (\( V_{rms} \)) of an ideal gas when its temperature is increased from \( 27^\circ C \) to \( 127^\circ C \), we can follow these steps:

### Step 1: Convert temperatures from Celsius to Kelvin
- The initial temperature \( T_i \) is:
\[ T_i = 27 + 273 = 300 \, K \]
- The final temperature \( T_f \) is:
\[ T_f = 127 + 273 = 400 \, K \]

### Step 2: Write the formula for \( V_{rms} \)
The formula for the root mean square velocity (\( V_{rms} \)) of an ideal gas is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{m}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( m \) is the molar mass of the gas.

### Step 3: Calculate initial and final \( V_{rms} \)
- The initial \( V_{rms} \) at \( T_i \):
\[ V_{rms,i} = \sqrt{\frac{3R \cdot 300}{m}} \]
- The final \( V_{rms} \) at \( T_f \):
\[ V_{rms,f} = \sqrt{\frac{3R \cdot 400}{m}} \]

### Step 4: Find the change in \( V_{rms} \)
To find the percentage increase, we first need to calculate the change in \( V_{rms} \):
\[ \Delta V_{rms} = V_{rms,f} - V_{rms,i} \]
Substituting the expressions for \( V_{rms} \):
\[ \Delta V_{rms} = \sqrt{\frac{3R \cdot 400}{m}} - \sqrt{\frac{3R \cdot 300}{m}} \]

### Step 5: Factor out common terms
We can factor out \( \sqrt{\frac{3R}{m}} \):
\[ \Delta V_{rms} = \sqrt{\frac{3R}{m}} \left( \sqrt{400} - \sqrt{300} \right) \]

### Step 6: Calculate \( \sqrt{400} \) and \( \sqrt{300} \)
- \( \sqrt{400} = 20 \)
- \( \sqrt{300} \approx 17.32 \)

Thus,
\[ \Delta V_{rms} = \sqrt{\frac{3R}{m}} (20 - 17.32) = \sqrt{\frac{3R}{m}} \cdot 2.68 \]

### Step 7: Calculate the percentage increase
The percentage increase in \( V_{rms} \) is given by:
\[ \text{Percentage Increase} = \frac{\Delta V_{rms}}{V_{rms,i}} \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \frac{\sqrt{\frac{3R}{m}} \cdot 2.68}{\sqrt{\frac{3R}{m}} \cdot \sqrt{300}} \times 100 \]
The \( \sqrt{\frac{3R}{m}} \) cancels out:
\[ \text{Percentage Increase} = \frac{2.68}{\sqrt{300}} \times 100 \]

### Step 8: Calculate \( \sqrt{300} \)
- \( \sqrt{300} \approx 17.32 \)

Thus,
\[ \text{Percentage Increase} = \frac{2.68}{17.32} \times 100 \approx 15.5\% \]

### Final Answer
The percentage increase in \( V_{rms} \) is approximately \( 15.5\% \).

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