At what temperature most probable speed of `SO_(2)` molecule have the same value as root mean square speed of `O_(2)` molecules at 300 K?

Options

A. 150 K
B. 600 K
C. 750 K
D. 900 K

Explanation

To solve the problem, we need to find the temperature at which the most probable speed of \( SO_2 \) molecules equals the root mean square speed of \( O_2 \) molecules at 300 K.

### Step-by-step Solution:

1. **Understand the formulas**:
- The most probable speed (\( v_{mp} \)) of a gas is given by the formula:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kg/mol.
- The root mean square speed (\( v_{rms} \)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

2. **Identify the molar masses**:
- For \( O_2 \), the molar mass \( M_{O_2} = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol} \).
- For \( SO_2 \), the molar mass \( M_{SO_2} = 32 + 16 = 48 \, \text{g/mol} = 0.048 \, \text{kg/mol} \).

3. **Calculate \( v_{rms} \) for \( O_2 \) at 300 K**:
\[ v_{rms} = \sqrt{\frac{3R \cdot 300}{M_{O_2}}} \]
Substituting \( R = 8.314 \, \text{J/(mol K)} \):
\[ v_{rms} = \sqrt{\frac{3 \cdot 8.314 \cdot 300}{0.032}} = \sqrt{\frac{7482.6}{0.032}} = \sqrt{233,828.125} \approx 483.5 \, \text{m/s} \]

4. **Set the most probable speed of \( SO_2 \) equal to \( v_{rms} \) of \( O_2 \)**:
\[ \sqrt{\frac{2RT_{SO_2}}{M_{SO_2}}} = v_{rms} \]
Substituting \( M_{SO_2} = 0.048 \, \text{kg/mol} \) and \( v_{rms} \approx 483.5 \, \text{m/s} \):
\[ \sqrt{\frac{2R T_{SO_2}}{0.048}} = 483.5 \]

5. **Square both sides**:
\[ \frac{2R T_{SO_2}}{0.048} = (483.5)^2 \]
\[ 2R T_{SO_2} = 0.048 \cdot 233,828.125 \]
\[ 2R T_{SO_2} = 11,202.75 \]

6. **Solve for \( T_{SO_2} \)**:
\[ T_{SO_2} = \frac{11,202.75}{2R} \]
Substituting \( R = 8.314 \):
\[ T_{SO_2} = \frac{11,202.75}{2 \cdot 8.314} \approx \frac{11,202.75}{16.628} \approx 674.5 \, \text{K} \]

7. **Convert to Celsius**:
\[ T_{SO_2} = 674.5 - 273.15 \approx 401.35 \, \text{°C} \]

### Final Answer:
The temperature at which the most probable speed of \( SO_2 \) molecules equals the root mean square speed of \( O_2 \) molecules at 300 K is approximately **401.35 °C**.