Calculate the temperature at which the rms velocity of `SO_(2)` is the same as that of oxygen at `27^(@)C`.

Options

A. 273 K
B. 600 K
C. 303 K
D. 403 K

Answer

To solve the problem of calculating the temperature at which the root mean square (rms) velocity of sulfur dioxide (SO₂) is the same as that of oxygen (O₂) at 27°C, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for rms Velocity**:
The root mean square velocity (v_rms) is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

2. **Convert the Given Temperature**:
The temperature of oxygen is given as 27°C. We need to convert this to Kelvin:
\[ T_{O_2} = 27 + 273 = 300 \, K \]

3. **Identify Molar Masses**:
- The molar mass of oxygen (O₂) is \( 32 \, g/mol \) (since each oxygen atom has a molar mass of 16 g/mol).
- The molar mass of sulfur dioxide (SO₂) is \( 64 \, g/mol \) (16 g/mol for sulfur and 32 g/mol for two oxygen atoms).

4. **Set Up the Equation**:
Since we want the rms velocities of SO₂ and O₂ to be equal, we can set their equations equal to each other:
\[ \sqrt{\frac{3RT_{SO_2}}{M_{SO_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

5. **Cancel Common Terms**:
The \( \sqrt{3R} \) terms can be canceled from both sides:
\[ \frac{T_{SO_2}}{M_{SO_2}} = \frac{T_{O_2}}{M_{O_2}} \]

6. **Substitute Known Values**:
Substitute the known values for the molar masses and the temperature of O₂:
\[ \frac{T_{SO_2}}{64} = \frac{300}{32} \]

7. **Cross Multiply to Solve for \( T_{SO_2} \)**:
Cross-multiplying gives:
\[ T_{SO_2} = \frac{300 \times 64}{32} \]

8. **Calculate \( T_{SO_2} \)**:
Simplifying the right side:
\[ T_{SO_2} = 300 \times 2 = 600 \, K \]

9. **Convert to Celsius (if needed)**:
If you want to convert the temperature back to Celsius:
\[ T_{SO_2} = 600 - 273 = 327 \, °C \]

### Final Answer:
The temperature at which the rms velocity of SO₂ is the same as that of oxygen at 27°C is **600 K** or **327 °C**.