The temperature at which the root mean square speed of `SO_2`molecule is same as that of methane at `27^@` C.

Options

A. 600 K
B. 900 K
C. 1200 K
D. 400 K

Answer

To find the temperature at which the root mean square (RMS) speed of sulfur dioxide (SO₂) molecules is the same as that of methane (CH₄) at 27°C, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the Temperature of Methane to Kelvin:**
- The temperature given is 27°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
- Therefore,
\[ T_{CH_4} = 27 + 273.15 = 300.15 \text{ K} \approx 300 \text{ K} \]

2. **Write the Formula for Root Mean Square Speed (VRMS):**
- The formula for the root mean square speed (VRMS) of a gas is given by:
\[ V_{RMS} = \sqrt{\frac{3RT}{M}} \]
- Where:
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

3. **Calculate VRMS for Methane (CH₄):**
- The molar mass of methane (CH₄) is approximately 16 g/mol, which is 0.016 kg/mol.
- Using the formula:
\[ V_{CH_4} = \sqrt{\frac{3R \cdot 300}{0.016}} \]

4. **Calculate VRMS for Sulfur Dioxide (SO₂):**
- The molar mass of sulfur dioxide (SO₂) is approximately 64 g/mol, which is 0.064 kg/mol.
- Let the temperature of SO₂ be \( T \). Then:
\[ V_{SO_2} = \sqrt{\frac{3R \cdot T}{0.064}} \]

5. **Set the Two VRMS Equations Equal to Each Other:**
- Since the RMS speeds are equal:
\[ \sqrt{\frac{3R \cdot T}{0.064}} = \sqrt{\frac{3R \cdot 300}{0.016}} \]

6. **Square Both Sides to Eliminate the Square Root:**
- This gives:
\[ \frac{3R \cdot T}{0.064} = \frac{3R \cdot 300}{0.016} \]

7. **Cancel Out Common Terms:**
- The \( 3R \) cancels out:
\[ \frac{T}{0.064} = \frac{300}{0.016} \]

8. **Cross-Multiply to Solve for T:**
- Cross-multiplying gives:
\[ T \cdot 0.016 = 300 \cdot 0.064 \]
- Simplifying:
\[ T \cdot 0.016 = 19.2 \]
- Therefore:
\[ T = \frac{19.2}{0.016} = 1200 \text{ K} \]

### Final Answer:
The temperature at which the root mean square speed of SO₂ is the same as that of methane at 27°C is **1200 K**.