At what temperature will most probable speed of the molecules of the second member of homologous series `C_(n)H_(2n-2)` be the same as that of `SO_(2)` at `527^(@)C`.

Options

A. `500^(@)C`
B. `727^(@)C`
C. `227^(@)C`
D. None of these

Explanation

To solve the problem, we need to determine the temperature at which the most probable speed of the molecules of the second member of the homologous series \( C_nH_{2n-2} \) is the same as that of \( SO_2 \) at \( 527^\circ C \).

### Step-by-Step Solution:

1. **Identify the second member of the series \( C_nH_{2n-2} \)**:
- The first member is \( C_2H_2 \) (ethyne), so the second member is \( C_3H_4 \) (propene).
- Therefore, \( n = 3 \) gives us \( C_3H_4 \).

2. **Calculate the molar mass of \( C_3H_4 \)**:
- The molar mass \( M_1 \) of \( C_3H_4 \) is calculated as:
\[ M_1 = (3 \times 12) + (4 \times 1) = 36 + 4 = 40 \, \text{g/mol} \]

3. **Determine the molar mass of \( SO_2 \)**:
- The molar mass \( M_2 \) of \( SO_2 \) is:
\[ M_2 = (32) + (2 \times 16) = 32 + 32 = 64 \, \text{g/mol} \]

4. **Convert the temperature of \( SO_2 \) from Celsius to Kelvin**:
- The given temperature is \( 527^\circ C \). To convert to Kelvin:
\[ T_1 = 527 + 273 = 800 \, \text{K} \]

5. **Use the formula for most probable speed**:
- The most probable speed \( v \) is given by the formula:
\[ v = \sqrt{\frac{2RT}{M}} \]
- For \( SO_2 \):
\[ v_1 = \sqrt{\frac{2RT_1}{M_2}} \]
- For \( C_3H_4 \):
\[ v_2 = \sqrt{\frac{2RT_2}{M_1}} \]

6. **Set the speeds equal to each other**:
- Since we want \( v_1 = v_2 \):
\[ \sqrt{\frac{2RT_1}{M_2}} = \sqrt{\frac{2RT_2}{M_1}} \]
- Squaring both sides and simplifying gives:
\[ \frac{T_1}{M_2} = \frac{T_2}{M_1} \]

7. **Rearranging the equation**:
- Rearranging gives:
\[ T_2 = T_1 \cdot \frac{M_1}{M_2} \]

8. **Substituting the known values**:
- Substitute \( T_1 = 800 \, K \), \( M_1 = 40 \, g/mol \), and \( M_2 = 64 \, g/mol \):
\[ T_2 = 800 \cdot \frac{40}{64} \]
- Calculate \( T_2 \):
\[ T_2 = 800 \cdot 0.625 = 500 \, K \]

9. **Convert \( T_2 \) from Kelvin to Celsius**:
- To convert back to Celsius:
\[ T_2 = 500 - 273 = 227^\circ C \]

### Final Answer:
The temperature at which the most probable speed of the molecules of \( C_3H_4 \) will be the same as that of \( SO_2 \) at \( 527^\circ C \) is \( 227^\circ C \).