In alkaline solution, the following equilibria exist
a. `S^(2-)+S rarr S_(2)^(2-)` equilibrium constant `K_(1)`
b. `S_(2)^(2-)+S rarr S_(3)^(2-)` equilibrium constant `K_(2)`
`K_(1)` and `K_(2)` have values `12` and `11`, respectively.
`S_(3)^(2-) rarr S^(2-)+2S`. What is equilibrium constant for the reaction

Options

A. `132`
B. `7.58xx10^(-3)`
C. `1.09`
D. `0.918`

Explanation

To find the equilibrium constant for the reaction \( S_3^{2-} \rightleftharpoons S^{2-} + 2S \), we will use the given reactions and their equilibrium constants.

### Step-by-Step Solution:

1. **Identify the Given Reactions and Constants:**
- Reaction 1: \( S^{2-} + S \rightleftharpoons S_2^{2-} \) with \( K_1 = 12 \)
- Reaction 2: \( S_2^{2-} + S \rightleftharpoons S_3^{2-} \) with \( K_2 = 11 \)

2. **Reverse the Reactions:**
- For Reaction 1 (reversed):
\[ S_2^{2-} \rightleftharpoons S^{2-} + S \]
The equilibrium constant for the reversed reaction is:
\[ K' = \frac{1}{K_1} = \frac{1}{12} \]

- For Reaction 2 (reversed):
\[ S_3^{2-} \rightleftharpoons S_2^{2-} + S \]
The equilibrium constant for the reversed reaction is:
\[ K'' = \frac{1}{K_2} = \frac{1}{11} \]

3. **Add the Reversed Reactions:**
- Now, we add the two reversed reactions:
\[ S_2^{2-} \rightleftharpoons S^{2-} + S \quad (K' = \frac{1}{12}) \]
\[ S_3^{2-} \rightleftharpoons S_2^{2-} + S \quad (K'' = \frac{1}{11}) \]
- When we add these reactions, the \( S_2^{2-} \) cancels out:
\[ S_3^{2-} \rightleftharpoons S^{2-} + 2S \]

4. **Calculate the Overall Equilibrium Constant:**
- The equilibrium constant for the overall reaction is the product of the equilibrium constants of the individual reactions:
\[ K_{\text{overall}} = K' \times K'' = \left(\frac{1}{12}\right) \times \left(\frac{1}{11}\right) = \frac{1}{132} \]

5. **Final Calculation:**
- Now, calculate \( K_{\text{overall}} \):
\[ K_{\text{overall}} = \frac{1}{132} \approx 0.00758 \]

### Conclusion:
The equilibrium constant for the reaction \( S_3^{2-} \rightleftharpoons S^{2-} + 2S \) is approximately \( 7.58 \times 10^{-3} \).