Solubility of `M_(2)S` type salt is `3.5xx10^(-3)` then find out its solubility product

Options

A. `1.7xx10^(-6)`
B. `1.7xx10^(-10)`
C. `1.7xx10^(-18)`
D. `1.7xx10^(-12)`

Explanation

To find the solubility product (Ksp) of the salt \( M_2S \) given its solubility, follow these steps:

### Step 1: Write the dissociation equation
The salt \( M_2S \) dissociates in water as follows:
\[ M_2S \rightleftharpoons 2M^+ + S^{2-} \]

### Step 2: Define the solubility
Let the solubility of \( M_2S \) be \( S \). According to the dissociation:
- For every 1 mole of \( M_2S \) that dissolves, it produces 2 moles of \( M^+ \) ions and 1 mole of \( S^{2-} \) ions.
- Therefore, the concentration of \( M^+ \) ions will be \( 2S \) and the concentration of \( S^{2-} \) ions will be \( S \).

### Step 3: Write the expression for Ksp
The solubility product \( K_{sp} \) is given by the formula:
\[ K_{sp} = [M^+]^2 [S^{2-}] \]
Substituting the concentrations from the previous step:
\[ K_{sp} = (2S)^2 \cdot (S) = 4S^2 \cdot S = 4S^3 \]

### Step 4: Substitute the given solubility
Given that the solubility \( S = 3.5 \times 10^{-3} \):
\[ K_{sp} = 4(3.5 \times 10^{-3})^3 \]

### Step 5: Calculate \( Ksp \)
First, calculate \( (3.5 \times 10^{-3})^3 \):
\[ (3.5)^3 = 42.875 \quad \text{and} \quad (10^{-3})^3 = 10^{-9} \]
Thus,
\[ (3.5 \times 10^{-3})^3 = 42.875 \times 10^{-9} \]
Now, multiply by 4:
\[ K_{sp} = 4 \times 42.875 \times 10^{-9} = 171.5 \times 10^{-9} \]

### Final Answer
\[ K_{sp} = 171.5 \times 10^{-9} \]

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