The relation between S (solubility) and `K_(sp)` (solubility product) for a sparingly soluble binary electrolyte

Options

A. `S=K_(sp)^(2)`
B. `S=K_(sp)`
C. `sqrt(S)=sqrt(K_(sp))`
D. `S=1/2 K_(sp)`

Answer

To understand the relationship between solubility (S) and the solubility product constant (Ksp) for a sparingly soluble binary electrolyte, we can break down the concept step by step.

### Step 1: Define the Sparingly Soluble Binary Electrolyte
A sparingly soluble binary electrolyte is a compound that dissociates into two ions when it dissolves in water. For example, consider a salt AB that dissociates into cation A⁺ and anion B⁻.

### Step 2: Write the Dissociation Equation
When the salt AB dissolves in water, it can be represented by the following equation:
\[ AB (s) \rightleftharpoons A^+ (aq) + B^- (aq) \]

### Step 3: Define Solubility (S)
Let S be the solubility of the salt AB in moles per liter (mol/L). This means that when the salt dissolves, S moles of AB will produce S moles of A⁺ and S moles of B⁻ in solution.

### Step 4: Write the Expression for Ksp
The solubility product constant (Ksp) for the dissociation of the salt can be expressed as:
\[ K_{sp} = [A^+][B^-] \]
Since both ions are produced in equal amounts, we can substitute S into the expression:
\[ K_{sp} = [A^+][B^-] = S \cdot S = S^2 \]

### Step 5: Relate Solubility to Ksp
From the Ksp expression, we can derive the relationship between solubility and Ksp:
\[ S^2 = K_{sp} \]
To find S, we take the square root of both sides:
\[ S = \sqrt{K_{sp}} \]

### Conclusion
The relationship between solubility (S) and the solubility product (Ksp) for a sparingly soluble binary electrolyte is given by:
\[ S = \sqrt{K_{sp}} \]