Three sparingly soluble salts that have same solubility products as given below
I. `A_(2)X` II. `AX` III. `A_(2)X_(3)`
Their solubilities in a saturated solution will be such that

Options

A. `III gt II gt I`
B. III gt I gt II
C. II gt III gt I
D. II gt I gt III

Answer

To determine the order of solubility of the three sparingly soluble salts \( A_2X \), \( AX \), and \( A_2X_3 \) with the same solubility product (\( K_{sp} \)), we will analyze their solubility expressions step by step.

### Step 1: Write the dissociation equations for each salt

1. **For \( A_2X \)**:
\[ A_2X \rightleftharpoons 2A^+ + X^{2-} \]
Let the solubility of \( A_2X \) be \( s_1 \). Thus, at equilibrium:
- Concentration of \( A^+ = 2s_1 \)
- Concentration of \( X^{2-} = s_1 \)

2. **For \( AX \)**:
\[ AX \rightleftharpoons A^+ + X^- \]
Let the solubility of \( AX \) be \( s_2 \). Thus, at equilibrium:
- Concentration of \( A^+ = s_2 \)
- Concentration of \( X^- = s_2 \)

3. **For \( A_2X_3 \)**:
\[ A_2X_3 \rightleftharpoons 2A^+ + 3X^{2-} \]
Let the solubility of \( A_2X_3 \) be \( s_3 \). Thus, at equilibrium:
- Concentration of \( A^+ = 2s_3 \)
- Concentration of \( X^{2-} = 3s_3 \)

### Step 2: Write the expressions for \( K_{sp} \)

1. **For \( A_2X \)**:
\[ K_{sp} = [A^+]^2[X^{2-}] = (2s_1)^2(s_1) = 4s_1^3 \]

2. **For \( AX \)**:
\[ K_{sp} = [A^+][X^-] = (s_2)(s_2) = s_2^2 \]

3. **For \( A_2X_3 \)**:
\[ K_{sp} = [A^+]^2[X^{2-}]^3 = (2s_3)^2(3s_3)^3 = 4s_3^2 \cdot 27s_3^3 = 108s_3^5 \]

### Step 3: Solve for solubility in terms of \( K_{sp} \)

1. **For \( A_2X \)**:
\[ s_1 = \left(\frac{K_{sp}}{4}\right)^{\frac{1}{3}} \]

2. **For \( AX \)**:
\[ s_2 = \sqrt{K_{sp}} \]

3. **For \( A_2X_3 \)**:
\[ s_3 = \left(\frac{K_{sp}}{108}\right)^{\frac{1}{5}} \]

### Step 4: Compare the solubilities \( s_1, s_2, s_3 \)

To compare the values of \( s_1, s_2, \) and \( s_3 \):

- \( s_1 = \left(\frac{K_{sp}}{4}\right)^{\frac{1}{3}} \)
- \( s_2 = K_{sp}^{\frac{1}{2}} \)
- \( s_3 = \left(\frac{K_{sp}}{108}\right)^{\frac{1}{5}} \)

### Step 5: Determine the order of solubility

To find the order of solubility, we can analyze the powers of \( K_{sp} \):

- \( s_1 \) has a power of \( \frac{1}{3} \)
- \( s_2 \) has a power of \( \frac{1}{2} \)
- \( s_3 \) has a power of \( \frac{1}{5} \)

Since \( \frac{1}{2} > \frac{1}{3} > \frac{1}{5} \), we can conclude:
\[ s_2 > s_1 > s_3 \]

### Final Order of Solubility
Thus, the order of solubility is:
\[ AX > A_2X > A_2X_3 \]