The solubility product of different sparingly soluble salts are
1. `XY = 4 xx10^(-20)`
2. `X_2Y = 3.2 xx10^(-11)`
3. `XY_3 = 2.7 xx10^(-31)`
The increasing order of solubility is

Options

A. 1,3,2
B. 2,1,3
C. 1,2,3
D. 3,1,2

Answer

To determine the increasing order of solubility for the given sparingly soluble salts based on their solubility products (Ksp), we will analyze each salt step by step.

### Step 1: Analyze the first salt, XY
- The dissociation of salt XY can be represented as:
\[ XY \rightleftharpoons X^+ + Y^- \]
- Let the solubility of XY be \( S \). Therefore, at equilibrium:
- \([X^+] = S\)
- \([Y^-] = S\)
- The solubility product \( K_{sp} \) for XY is given by:
\[ K_{sp} = [X^+][Y^-] = S \cdot S = S^2 \]
- Given \( K_{sp} = 4 \times 10^{-20} \):
\[ S^2 = 4 \times 10^{-20} \]
- Solving for \( S \):
\[ S = \sqrt{4 \times 10^{-20}} = 2 \times 10^{-10} \]

### Step 2: Analyze the second salt, \( X_2Y \)
- The dissociation of salt \( X_2Y \) can be represented as:
\[ X_2Y \rightleftharpoons 2X^+ + Y^- \]
- Let the solubility of \( X_2Y \) be \( s \). Therefore, at equilibrium:
- \([X^+] = 2s\)
- \([Y^-] = s\)
- The solubility product \( K_{sp} \) for \( X_2Y \) is given by:
\[ K_{sp} = [X^+]^2[Y^-] = (2s)^2 \cdot s = 4s^3 \]
- Given \( K_{sp} = 3.2 \times 10^{-11} \):
\[ 4s^3 = 3.2 \times 10^{-11} \]
- Solving for \( s \):
\[ s^3 = \frac{3.2 \times 10^{-11}}{4} = 8 \times 10^{-12} \]
\[ s = \sqrt[3]{8 \times 10^{-12}} = 2 \times 10^{-4} \]

### Step 3: Analyze the third salt, \( XY_3 \)
- The dissociation of salt \( XY_3 \) can be represented as:
\[ XY_3 \rightleftharpoons X^+ + 3Y^- \]
- Let the solubility of \( XY_3 \) be \( t \). Therefore, at equilibrium:
- \([X^+] = t\)
- \([Y^-] = 3t\)
- The solubility product \( K_{sp} \) for \( XY_3 \) is given by:
\[ K_{sp} = [X^+][Y^-]^3 = t \cdot (3t)^3 = 27t^4 \]
- Given \( K_{sp} = 2.7 \times 10^{-31} \):
\[ 27t^4 = 2.7 \times 10^{-31} \]
- Solving for \( t \):
\[ t^4 = \frac{2.7 \times 10^{-31}}{27} = 1 \times 10^{-32} \]
\[ t = \sqrt[4]{1 \times 10^{-32}} = 10^{-8} \]

### Step 4: Compare the solubilities
- The calculated solubilities are:
- For \( XY \): \( S = 2 \times 10^{-10} \)
- For \( X_2Y \): \( s = 2 \times 10^{-4} \)
- For \( XY_3 \): \( t = 10^{-8} \)

### Step 5: Determine the increasing order of solubility
- To find the increasing order of solubility:
- \( XY \) has the lowest solubility: \( 2 \times 10^{-10} \)
- \( XY_3 \) has a higher solubility: \( 10^{-8} \)
- \( X_2Y \) has the highest solubility: \( 2 \times 10^{-4} \)

Thus, the increasing order of solubility is:
\[ XY < XY_3 < X_2Y \]

### Final Answer:
The increasing order of solubility is:
\[ XY < XY_3 < X_2Y \]