The solubility of different springly soluble salts are given as under :
`{:(S. No,"Formula Type","Solubility product"),((1),"AB" ,4.0xx10^(-20)),((2),A_(2)B,3.2xx10^(-11)),((3),AB_(3),2.7xx10^(-31)):}`
The correct 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 follow these steps:

### Step 1: Write down the given solubility products
We have the following salts and their respective Ksp values:
1. For AB: Ksp = 4.0 x 10^(-20)
2. For A2B: Ksp = 3.2 x 10^(-11)
3. For AB3: Ksp = 2.7 x 10^(-31)

### Step 2: Calculate the solubility for each salt

#### For AB:
- The dissociation can be represented as:
\[ AB \rightleftharpoons A^+ + B^- \]
- Let the solubility of AB be \( S \). At equilibrium:
\[ [A^+] = S \quad \text{and} \quad [B^-] = S \]
- The Ksp expression is:
\[ Ksp = [A^+][B^-] = S \cdot S = S^2 \]
- Given \( Ksp = 4.0 \times 10^{-20} \):
\[ S^2 = 4.0 \times 10^{-20} \implies S = \sqrt{4.0 \times 10^{-20}} = 2.0 \times 10^{-10} \]

#### For A2B:
- The dissociation can be represented as:
\[ A2B \rightleftharpoons 2A^+ + B^{2-} \]
- Let the solubility of A2B be \( S \). At equilibrium:
\[ [A^+] = 2S \quad \text{and} \quad [B^{2-}] = S \]
- The Ksp expression is:
\[ Ksp = [A^+]^2[B^{2-}] = (2S)^2 \cdot S = 4S^3 \]
- Given \( Ksp = 3.2 \times 10^{-11} \):
\[ 4S^3 = 3.2 \times 10^{-11} \implies S^3 = \frac{3.2 \times 10^{-11}}{4} = 8.0 \times 10^{-12} \implies S = \sqrt[3]{8.0 \times 10^{-12}} = 2.0 \times 10^{-4} \]

#### For AB3:
- The dissociation can be represented as:
\[ AB3 \rightleftharpoons A^{3+} + 3B^- \]
- Let the solubility of AB3 be \( S \). At equilibrium:
\[ [A^{3+}] = S \quad \text{and} \quad [B^-] = 3S \]
- The Ksp expression is:
\[ Ksp = [A^{3+}][B^-]^3 = S \cdot (3S)^3 = 27S^4 \]
- Given \( Ksp = 2.7 \times 10^{-31} \):
\[ 27S^4 = 2.7 \times 10^{-31} \implies S^4 = \frac{2.7 \times 10^{-31}}{27} = 1.0 \times 10^{-32} \implies S = \sqrt[4]{1.0 \times 10^{-32}} = 1.0 \times 10^{-8} \]

### Step 3: Compare the solubilities
Now we have the solubilities:
1. \( S_{AB} = 2.0 \times 10^{-10} \)
2. \( S_{A2B} = 2.0 \times 10^{-4} \)
3. \( S_{AB3} = 1.0 \times 10^{-8} \)

### Step 4: Arrange in increasing order
To find the increasing order of solubility:
- \( S_{AB} = 2.0 \times 10^{-10} \) (least solubility)
- \( S_{AB3} = 1.0 \times 10^{-8} \)
- \( S_{A2B} = 2.0 \times 10^{-4} \) (most solubility)

Thus, the increasing order of solubility is:
1. AB (2.0 x 10^(-10))
2. AB3 (1.0 x 10^(-8))
3. A2B (2.0 x 10^(-4))

### Final Answer
The correct increasing order of solubility is: **AB < AB3 < A2B**

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