Solubility product `(K_(sp))` of salts of types MX, `MX_(2)` and `MX_(3)` at temperature ‘T’ are `4.0xx10^(-8), 3.0xx10^(-14)` and `2.7xx10^(-15)` , respectively. Solubilities (in `mol dm^(3)`) of the salts at temperature ‘T’ are in the order

Options

A. `MX gt MX_(2) gt M_(3)X`
B. `M_(3)X gt MX_(2) gt MX`
C. `MX_(2) gt M_(3)X lt MX`
D. `MX gt M_(3)X gt MX_(2)`

Explanation

To solve the problem of determining the order of solubilities of the salts MX, MX₂, and MX₃ based on their solubility products (Ksp), we can follow these steps:

### Step 1: Write the dissociation equations and expressions for Ksp

1. **For MX:**
- Dissociation: \( MX \rightleftharpoons M^+ + X^- \)
- If the solubility is \( s \), then:
- \( [M^+] = s \)
- \( [X^-] = s \)
- Ksp expression:
\[ K_{sp} = [M^+][X^-] = s \cdot s = s^2 \]
- Given \( K_{sp} = 4.0 \times 10^{-8} \):
\[ s^2 = 4.0 \times 10^{-8} \implies s = \sqrt{4.0 \times 10^{-8}} = 2.0 \times 10^{-4} \, \text{mol dm}^{-3} \]

### Step 2: Calculate the solubility for MX₂

2. **For MX₂:**
- Dissociation: \( MX_2 \rightleftharpoons M^{2+} + 2X^- \)
- If the solubility is \( s \), then:
- \( [M^{2+}] = s \)
- \( [X^-] = 2s \)
- Ksp expression:
\[ K_{sp} = [M^{2+}][X^-]^2 = s \cdot (2s)^2 = s \cdot 4s^2 = 4s^3 \]
- Given \( K_{sp} = 3.0 \times 10^{-14} \):
\[ 4s^3 = 3.0 \times 10^{-14} \implies s^3 = \frac{3.0 \times 10^{-14}}{4} = 7.5 \times 10^{-15} \]
\[ s = \sqrt[3]{7.5 \times 10^{-15}} \approx 2.0 \times 10^{-5} \, \text{mol dm}^{-3} \]

### Step 3: Calculate the solubility for MX₃

3. **For MX₃:**
- Dissociation: \( MX_3 \rightleftharpoons M^{3+} + 3X^- \)
- If the solubility is \( s \), then:
- \( [M^{3+}] = s \)
- \( [X^-] = 3s \)
- Ksp expression:
\[ K_{sp} = [M^{3+}][X^-]^3 = s \cdot (3s)^3 = s \cdot 27s^3 = 27s^4 \]
- Given \( K_{sp} = 2.7 \times 10^{-15} \):
\[ 27s^4 = 2.7 \times 10^{-15} \implies s^4 = \frac{2.7 \times 10^{-15}}{27} = 1.0 \times 10^{-16} \]
\[ s = \sqrt[4]{1.0 \times 10^{-16}} \approx 1.0 \times 10^{-4} \, \text{mol dm}^{-3} \]

### Step 4: Compare the solubilities

Now we have the solubilities:
- \( s_{MX} = 2.0 \times 10^{-4} \, \text{mol dm}^{-3} \)
- \( s_{MX_2} = 2.0 \times 10^{-5} \, \text{mol dm}^{-3} \)
- \( s_{MX_3} = 1.0 \times 10^{-4} \, \text{mol dm}^{-3} \)

### Step 5: Order of solubility

Arranging these values in decreasing order:
1. \( MX \) (2.0 × 10⁻⁴)
2. \( MX_3 \) (1.0 × 10⁻⁴)
3. \( MX_2 \) (2.0 × 10⁻⁵)

### Final Answer
The order of solubilities at temperature T is:
\[ MX > MX_3 > MX_2 \]