The solubility product constant `(K_(sp))` of salts of types `MX, MX_(2)`, and `M_(3)X` at temperature `T` are `4.0 xx 10^(-8), 3.2 xx 10^(-14)`, and `2.7 xx 10^(-15)`, respectively. The solubilities 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 gt MX`
D. `MX gt M_(3)X gt MX_(2)`

Explanation

To determine the order of solubility of the salts `MX`, `MX2`, and `M3X` based on their solubility product constants `(K_sp)`, we can follow these steps:

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

1. For `MX`:
\[ MX \rightleftharpoons M^+ + X^- \]
The solubility product \( K_{sp} \) is given by:
\[ K_{sp} = [M^+][X^-] = s \cdot s = s^2 \]
where \( s \) is the solubility of `MX`.

2. For `MX2`:
\[ MX_2 \rightleftharpoons M^{2+} + 2X^- \]
The solubility product \( K_{sp} \) is given by:
\[ K_{sp} = [M^{2+}][X^-]^2 = s \cdot (2s)^2 = 4s^3 \]
where \( s \) is the solubility of `MX2`.

3. For `M3X`:
\[ M_3X \rightleftharpoons 3M^{3+} + X^- \]
The solubility product \( K_{sp} \) is given by:
\[ K_{sp} = [M^{3+}]^3[X^-] = (3s)^3 \cdot s = 27s^4 \]
where \( s \) is the solubility of `M3X`.

### Step 2: Relate solubility to \( K_{sp} \).

Now we can express the solubility \( s \) in terms of \( K_{sp} \) for each salt:

1. For `MX`:
\[ K_{sp} = s^2 \implies s = \sqrt{K_{sp}} = \sqrt{4.0 \times 10^{-8}} = 2.0 \times 10^{-4} \text{ mol/L} \]

2. For `MX2`:
\[ K_{sp} = 4s^3 \implies s = \left(\frac{K_{sp}}{4}\right)^{1/3} = \left(\frac{3.2 \times 10^{-14}}{4}\right)^{1/3} = \left(8.0 \times 10^{-15}\right)^{1/3} \approx 2.0 \times 10^{-5} \text{ mol/L} \]

3. For `M3X`:
\[ K_{sp} = 27s^4 \implies s = \left(\frac{K_{sp}}{27}\right)^{1/4} = \left(\frac{2.7 \times 10^{-15}}{27}\right)^{1/4} = \left(1.0 \times 10^{-16}\right)^{1/4} \approx 1.78 \times 10^{-4} \text{ mol/L} \]

### Step 3: Compare the solubilities.

Now we can compare the calculated solubilities:

- Solubility of `MX`: \( 2.0 \times 10^{-4} \) mol/L
- Solubility of `MX2`: \( 2.0 \times 10^{-5} \) mol/L
- Solubility of `M3X`: \( 1.78 \times 10^{-4} \) mol/L

### Step 4: Order the solubilities.

From the values calculated, we can rank the solubilities from highest to lowest:

1. `MX`: \( 2.0 \times 10^{-4} \) mol/L
2. `M3X`: \( 1.78 \times 10^{-4} \) mol/L
3. `MX2`: \( 2.0 \times 10^{-5} \) mol/L

### Final Answer:
The order of solubilities at temperature \( T \) is:
\[ \text{MX} > \text{M3X} > \text{MX2} \]