If the salts `M_(2)X,QY_(2)`, and `PZ_(3)` have the same solubilities `(lt(4)/(27))`, their `K_(sp)` values are related

Options

A. `K_(sp) (M_(2)X) = K_(sp)(QY_(2)) gt K_(sp) (PZ_(3))`
B. `K_(sp)(M_(2)X) gt K_(sp) (QY_(2)) =K_(sp) (PZ_(3))`
C. `K_(sp) (M_(2)X) = K_(sp) (QY_(2)) = K_(sp) (PZ_(3))`
D. `K_(sp)(M_(2)X) gt K_(sp) (QY_(2)) gt K_(sp) (PZ_(3))`

Explanation

To solve the problem, we need to determine the relationship between the solubility product constants (Ksp) of the salts \( M_2X \), \( QY_2 \), and \( PZ_3 \) given that they have the same solubility \( S \) (which is \( \frac{4}{27} \)).

### Step-by-Step Solution:

1. **Identify the dissociation of each salt:**
- For \( M_2X \):
\[ M_2X \rightleftharpoons 2M^+ + X^{2-} \]
If the solubility is \( S_1 \), then:
- Concentration of \( M^+ = 2S_1 \)
- Concentration of \( X^{2-} = S_1 \)

- For \( QY_2 \):
\[ QY_2 \rightleftharpoons Q^{2+} + 2Y^{-} \]
If the solubility is \( S_2 \), then:
- Concentration of \( Q^{2+} = S_2 \)
- Concentration of \( Y^{-} = 2S_2 \)

- For \( PZ_3 \):
\[ PZ_3 \rightleftharpoons P^{3+} + 3Z^{-} \]
If the solubility is \( S_3 \), then:
- Concentration of \( P^{3+} = S_3 \)
- Concentration of \( Z^{-} = 3S_3 \)

2. **Write the Ksp expressions for each salt:**
- For \( M_2X \):
\[ K_{sp}(M_2X) = [M^+]^2[X^{2-}] = (2S_1)^2(S_1) = 4S_1^3 \]

- For \( QY_2 \):
\[ K_{sp}(QY_2) = [Q^{2+}][Y^{-}]^2 = (S_2)(2S_2)^2 = S_2 \cdot 4S_2^2 = 4S_2^3 \]

- For \( PZ_3 \):
\[ K_{sp}(PZ_3) = [P^{3+}][Z^{-}]^3 = (S_3)(3S_3)^3 = S_3 \cdot 27S_3^3 = 27S_3^4 \]

3. **Since the solubilities are the same:**
- Let \( S_1 = S_2 = S_3 = S \)
- Thus:
\[ K_{sp}(M_2X) = 4S^3 \]
\[ K_{sp}(QY_2) = 4S^3 \]
\[ K_{sp}(PZ_3) = 27S^4 \]

4. **Substituting the value of solubility \( S = \frac{4}{27} \):**
- Calculate \( K_{sp}(M_2X) \) and \( K_{sp}(QY_2) \):
\[ K_{sp}(M_2X) = K_{sp}(QY_2) = 4\left(\frac{4}{27}\right)^3 = 4 \cdot \frac{64}{19683} = \frac{256}{19683} \]

- Calculate \( K_{sp}(PZ_3) \):
\[ K_{sp}(PZ_3) = 27\left(\frac{4}{27}\right)^4 = 27 \cdot \frac{256}{531441} = \frac{6912}{531441} \]

5. **Compare the Ksp values:**
- Since \( K_{sp}(M_2X) = K_{sp}(QY_2) = \frac{256}{19683} \) and \( K_{sp}(PZ_3) = \frac{6912}{531441} \), we need to compare these values.
- It can be observed that \( K_{sp}(M_2X) \) and \( K_{sp}(QY_2) \) are greater than \( K_{sp}(PZ_3) \).

### Final Conclusion:
The relationship among the Ksp values is:
\[ K_{sp}(M_2X) = K_{sp}(QY_2) > K_{sp}(PZ_3) \]