Zirconium phosphate [Zr$$_3$$(PO$$_4$$)$$_4$$] dissociates into three zirconium cations of charge +4 and four phosphate anions of charge $$-3$$. If molar solubility of zirconium phosphate is denoted by s and its solubility product by K$$_{sp}$$ then which of the following relationship between s and K$$_{sp}$$ is correct?

The dissociation reaction for zirconium phosphate, Zr₃(PO₄)₄, is given by:

$$\text{Zr}_3(\text{PO}_4)_4(s) \rightarrow 3\text{Zr}^{4+}(aq) + 4\text{PO}_4^{3-}(aq)$$

Let the molar solubility of zirconium phosphate be denoted by $$s$$ mol/L. This means that when $$s$$ moles of Zr₃(PO₄)₄ dissolve per liter of solution, the concentration of Zr⁴⁺ ions produced is $$3s$$ mol/L (since each formula unit produces 3 zirconium ions), and the concentration of PO₄³⁻ ions produced is $$4s$$ mol/L (since each formula unit produces 4 phosphate ions).

The solubility product constant, $$K_{sp}$$, is defined as the product of the concentrations of the ions raised to their respective stoichiometric coefficients. Therefore, for this reaction:

$$K_{sp} = [\text{Zr}^{4+}]^3 \times [\text{PO}_4^{3-}]^4$$

Substituting the concentrations in terms of $$s$$:

$$K_{sp} = (3s)^3 \times (4s)^4$$

Now, compute each part separately. First, $$(3s)^3$$:

$$(3s)^3 = 3^3 \times s^3 = 27s^3$$

Next, $$(4s)^4$$:

$$(4s)^4 = 4^4 \times s^4 = 256s^4$$

Now, multiply these together:

$$K_{sp} = 27s^3 \times 256s^4$$

Combine the constants and the powers of $$s$$. The constants are $$27 \times 256$$, and the powers of $$s$$ are $$s^3 \times s^4 = s^{3+4} = s^7$$. So:

$$K_{sp} = (27 \times 256) \times s^7$$

Calculate $$27 \times 256$$. Break it down: $$20 \times 256 = 5120$$, and $$7 \times 256 = 1792$$. Adding these gives:

$$5120 + 1792 = 6912$$

Thus:

$$K_{sp} = 6912 \cdot s^7$$

To find the relationship between $$s$$ and $$K_{sp}$$, solve for $$s$$:

$$s^7 = \frac{K_{sp}}{6912}$$

Taking the seventh root of both sides:

$$s = \left( \frac{K_{sp}}{6912} \right)^{1/7}$$

Now, compare this with the given options:

• Option A: $$S = \{K_{sp}/6912\}^7$$ → This is $$s = \left( \frac{K_{sp}}{6912} \right)^7$$, which is incorrect.
• Option B: $$S = \{K_{sp}/144\}^{1/7}$$ → The denominator is 144 instead of 6912, so incorrect.
• Option C: $$S = \{K_{sp}/(6912)^{1/7}\}$$ → This is $$s = \frac{K_{sp}}{6912^{1/7}}$$, which is not equivalent.
• Option D: $$S = (K_{sp}/6912)^{1/7}$$ → This matches exactly: $$s = \left( \frac{K_{sp}}{6912} \right)^{1/7}$$.

Hence, the correct answer is Option D.