The solubility product of iron (III) hydroxide is `1.6xx10^(-19)`. If X is the solubility of iron (III) hydroxide, which one of the following expression can be used to calculate X ?

Options

A. `K_(sp)=X^(4)`
B. `K_(sp)=9X^(4)`
C. `K_(sp)=27 X^(3)`
D. `K_(sp)=27X^(4)`

Explanation

To solve the problem, we need to determine the expression that can be used to calculate the solubility (X) of iron (III) hydroxide (Fe(OH)₃) based on its solubility product (Ksp).

### Step-by-Step Solution:

1. **Understanding the Dissociation of Iron (III) Hydroxide:**
Iron (III) hydroxide, Fe(OH)₃, dissociates in water according to the following equation:
\[ \text{Fe(OH)}_3 (s) \rightleftharpoons \text{Fe}^{3+} (aq) + 3 \text{OH}^- (aq) \]

2. **Defining the Solubility (X):**
Let the solubility of Fe(OH)₃ be represented by \(X\). When Fe(OH)₃ dissolves:
- The concentration of \(\text{Fe}^{3+}\) ions will be \(X\).
- The concentration of \(\text{OH}^-\) ions will be \(3X\) (since three hydroxide ions are produced for every one formula unit of Fe(OH)₃).

3. **Writing the Expression for the Solubility Product (Ksp):**
The solubility product (Ksp) is defined as:
\[ K_{sp} = [\text{Fe}^{3+}][\text{OH}^-]^3 \]
Substituting the concentrations in terms of \(X\):
\[ K_{sp} = (X)(3X)^3 \]

4. **Calculating the Expression:**
Expanding the expression:
\[ K_{sp} = X \cdot (27X^3) = 27X^4 \]

5. **Setting Up the Equation:**
Given that the solubility product \(K_{sp}\) of iron (III) hydroxide is \(1.6 \times 10^{-19}\), we can set up the equation:
\[ 27X^4 = 1.6 \times 10^{-19} \]

6. **Solving for X:**
To find \(X\), we rearrange the equation:
\[ X^4 = \frac{1.6 \times 10^{-19}}{27} \]
Then, take the fourth root to find \(X\):
\[ X = \left(\frac{1.6 \times 10^{-19}}{27}\right)^{1/4} \]

### Final Expression:
Thus, the expression that can be used to calculate the solubility \(X\) of iron (III) hydroxide is:
\[ X = \left(\frac{1.6 \times 10^{-19}}{27}\right)^{1/4} \]