`K_(a)` for the reaction,
`Fe^(3+)(aq)+H_(2)O(l)hArr Fe(OH)^(2+)(aq) +H_(3)O^(o+)(aq)` is `6.5 xx 10^(-3)`, what is the maximum `pH` value which could be used so that at least `80%` of the total iron `(III)` in a dilute solution exsists as `Fe^(3+)`?

Options

A. `2`
B. `2.41`
C. `2.79`
D. `1.60`

Answer

To solve the problem, we need to determine the maximum pH value at which at least 80% of the total iron (III) exists as \( \text{Fe}^{3+} \) in a dilute solution. We are given the dissociation constant \( K_a \) for the hydrolysis reaction of \( \text{Fe}^{3+} \).

### Step-by-Step Solution:

1. **Write the hydrolysis reaction**:
\[ \text{Fe}^{3+}(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{Fe(OH)}^{2+}(aq) + \text{H}_3\text{O}^+(aq) \]

2. **Define the initial concentration**:
Let the initial concentration of \( \text{Fe}^{3+} \) be \( C \).

3. **Set up the equilibrium concentrations**:
At equilibrium:
- Concentration of \( \text{Fe}^{3+} \) = \( C(1 - \alpha) \)
- Concentration of \( \text{Fe(OH)}^{2+} \) = \( C\alpha \)
- Concentration of \( \text{H}_3\text{O}^+ \) = \( C\alpha \)

4. **Use the condition for 80% of \( \text{Fe}^{3+} \)**:
We need at least 80% of the total iron to exist as \( \text{Fe}^{3+} \):
\[ 1 - \alpha \geq 0.8 \implies \alpha \leq 0.2 \]

5. **Express \( K_a \)**:
The expression for \( K_a \) is given by:
\[ K_a = \frac{[\text{Fe(OH)}^{2+}][\text{H}_3\text{O}^+]}{[\text{Fe}^{3+}]} = \frac{C\alpha \cdot C\alpha}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

6. **Substitute the known values**:
Given \( K_a = 6.5 \times 10^{-3} \) and \( \alpha = 0.2 \):
\[ 6.5 \times 10^{-3} = \frac{C(0.2)^2}{1 - 0.2} = \frac{C(0.04)}{0.8} \]
Rearranging gives:
\[ C = \frac{6.5 \times 10^{-3} \times 0.8}{0.04} = 0.13 \, \text{mol/L} \]

7. **Calculate the concentration of \( \text{H}_3\text{O}^+ \)**:
\[ [\text{H}_3\text{O}^+] = C \alpha = 0.13 \times 0.2 = 0.026 \, \text{mol/L} \]

8. **Calculate the pH**:
\[ \text{pH} = -\log[\text{H}_3\text{O}^+] = -\log(0.026) \approx 1.585 \]

9. **Final answer**:
The maximum pH value is approximately \( 1.6 \).