`E_(Al^(3+)//Al)^(*)=-1.66V` and `K_(SP)` of `Al(OH)_(3)=1.0xx10^(-33)` . Reduction potential of the above couple at `pH =14` is `:`

Options

A. `-2.31 V`
B. `+2.31`
C. `-1.01V`
D. `+1.01V`

Explanation

To find the reduction potential of the Al³⁺/Al couple at pH 14, we will follow these steps:

### Step 1: Understand the Ksp of Al(OH)₃
The solubility product (Ksp) for Al(OH)₃ can be expressed as:
\[ K_{sp} = [Al^{3+}][OH^-]^3 \]

### Step 2: Calculate the concentration of OH⁻ at pH 14
At pH 14, we can find the pOH:
\[ pOH = 14 - pH = 14 - 14 = 0 \]
Thus, we can calculate the concentration of hydroxide ions:
\[ [OH^-] = 10^{-pOH} = 10^{0} = 1 \, \text{M} \]

### Step 3: Substitute into the Ksp expression
Now we substitute the concentration of OH⁻ into the Ksp expression:
\[ K_{sp} = [Al^{3+}][1]^3 = [Al^{3+}] \]
Given that \( K_{sp} = 1.0 \times 10^{-33} \):
\[ 1.0 \times 10^{-33} = [Al^{3+}] \]
Thus, the concentration of Al³⁺ is:
\[ [Al^{3+}] = 1.0 \times 10^{-33} \, \text{M} \]

### Step 4: Use the Nernst equation
The Nernst equation is given by:
\[ E = E^0 - \frac{0.059}{n} \log \left( \frac{1}{[Al^{3+}]} \right) \]
Where:
- \( E^0 \) is the standard reduction potential, which is given as -1.66 V.
- \( n \) is the number of electrons transferred in the half-reaction. For Al³⁺ to Al, \( n = 3 \).

### Step 5: Substitute values into the Nernst equation
Now we substitute the known values into the Nernst equation:
\[ E = -1.66 - \frac{0.059}{3} \log \left( \frac{1}{1.0 \times 10^{-33}} \right) \]

### Step 6: Simplify the logarithm
Using the property of logarithms:
\[ \log \left( \frac{1}{1.0 \times 10^{-33}} \right) = -\log(1.0 \times 10^{-33}) = 33 \]
So we have:
\[ E = -1.66 - \frac{0.059}{3} \times 33 \]

### Step 7: Calculate the final value
Calculating the second term:
\[ \frac{0.059}{3} \times 33 = 0.059 \times 11 = 0.649 \]
Now substituting back:
\[ E = -1.66 - 0.649 = -2.309 \, \text{V} \]

### Final Answer
Thus, the reduction potential of the Al³⁺/Al couple at pH 14 is approximately:
\[ E \approx -2.31 \, \text{V} \]