In a Galvanic cell involving the redox reaction:
`A^(2+)+ B_((s)) Leftrightarrow A_((s))+B^(2+)`
The `E_("cell")` will not be equal to `E_("cell")^(0)` when

Options

A. `[A^(2+)]=[B^(2+]=1M`
B. `[A^(2+)]=[B^(2+)]=0.1M`
C. `[A^(2+)]=[B^(2+)]=0.5M`
D. None of the above

Answer

To determine when the cell potential \( E_{\text{cell}} \) is not equal to the standard cell potential \( E_{\text{cell}}^{0} \) for the given redox reaction:

\[ A^{2+} + B_{(s)} \leftrightarrow A_{(s)} + B^{2+} \]

we can use the Nernst equation, which relates the cell potential to the concentrations of the reactants and products.

### Step 1: Write the Nernst Equation
The Nernst equation is given by:

\[ E_{\text{cell}} = E_{\text{cell}}^{0} - \frac{0.0591}{n} \log \left( \frac{[B^{2+}]}{[A^{2+}]} \right) \]

where:
- \( E_{\text{cell}} \) is the cell potential under non-standard conditions,
- \( E_{\text{cell}}^{0} \) is the standard cell potential,
- \( n \) is the number of moles of electrons transferred in the balanced equation,
- \([B^{2+}]\) and \([A^{2+}]\) are the concentrations of the products and reactants, respectively.

### Step 2: Identify Conditions for \( E_{\text{cell}} \neq E_{\text{cell}}^{0} \)
For \( E_{\text{cell}} \) to be equal to \( E_{\text{cell}}^{0} \), the term:

\[ \frac{0.0591}{n} \log \left( \frac{[B^{2+}]}{[A^{2+}]} \right) \]

must equal zero. This occurs when:

\[ \log \left( \frac{[B^{2+}]}{[A^{2+}]} \right) = 0 \]

This is true when:

\[ \frac{[B^{2+}]}{[A^{2+}]} = 1 \quad \Rightarrow \quad [B^{2+}] = [A^{2+}] \]

### Step 3: Conclusion
Thus, \( E_{\text{cell}} \) will not be equal to \( E_{\text{cell}}^{0} \) when the concentrations of \( A^{2+} \) and \( B^{2+} \) are not equal. Therefore, the answer is that \( E_{\text{cell}} \) will not equal \( E_{\text{cell}}^{0} \ when \) the concentrations of the reactants and products differ from each other.

### Final Answer
The cell potential \( E_{\text{cell}} \) will not be equal to \( E_{\text{cell}}^{0} \) when the concentrations of \( A^{2+} \) and \( B^{2+} \) are not equal.

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