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 \( E_{\text{cell}} \) will not be 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 will analyze the conditions under which the Nernst equation applies.

### Step 1: Write the Nernst Equation
The Nernst equation relates the cell potential \( E_{\text{cell}} \) to the standard cell potential \( E_{\text{cell}}^{0} \) and the reaction quotient \( Q \):

\[ E_{\text{cell}} = E_{\text{cell}}^{0} - \frac{0.0591}{n} \log Q \]

where \( n \) is the number of moles of electrons transferred in the redox reaction.

### Step 2: Identify the Reaction Quotient \( Q \)
For the given reaction, the reaction quotient \( Q \) is defined as:

\[ Q = \frac{[B^{2+}]}{[A^{2+}]} \]

### Step 3: Determine the Value of \( Q \)
The values of \( [A^{2+}] \) and \( [B^{2+}] \) will affect \( Q \). If both concentrations are equal, \( Q \) will equal 1, and the logarithm of 1 is 0.

### Step 4: Analyze Each Condition
1. **Condition 1: \( [A^{2+}] = [B^{2+}] = 1 \, \text{M} \)**
- Here, \( Q = \frac{1}{1} = 1 \)
- Thus, \( E_{\text{cell}} = E_{\text{cell}}^{0} - \frac{0.0591}{n} \cdot 0 = E_{\text{cell}}^{0} \)

2. **Condition 2: \( [A^{2+}] = [B^{2+}] = 0.1 \, \text{M} \)**
- Again, \( Q = \frac{0.1}{0.1} = 1 \)
- Thus, \( E_{\text{cell}} = E_{\text{cell}}^{0} - \frac{0.0591}{n} \cdot 0 = E_{\text{cell}}^{0} \)

3. **Condition 3: \( [A^{2+}] = [B^{2+}] = 0.5 \, \text{M} \)**
- Similarly, \( Q = \frac{0.5}{0.5} = 1 \)
- Thus, \( E_{\text{cell}} = E_{\text{cell}}^{0} - \frac{0.0591}{n} \cdot 0 = E_{\text{cell}}^{0} \)

### Conclusion
In all three conditions, \( E_{\text{cell}} \) is equal to \( E_{\text{cell}}^{0} \). Therefore, the correct answer to the question is:

**None of these**