At equilibrium :

Options

A. `E_("cell")^(@)=0,DeltaG^(@)=0`
B. `E_("cell")^(@)=0,DeltaG=0`
C. both are correct
D. none is correct

Answer

To solve the question regarding the conditions at equilibrium in electrochemistry, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship between ΔG and E_cell**:
- At equilibrium, the change in Gibbs free energy (ΔG) is zero. This is a fundamental principle in thermodynamics.
- The relationship between Gibbs free energy and cell potential (E_cell) is given by the equation:
\[ \Delta G = -nFE_{cell} \]
- Where:
- \( \Delta G \) = change in Gibbs free energy
- \( n \) = number of moles of electrons transferred in the reaction
- \( F \) = Faraday's constant (approximately 96485 C/mol)
- \( E_{cell} \) = cell potential

2. **Applying the Equilibrium Condition**:
- At equilibrium, since ΔG = 0, we can substitute this into our equation:
\[ 0 = -nFE_{cell} \]
- This implies that:
\[ E_{cell} = 0 \]
- Therefore, at equilibrium, the cell potential (E_cell) is also zero.

3. **Understanding the Reaction Quotient (Q) and Equilibrium Constant (K)**:
- The reaction quotient (Q) is defined as:
\[ Q = \frac{[\text{products}]^{\text{coefficients}}}{[\text{reactants}]^{\text{coefficients}}} \]
- At equilibrium, Q becomes the equilibrium constant (K). Thus, at equilibrium, we can say:
\[ \Delta G = \Delta G^0 + RT \ln K \]
- Since ΔG = 0 at equilibrium, we have:
\[ 0 = \Delta G^0 + RT \ln K \]
- Rearranging gives:
\[ \Delta G^0 = -RT \ln K \]
- This indicates that ΔG^0 is not necessarily zero; it depends on the value of K.

4. **Analyzing the Options**:
- The options provided in the question likely relate to whether ΔG^0 and E^0_cell are zero or not.
- We have established that at equilibrium:
- ΔG = 0
- E_cell = 0
- However, ΔG^0 and E^0_cell are standard values and are not zero in general.

5. **Conclusion**:
- Since both ΔG^0 and E^0_cell are not equal to zero, the correct conclusion is that none of the options provided are correct.

### Final Answer:
The correct answer is: **None is correct**.