Consider the following equations for a cell reaction
`{:(A+Biff C + D,E^(o)=x" volt, K"_(eq)="k"_(1)),(2A+2Biff2C+2D,E^(o)=y" volt, K"_(eq)="k"_(2)):}`
then which of these is the correct relation ?

Options

A. `x = y , K_(1) = K_(2)`
B. `x = 2y, K_(1) = 2K_(2)`
C. `x=y, K_(1)^(2)= K_(2)`
D. `x^(2)=y, k_(1)^(2)=K_(2)`

Answer

To solve the problem, we need to analyze the two given cell reactions and their corresponding standard cell potentials (E°) and equilibrium constants (K_eq).

### Step-by-Step Solution:

1. **Identify the Reactions**:
- The first reaction is:
\[ A + B \rightleftharpoons C + D \]
with \( E^\circ = x \) volts and \( K_{eq} = k_1 \).
- The second reaction is:
\[ 2A + 2B \rightleftharpoons 2C + 2D \]
with \( E^\circ = y \) volts and \( K_{eq} = k_2 \).

2. **Understand the Relationship Between the Reactions**:
- The second reaction is simply the first reaction multiplied by 2. This means that the stoichiometry of the second reaction is double that of the first reaction.

3. **Standard Cell Potential (E°)**:
- The standard cell potential (E°) is independent of the stoichiometry of the reaction. Therefore, for both reactions, we have:
\[ E^\circ_{cell1} = E^\circ_{cell2} \]
- This implies:
\[ x = y \]

4. **Equilibrium Constant (K_eq)**:
- The equilibrium constant (K_eq) is dependent on the stoichiometry of the reaction. When a reaction is multiplied by a factor, the equilibrium constant is raised to the power of that factor.
- Since the second reaction is the first reaction multiplied by 2, we have:
\[ K_{eq,2} = (K_{eq,1})^2 \]
- This translates to:
\[ k_2 = k_1^2 \]

5. **Final Relations**:
- From the above analysis, we conclude:
\[ x = y \]
\[ k_2 = k_1^2 \]

### Conclusion:
The correct relation between the equilibrium constants and the standard cell potentials for the given reactions is:
- \( x = y \) (E° values are equal)
- \( k_2 = k_1^2 \) (Equilibrium constants relate as the square of the first)