The correct relationship between free energy change in a reaction and the corresponding equilibrium constant `K_(c)` is:

Options

A. `DeltaG=RTlnK_(c)`
B. `-DeltaG=RTlnK_(c)`
C. `DeltaG^(@)=RTlnK_(c)`
D. `-DeltaG^(@)=RTlnK_(c)`

Explanation

To find the correct relationship between the free energy change in a reaction and the corresponding equilibrium constant \( K_c \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Free Energy Change (\( \Delta G^\circ \))**:
- The standard Gibbs free energy change (\( \Delta G^\circ \)) is a measure of the spontaneity of a reaction. A negative value indicates that the reaction can occur spontaneously under standard conditions.

2. **Equilibrium Constant (\( K_c \))**:
- The equilibrium constant \( K_c \) is a dimensionless number that expresses the ratio of the concentrations of products to reactants at equilibrium.

3. **Relationship Between \( \Delta G^\circ \) and \( K_c \)**:
- The relationship between the standard Gibbs free energy change and the equilibrium constant is given by the equation:
\[ \Delta G^\circ = -RT \ln K_c \]
- Here, \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/(mol·K)} \)), and \( T \) is the absolute temperature in Kelvin.

4. **Rearranging the Equation**:
- This equation can also be rearranged to express \( K_c \) in terms of \( \Delta G^\circ \):
\[ -\Delta G^\circ = RT \ln K_c \]

5. **Conclusion**:
- Therefore, the correct relationship between the free energy change in a reaction and the corresponding equilibrium constant \( K_c \) is:
\[ \Delta G^\circ = -RT \ln K_c \]
- This indicates that if \( K_c > 1 \), \( \Delta G^\circ \) will be negative, suggesting that the reaction is spontaneous in the forward direction.

### Final Answer:
The correct relationship is \( \Delta G^\circ = -RT \ln K_c \).

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