If the `E_("cell")^(@)` for a given reaction has a negative value, then which of the following gives the correct relationships for the values of `DeltaG^(0)` and `K_(eq)`-

Options

A. `DeltaG^(@) gt 0, K_(eq) lt 1`
B. `DeltaG^(@) gt0, K_(eq) gt 1`
C. `DeltaG^(@) lt 0, K_(eq) gt 1`
D. `DeltaG^(@) lt 0, K_(eq) lt 1`

Answer

To solve the question regarding the relationship between \( \Delta G^{0} \) and \( K_{eq} \) when \( E_{cell}^{0} \) is negative, we can follow these steps:

### Step 1: Understand the relationship between \( E_{cell}^{0} \) and \( \Delta G^{0} \)
We start with the equation that relates the standard cell potential to the Gibbs free energy change:
\[ \Delta G^{0} = -nFE_{cell}^{0} \]
where:
- \( n \) = number of moles of electrons transferred in the reaction
- \( F \) = Faraday's constant (approximately 96485 C/mol)
- \( E_{cell}^{0} \) = standard cell potential

### Step 2: Analyze the case when \( E_{cell}^{0} \) is negative
Given that \( E_{cell}^{0} < 0 \):
- The equation becomes:
\[ \Delta G^{0} = -nF(E_{cell}^{0}) \quad \text{(where \( E_{cell}^{0} < 0 \))} \]
- Since \( E_{cell}^{0} \) is negative, \( -E_{cell}^{0} \) is positive, leading to:
\[ \Delta G^{0} > 0 \]
This indicates that the reaction is non-spontaneous under standard conditions.

### Step 3: Relate \( \Delta G^{0} \) to \( K_{eq} \)
The relationship between the Gibbs free energy change and the equilibrium constant is given by:
\[ \Delta G^{0} = -RT \ln K_{eq} \]
where:
- \( R \) = universal gas constant (approximately 8.314 J/(mol·K))
- \( T \) = temperature in Kelvin

### Step 4: Analyze the implications for \( K_{eq} \)
Since we have established that \( \Delta G^{0} > 0 \):
- From the equation \( \Delta G^{0} = -RT \ln K_{eq} \), if \( \Delta G^{0} \) is positive, then:
\[ -RT \ln K_{eq} > 0 \]
This implies that:
\[ \ln K_{eq} < 0 \]
Thus, \( K_{eq} < 1 \).

### Conclusion
From the analysis, we conclude:
- \( \Delta G^{0} > 0 \)
- \( K_{eq} < 1 \)

### Final Answer
If \( E_{cell}^{0} \) is negative, then \( \Delta G^{0} > 0 \) and \( K_{eq} < 1 \).