Calculate `DeltaG^(@)` (kJ/mol) at `127^(@)C` for a reaction with `K_("equilibrium")=10^(5)` :

Options

A. `-38.294`
B. `-16.628`
C. `-9.16`
D. None of these

Explanation

To calculate the standard Gibbs free energy change (ΔG°) at 127°C for a reaction with an equilibrium constant (K_equilibrium) of 10^5, we can follow these steps:

### Step-by-Step Solution:

1. **Convert Temperature to Kelvin**:
\[ T(K) = 127°C + 273 = 400 K \]

2. **Use the Gibbs Free Energy Equation**:
At equilibrium, the change in Gibbs free energy (ΔG) is zero. Therefore, we can use the equation:
\[ ΔG° = -RT \ln K \]
where:
- R = 8.314 J/(mol·K) (universal gas constant)
- T = 400 K (temperature in Kelvin)
- K = 10^5 (equilibrium constant)

3. **Convert Natural Logarithm to Base 10 Logarithm**:
We can convert the natural logarithm (ln) to base 10 logarithm (log) using the relationship:
\[ \ln K = 2.303 \cdot \log K \]
Thus, we can rewrite the equation as:
\[ ΔG° = -RT \cdot 2.303 \cdot \log K \]

4. **Calculate log K**:
Since K = 10^5, we have:
\[ \log K = 5 \]

5. **Substitute Values into the Equation**:
Now substitute the values into the equation:
\[ ΔG° = - (8.314 \, \text{J/(mol·K)}) \cdot (400 \, \text{K}) \cdot (2.303) \cdot (5) \]

6. **Perform the Calculation**:
First, calculate the product:
\[ ΔG° = - (8.314 \cdot 400 \cdot 2.303 \cdot 5) \]
\[ = - (8.314 \cdot 400 \cdot 11.515) \]
\[ = - (38294.2 \, \text{J/mol}) \]

7. **Convert Joules to Kilojoules**:
Since the answer is required in kJ/mol, convert Joules to Kilojoules:
\[ ΔG° = - \frac{38294.2 \, \text{J/mol}}{1000} = -38.294 \, \text{kJ/mol} \]

### Final Answer:
\[ ΔG° = -38.294 \, \text{kJ/mol} \]