To determine the temperature at which the reaction took place and whether the reaction is spontaneous or nonspontaneous, we can use the Gibbs free energy equation:
[tex]\[ \Delta G = \Delta H - T \Delta S \][/tex]
### Step-by-Step Solution:
1. Given Data:
- [tex]\(\Delta H = 178.3 \,\text{kJ/mol}\)[/tex]
- [tex]\(\Delta S = 160.5 \,\text{J/(mol·K)}\)[/tex]
- [tex]\(\Delta G = 130.5 \,\text{kJ/mol}\)[/tex]
2. Convert [tex]\(\Delta S\)[/tex] into kJ/(mol·K):
- Since 1 kJ = 1000 J, we convert [tex]\(\Delta S\)[/tex] from J/(mol·K) to kJ/(mol·K).
[tex]\[ \Delta S = 160.5 \,\text{J/(mol·K)} = 0.1605 \,\text{kJ/(mol·K)} \][/tex]
3. Rearrange the Gibbs Free Energy equation to solve for temperature, [tex]\( T \)[/tex]:
[tex]\[ T = \frac{\Delta H - \Delta G}{\Delta S} \][/tex]
4. Substitute the values into the equation:
[tex]\[
T = \frac{178.3 \,\text{kJ/mol} - 130.5 \,\text{kJ/mol}}{0.1605 \,\text{kJ/(mol·K)}}
\][/tex]
5. Calculate the numerator:
[tex]\[ 178.3 - 130.5 = 47.8 \,\text{kJ/mol} \][/tex]
6. Divide by [tex]\(\Delta S\)[/tex]:
[tex]\[
T = \frac{47.8 \,\text{kJ/mol}}{0.1605 \,\text{kJ/(mol·K)}}
= 297.8 \,\text{K}
\][/tex]
7. Determine the spontaneity of the reaction:
- A reaction is spontaneous if [tex]\(\Delta G < 0\)[/tex], and nonspontaneous if [tex]\(\Delta G > 0\)[/tex].
- Given [tex]\(\Delta G = 130.5 \,\text{kJ/mol}\)[/tex], the reaction is nonspontaneous as [tex]\(\Delta G > 0\)[/tex].
### Conclusion:
The correct temperature at which the reaction took place is approximately [tex]\(297.8 \,\text{K}\)[/tex], and since [tex]\(\Delta G\)[/tex] is greater than zero, the reaction is nonspontaneous. Therefore, the correct statement is:
- [tex]\(297.8 \,\text{K}\)[/tex], nonspontaneous
