To solve for the entropy change [tex]\(\Delta S\)[/tex] of the reaction, we will use the given Gibbs free energy change [tex]\(\Delta G\)[/tex], enthalpy change [tex]\(\Delta H\)[/tex], and the temperature [tex]\(T\)[/tex] along with the formula [tex]\(\Delta G = \Delta H - T\Delta S\)[/tex],
Here are the given values:
[tex]\[
\Delta G = 130.5 \, \text{kJ/mol}
\][/tex]
[tex]\[
\Delta H = 178.3 \, \text{kJ/mol}
\][/tex]
[tex]\[
T = 25.0^\circ \text{C} = 25.0 + 273.15 \, \text{K} = 298.15 \, \text{K}
\][/tex]
We need to rearrange the formula [tex]\(\Delta G = \Delta H - T\Delta S\)[/tex] to solve for [tex]\(\Delta S\)[/tex]:
[tex]\[
\Delta S = \frac{\Delta H - \Delta G}{T}
\][/tex]
First, substitute the given values into the equation. Since [tex]\(\Delta H\)[/tex] and [tex]\(\Delta G\)[/tex] are given in kJ/mol, we need to convert the result into J/(mol·K) by multiplying by 1000.
[tex]\[
\Delta S = \frac{(178.3 - 130.5) \, \text{kJ/mol}}{298.15 \, \text{K}} \cdot 1000 \, \frac{\text{J}}{\text{kJ}}
\][/tex]
Calculate the difference in enthalpy and free energy:
[tex]\[
178.3 \, \text{kJ/mol} - 130.5 \, \text{kJ/mol} = 47.8 \, \text{kJ/mol}
\][/tex]
Now, substitute this value into the equation:
[tex]\[
\Delta S = \frac{47.8 \, \text{kJ/mol}}{298.15 \, \text{K}} \cdot 1000 \, \frac{\text{J}}{\text{kJ}}
\][/tex]
[tex]\[
\Delta S = \frac{47800 \, \text{J/mol}}{298.15 \, \text{K}}
\][/tex]
Finally, we perform the division to find [tex]\(\Delta S\)[/tex]:
[tex]\[
\Delta S \approx 160.32 \, \text{J/(mol·K)}
\][/tex]
Therefore, the entropy change [tex]\(\Delta S\)[/tex] of the reaction is approximately [tex]\(160.3 \, \text{J/(mol·K)}\)[/tex].
Hence, the correct answer is:
[tex]\[\boxed{160.3 \, \text{J/(mol·K)}}\][/tex]
