To solve the given problem, we will calculate the Gibbs free energy change ([tex]\(\Delta G\)[/tex]) using the provided values for enthalpy change ([tex]\(\Delta H\)[/tex]), entropy change ([tex]\(\Delta S\)[/tex]), and temperature ([tex]\(T\)[/tex]).
We are given the following data:
[tex]\[
\Delta H = -495.2 \ \text{kJ} \\
\Delta S = -139 \ \text{J/K} \\
T = 298 \ \text{K}
\][/tex]
We need to use the formula:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
### Step-by-Step Solution
1. Convert [tex]\(\Delta S\)[/tex] from J/K to kJ/K:
Recall that [tex]\(1 \ \text{kJ} = 1000 \ \text{J}\)[/tex]. Hence, to convert [tex]\(\Delta S\)[/tex] to kJ/K, we divide by 1000:
[tex]\[
\Delta S = -139 \ \text{J/K} \times \frac{1 \ \text{kJ}}{1000 \ \text{J}} = -0.139 \ \text{kJ/K}
\][/tex]
2. Calculate [tex]\(\Delta G\)[/tex]:
Using the formula [tex]\(\Delta G = \Delta H - T \Delta S\)[/tex]:
[tex]\[
\Delta G = -495.2 \ \text{kJ} - (298 \ \text{K} \times -0.139 \ \text{kJ/K})
\][/tex]
3. Simplify the expression:
First, calculate the product [tex]\(T \Delta S\)[/tex]:
[tex]\[
T \Delta S = 298 \ \text{K} \times -0.139 \ \text{kJ/K} = -41.422 \ \text{kJ}
\][/tex]
Now, substitute this back into the expression for [tex]\(\Delta G\)[/tex]:
[tex]\[
\Delta G = -495.2 \ \text{kJ} - (-41.422 \ \text{kJ}) = -495.2 \ \text{kJ} + 41.422 \ \text{kJ} = -453.778 \ \text{kJ}
\][/tex]
4. Determine spontaneity:
A reaction is spontaneous if [tex]\(\Delta G\)[/tex] is less than 0. In this case:
[tex]\[
\Delta G = -453.778 \ \text{kJ}
\][/tex]
Since [tex]\(\Delta G\)[/tex] is negative, the reaction is spontaneous.
### Conclusion:
- [tex]\(\Delta S\)[/tex] after conversion: [tex]\(-0.139 \ \text{kJ/K}\)[/tex]
- [tex]\(\Delta G\)[/tex]: [tex]\(-453.778 \ \text{kJ}\)[/tex]
- The reaction is spontaneous.
