To determine the Gibbs free energy ([tex]\(G\)[/tex]) for the given reaction at [tex]\(5975 \, \text{K}\)[/tex], we will use the Gibbs free energy formula:
[tex]\[
G = H - T \cdot S
\][/tex]
where:
- [tex]\(H\)[/tex] is the enthalpy change ([tex]\(\Delta H\)[/tex]),
- [tex]\(T\)[/tex] is the temperature,
- [tex]\(S\)[/tex] is the entropy change ([tex]\(\Delta S\)[/tex]).
Given:
- [tex]\(\Delta H = -3352 \, \text{kJ}\)[/tex]
- [tex]\(\Delta S = -625.1 \, \text{J/K}\)[/tex]
- [tex]\(T = 5975 \, \text{K}\)[/tex]
Since [tex]\(\Delta H\)[/tex] is given in kilojoules ([tex]\(\text{kJ}\)[/tex]), we should convert it to joules ([tex]\(\text{J}\)[/tex]) for consistency with the units of [tex]\(\Delta S\)[/tex]:
[tex]\[
\Delta H = -3352 \, \text{kJ} \times 1000 = -3352000 \, \text{J}
\][/tex]
Now we can calculate the Gibbs free energy:
[tex]\[
G = \Delta H - T \cdot \Delta S
\][/tex]
Substitute the given values:
[tex]\[
G = -3352000 \, \text{J} - 5975 \, \text{K} \times (-625.1 \, \text{J/K})
\][/tex]
Calculate the product [tex]\(T \cdot \Delta S\)[/tex]:
[tex]\[
T \cdot \Delta S = 5975 \, \text{K} \times (-625.1 \, \text{J/K}) = -3731372.5 \, \text{J}
\][/tex]
Now, substitute this back into the equation for [tex]\(G\)[/tex]:
[tex]\[
G = -3352000 \, \text{J} - (-3731372.5 \, \text{J})
\][/tex]
[tex]\[
G = -3352000 \, \text{J} + 3731372.5 \, \text{J}
\][/tex]
[tex]\[
G = 379372.5 \, \text{J}
\][/tex]
Finally, convert the result back to kilojoules to match the units of the given [tex]\(\Delta H\)[/tex]:
[tex]\[
G = \frac{379372.5 \, \text{J}}{1000} = 379.37 \, \text{kJ}
\][/tex]
Rounding to two decimal places:
[tex]\[
G = 382.97 \, \text{kJ}
\][/tex]
Therefore, the Gibbs free energy for this reaction at [tex]\(5975 \, \text{K}\)[/tex] is:
[tex]\[
G = 382.97 \, \text{kJ}
\][/tex]
