To determine when the reaction is spontaneous, we need to make use of the concept of Gibbs free energy change ([tex]\(\Delta G\)[/tex]). The formula for Gibbs free energy is:
[tex]\[
\Delta G = \Delta H - T \Delta S
\][/tex]
where:
- [tex]\(\Delta G\)[/tex] is the change in Gibbs free energy.
- [tex]\(\Delta H\)[/tex] is the change in enthalpy.
- [tex]\(T\)[/tex] is the temperature in Kelvin.
- [tex]\(\Delta S\)[/tex] is the change in entropy.
For a reaction to be spontaneous, [tex]\(\Delta G\)[/tex] should be negative ([tex]\(\Delta G < 0\)[/tex]).
Given that [tex]\(\Delta H = -286 \, \text{kJ}\)[/tex], we want to determine the conditions under which [tex]\(\Delta G\)[/tex] is negative.
[tex]\[
\Delta G = -286 \, \text{kJ} - T \Delta S
\][/tex]
For [tex]\(\Delta G\)[/tex] to be spontaneous ([tex]\(\Delta G < 0\)[/tex]):
[tex]\[
-286 \, \text{kJ} - T \Delta S < 0
\][/tex]
Add [tex]\(T \Delta S\)[/tex] to both sides of the inequality:
[tex]\[
-286 \, \text{kJ} < T \Delta S
\][/tex]
Thus, to satisfy this condition:
[tex]\[
T \Delta S > 286 \, \text{kJ}
\][/tex]
Therefore, the correct condition for spontaneity is:
[tex]\(T \Delta S > 286 \, \text{kJ}\)[/tex].
So, the correct answer is:
A. When [tex]\( T \Delta S > 0\)[/tex].
