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Answer :
To determine for which value of [tex]\( T \Delta S \)[/tex] the reaction is spontaneous, we need to consider the relationship involving Gibbs free energy change, [tex]\(\Delta G\)[/tex]. A reaction is spontaneous when [tex]\(\Delta G < 0\)[/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 (given as 198 kJ in the problem),
- [tex]\(T \Delta S\)[/tex] is the product of temperature and the change in entropy.
For the reaction to be spontaneous, we want:
[tex]\[
\Delta H - T \Delta S < 0
\][/tex]
This inequality can be rearranged to:
[tex]\[
T \Delta S > \Delta H
\][/tex]
Given that [tex]\(\Delta H = 198 \text{ kJ}\)[/tex], we need [tex]\(T \Delta S\)[/tex] to be greater than 198 kJ for the reaction to be spontaneous.
Now, let's assess the options:
- Option A: [tex]\(T \Delta S = -198 \text{ kJ}\)[/tex]
This is not greater than 198 kJ, so the reaction would not be spontaneous.
- Option B: [tex]\(T \Delta S = 198 \text{ kJ}\)[/tex]
This is equal to [tex]\(\Delta H\)[/tex]; therefore, it is not greater. The reaction would not be spontaneous.
- Option C: [tex]\(T \Delta S = 396 \text{ kJ}\)[/tex]
This is greater than [tex]\(\Delta H\)[/tex], so the inequality [tex]\(T \Delta S > \Delta H\)[/tex] holds. The reaction is spontaneous.
- Option D: [tex]\(T \Delta S = 0 \text{ kJ}\)[/tex]
This is not greater than 198 kJ, so the reaction would not be spontaneous.
Therefore, the correct answer is Option C (396 kJ), as this value makes the reaction spontaneous.
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 (given as 198 kJ in the problem),
- [tex]\(T \Delta S\)[/tex] is the product of temperature and the change in entropy.
For the reaction to be spontaneous, we want:
[tex]\[
\Delta H - T \Delta S < 0
\][/tex]
This inequality can be rearranged to:
[tex]\[
T \Delta S > \Delta H
\][/tex]
Given that [tex]\(\Delta H = 198 \text{ kJ}\)[/tex], we need [tex]\(T \Delta S\)[/tex] to be greater than 198 kJ for the reaction to be spontaneous.
Now, let's assess the options:
- Option A: [tex]\(T \Delta S = -198 \text{ kJ}\)[/tex]
This is not greater than 198 kJ, so the reaction would not be spontaneous.
- Option B: [tex]\(T \Delta S = 198 \text{ kJ}\)[/tex]
This is equal to [tex]\(\Delta H\)[/tex]; therefore, it is not greater. The reaction would not be spontaneous.
- Option C: [tex]\(T \Delta S = 396 \text{ kJ}\)[/tex]
This is greater than [tex]\(\Delta H\)[/tex], so the inequality [tex]\(T \Delta S > \Delta H\)[/tex] holds. The reaction is spontaneous.
- Option D: [tex]\(T \Delta S = 0 \text{ kJ}\)[/tex]
This is not greater than 198 kJ, so the reaction would not be spontaneous.
Therefore, the correct answer is Option C (396 kJ), as this value makes the reaction spontaneous.
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