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Answer :
To solve this problem, it's important to understand the role of a catalyst in a chemical reaction. A catalyst speeds up the reaction by lowering the activation energy, but it does not affect the overall energy change of the reaction ([tex]\(\Delta H\)[/tex]).
Here's how to evaluate the possible values for [tex]\(\Delta H\)[/tex] and [tex]\(A_E\)[/tex] with a catalyst involved:
1. Understand the given values:
- The energy difference between reactants and products without a catalyst ([tex]\(\Delta H\)[/tex]) is [tex]\(-23.7 \text{ kJ/mol}\)[/tex].
- The forward activation energy without a catalyst ([tex]\(A_E\)[/tex]) is [tex]\(27.9 \text{ kJ/mol}\)[/tex].
2. Recognize catalyst effects:
- A catalyst does not change the energy difference between reactants and products, so [tex]\(\Delta H\)[/tex] remains [tex]\(-23.7 \text{ kJ/mol}\)[/tex] when a catalyst is present.
- A catalyst lowers the activation energy required for the reaction.
3. Evaluate each option:
- For the pair [tex]\(\Delta H = -9.27 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 50.4 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] does not match the required [tex]\(-23.7 \text{ kJ/mol}\)[/tex]. Hence, it is not possible.
- For the pair [tex]\(\Delta H = -12.1 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is incorrect.
- For the pair [tex]\(\Delta H = -43 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 50.4 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is incorrect.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 21.2 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct and [tex]\(A_E\)[/tex] is lower than the original [tex]\(27.9 \text{ kJ/mol}\)[/tex], so this is valid.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 99.1 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct, but [tex]\(A_E\)[/tex] is higher, not lower, so this is incorrect.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct and [tex]\(A_E\)[/tex] is lower, making this choice valid.
So, the possible values for [tex]\(\Delta H\)[/tex] and [tex]\(A_E\)[/tex] for the forward reaction in the presence of a catalyst are:
- [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 21.2 \text{ kJ/mol}\)[/tex]
- [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]
Here's how to evaluate the possible values for [tex]\(\Delta H\)[/tex] and [tex]\(A_E\)[/tex] with a catalyst involved:
1. Understand the given values:
- The energy difference between reactants and products without a catalyst ([tex]\(\Delta H\)[/tex]) is [tex]\(-23.7 \text{ kJ/mol}\)[/tex].
- The forward activation energy without a catalyst ([tex]\(A_E\)[/tex]) is [tex]\(27.9 \text{ kJ/mol}\)[/tex].
2. Recognize catalyst effects:
- A catalyst does not change the energy difference between reactants and products, so [tex]\(\Delta H\)[/tex] remains [tex]\(-23.7 \text{ kJ/mol}\)[/tex] when a catalyst is present.
- A catalyst lowers the activation energy required for the reaction.
3. Evaluate each option:
- For the pair [tex]\(\Delta H = -9.27 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 50.4 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] does not match the required [tex]\(-23.7 \text{ kJ/mol}\)[/tex]. Hence, it is not possible.
- For the pair [tex]\(\Delta H = -12.1 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is incorrect.
- For the pair [tex]\(\Delta H = -43 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 50.4 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is incorrect.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 21.2 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct and [tex]\(A_E\)[/tex] is lower than the original [tex]\(27.9 \text{ kJ/mol}\)[/tex], so this is valid.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 99.1 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct, but [tex]\(A_E\)[/tex] is higher, not lower, so this is incorrect.
- For the pair [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]: [tex]\(\Delta H\)[/tex] is correct and [tex]\(A_E\)[/tex] is lower, making this choice valid.
So, the possible values for [tex]\(\Delta H\)[/tex] and [tex]\(A_E\)[/tex] for the forward reaction in the presence of a catalyst are:
- [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 21.2 \text{ kJ/mol}\)[/tex]
- [tex]\(\Delta H = -23.7 \text{ kJ/mol}\)[/tex] and [tex]\(A_E = 10.3 \text{ kJ/mol}\)[/tex]
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