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# For reaction taking place in three steps, the rate constants ${{k}_{1}},\text{ }{{k}_{2}},\text{ }{{k}_{3}}$. The overall rate constant $k=\dfrac{{{k}_{1}}{{k}_{2}}}{{{k}_{3}}}$ . If the energy of activation values for the first, second and third stages are 40, 50, and 60 kJ/mol respectively, then the overall energy of activation in kJ/mol is:A. 30B. 40C. 60D. 50

Last updated date: 20th Jun 2024
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Hint: There is a relationship between activation energy and rate constant of a reaction and it is as follows.
$K=A{{e}^{\dfrac{-{{E}_{a}}}{RT}}}$
Here, K= rate constant of the reaction
A = Pre-exponential factor
${{E}_{a}}$ = activation energy
R = gas constant
T = temperature

Complete step by step solution:
- In the question it is given that there is a reaction containing three steps. The energy of activation values for the first second and third stages are 40, 50, and 60 kJ/mol respectively.
- We have to calculate the overall energy of activation.
- The formula to calculate the activation energy is as follows.
$K=A{{e}^{\dfrac{-{{E}_{a}}}{RT}}}\to (1)$
Here, K= rate constant of the reaction
A = Pre-exponential factor
${{E}_{a}}$ = activation energy
R = gas constant
T = temperature
- In the it is given that the rate constant is $k=\dfrac{{{k}_{1}}{{k}_{2}}}{{{k}_{3}}}$ .
- Substitute the rate constant value in equation (1) to get the overall activation energy.
\begin{align} & k=\dfrac{{{k}_{1}}{{k}_{2}}}{{{k}_{3}}} \\ & A{{e}^{\dfrac{-{{E}_{a}}}{RT}}}=\dfrac{\left( A{{e}^{\dfrac{-{{E}_{{{a}_{1}}}}}{RT}}} \right)\left( A{{e}^{\dfrac{-{{E}_{{{a}_{2}}}}}{RT}}} \right)}{A{{e}^{\dfrac{-{{E}_{{{a}_{3}}}}}{RT}}}} \\ & {{e}^{\dfrac{-{{E}_{a}}}{RT}}}=\dfrac{e(-{{E}_{{{a}_{1}}}}-{{E}_{a2}}+{{E}_{a3}})}{RT} \\ & {{E}_{a}}={{E}_{{{a}_{1}}}}+{{E}_{{{a}_{2}}}}-{{E}_{{{a}_{3}}}} \\ & {{E}_{a}}=40+50-60 \\ & {{E}_{a}}=30kJ/mol \\ \end{align}
- Therefore the overall activation energy of the reaction is 30kJ/mol.

- So, the correct option is A.

Note: The rate constant of the reaction is given in the question. If the relationship between the rate constants of the three steps of the reaction is going to change then the activation energy of the reaction also changes.