For a reaction taking place in three steps at the same temperature, the overall rate constant \( K = \frac{K_1 K_2}{K_3} \). If \( E_{a1} \), \( E_{a2} \), and \( E_{a3} \) are 40, 50, and 60 kJ/mol respectively, the overall \( E_a \) is ____ kJ/mol.
Correct Answer: 30
Approach Solution - 1
For the overall rate constant:
\[ K = \frac{K_1 \cdot K_2}{K_3} = \frac{A_1 \cdot A_2}{A_3} \cdot e^{\left(\frac{E_{a1} + E_{a2} - E_{a3}}{RT}\right)} \]
Therefore,
\[ K = \frac{A \cdot e^{-E_a/RT}}{A_3} = \frac{A_1 A_2}{A_3} \cdot e^{\left(\frac{E_{a1} + E_{a2} - E_{a3}}{RT}\right)} \]
Given:
\[ E_a = E_{a1} + E_{a2} - E_{a3} = 40 + 50 - 60 = 30 \, \text{kJ/mol} \]
So, the correct answer is: 30
Approach Solution -2
Step 1: Write the rate constant expressions using the Arrhenius equation.
\[ K_1 = A_1 e^{-E_{a1}/RT}, \quad K_2 = A_2 e^{-E_{a2}/RT}, \quad K_3 = A_3 e^{-E_{a3}/RT} \]
Step 2: Substitute into the given overall rate expression.
\[ K = \frac{K_1 K_2}{K_3} = \frac{A_1 A_2}{A_3} \, e^{-(E_{a1} + E_{a2} - E_{a3})/RT} \]
Step 3: Identify the overall activation energy.
From the exponent, \[ E_a = E_{a1} + E_{a2} - E_{a3} \]
Step 4: Substitute values.
\[ E_a = 40 + 50 - 60 = 30\,\text{kJ/mol} \]
Final Answer:
\[ \boxed{E_a = 30\,\text{kJ/mol}} \]
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