For a complex reaction `A overset(k) rarr `products
`E_(a1)=180kJ //mol e,E_(a2)=80kJ//mol ,E_(a3)=50kJ//mol`
Overall rate constant `k` is related to individual rate constant by the equation `k=((k_(1)k_(2))/(k_(3)))^(2//3)`. Activation energy `( kJ//mol )` for the overall reaction is `:`

Options

A. 100
B. 43.44
C. 150
D. 140

Explanation

To find the overall activation energy for the complex reaction given the individual activation energies \( E_{a1} \), \( E_{a2} \), and \( E_{a3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- \( E_{a1} = 180 \, \text{kJ/mol} \)
- \( E_{a2} = 80 \, \text{kJ/mol} \)
- \( E_{a3} = 50 \, \text{kJ/mol} \)

2. **Write the Equation for the Overall Rate Constant**:
The overall rate constant \( k \) is related to the individual rate constants by the equation:
\[ k = \left( \frac{k_1 k_2}{k_3} \right)^{\frac{2}{3}} \]

3. **Relate the Rate Constants to Activation Energies**:
The rate constant \( k \) can also be expressed in terms of activation energy using the Arrhenius equation:
\[ k = A e^{-\frac{E_a}{RT}} \]
where \( A \) is the pre-exponential factor, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

4. **Express Individual Rate Constants**:
For the individual rate constants, we can write:
\[ k_1 = A_1 e^{-\frac{E_{a1}}{RT}}, \quad k_2 = A_2 e^{-\frac{E_{a2}}{RT}}, \quad k_3 = A_3 e^{-\frac{E_{a3}}{RT}} \]

5. **Substitute into the Overall Rate Constant Equation**:
Substitute \( k_1 \), \( k_2 \), and \( k_3 \) into the equation for \( k \):
\[ k = \left( \frac{A_1 e^{-\frac{E_{a1}}{RT}} \cdot A_2 e^{-\frac{E_{a2}}{RT}}}{A_3 e^{-\frac{E_{a3}}{RT}}} \right)^{\frac{2}{3}} \]

6. **Simplify the Expression**:
This simplifies to:
\[ k = \left( \frac{A_1 A_2}{A_3} \right)^{\frac{2}{3}} e^{-\frac{E_{a1} + E_{a2} - E_{a3}}{RT} \cdot \frac{2}{3}} \]

7. **Determine the Overall Activation Energy**:
From the above expression, we can identify that the overall activation energy \( E_a \) for the reaction is:
\[ E_a = \frac{2}{3} (E_{a1} + E_{a2} - E_{a3}) \]

8. **Substitute the Values**:
Now substituting the values of \( E_{a1} \), \( E_{a2} \), and \( E_{a3} \):
\[ E_a = \frac{2}{3} (180 + 80 - 50) \]
\[ E_a = \frac{2}{3} (210) = 140 \, \text{kJ/mol} \]

### Final Answer:
The overall activation energy for the reaction is \( 140 \, \text{kJ/mol} \).

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