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 solve the problem of determining the activation energy for the overall reaction given the activation energies of individual steps and the relationship between the rate constants, we can follow these steps:

### Step 1: Understand the given data
We are given:
- Activation energies for three steps:
- \( E_{a1} = 180 \, \text{kJ/mol} \)
- \( E_{a2} = 80 \, \text{kJ/mol} \)
- \( E_{a3} = 50 \, \text{kJ/mol} \)
- The overall rate constant \( k \) is related to the individual rate constants by the equation:
\[ k = \left( \frac{k_1 k_2}{k_3} \right)^{\frac{2}{3}} \]

### Step 2: Write the Arrhenius equation for each rate constant
The Arrhenius equation relates the rate constant \( k \) to the activation energy \( E_a \):
\[ k = A e^{-\frac{E_a}{RT}} \]
Where:
- \( A \) is the pre-exponential factor,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin.

For each step, we can write:
- \( k_1 = A e^{-\frac{E_{a1}}{RT}} \)
- \( k_2 = A e^{-\frac{E_{a2}}{RT}} \)
- \( k_3 = A e^{-\frac{E_{a3}}{RT}} \)

### Step 3: Substitute the rate constants into the overall rate constant equation
Substituting \( k_1 \), \( k_2 \), and \( k_3 \) into the equation for \( k \):
\[ k = \left( \frac{A e^{-\frac{E_{a1}}{RT}} \cdot A e^{-\frac{E_{a2}}{RT}}}{A e^{-\frac{E_{a3}}{RT}}} \right)^{\frac{2}{3}} \]
This simplifies to:
\[ k = \left( A^2 e^{-\frac{E_{a1} + E_{a2} - E_{a3}}{RT}} \right)^{\frac{2}{3}} \]

### Step 4: Simplify the expression
Since \( A^2 \) raised to \( \frac{2}{3} \) is \( A^{\frac{4}{3}} \):
\[ k = A^{\frac{4}{3}} e^{-\frac{2}{3}\left( \frac{E_{a1} + E_{a2} - E_{a3}}{RT} \right)} \]

### Step 5: Compare with the Arrhenius equation for overall reaction
We can express the overall rate constant \( k \) as:
\[ k = A' e^{-\frac{E_a}{RT}} \]
Where \( E_a \) is the activation energy for the overall reaction.

### Step 6: Equate the exponents
From the two expressions for \( k \), we can equate the exponents:
\[ -\frac{E_a}{RT} = -\frac{2}{3}\left( \frac{E_{a1} + E_{a2} - E_{a3}}{RT} \right) \]

### Step 7: Solve for \( E_a \)
Cancelling \( -\frac{1}{RT} \):
\[ E_a = \frac{2}{3}(E_{a1} + E_{a2} - E_{a3}) \]

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

### Final Answer
The activation energy for the overall reaction is:
\[ \boxed{140 \, \text{kJ/mol}} \]