A reaction takes place in theee steps: the rate constant are `k_(1), k_(2), and k_(3)`. The overall rate constant `k=k_(1)k_(3)//k_(2)`. If the energies of activation are `40,30, and 20 KJ mol^(-1)`, the overall energy of activation is (assuming `A` to be constant for all)

Options

A. `10`
B. `15`
C. `30`
D. `60`

Explanation

To solve the problem, we need to find the overall energy of activation for a reaction that occurs in three steps with given activation energies and a specific relationship between the rate constants. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship between rate constants
The overall rate constant \( k \) is given by:
\[ k = \frac{k_1 \cdot k_3}{k_2} \]

### Step 2: Use the Arrhenius equation
According to the Arrhenius equation, the rate constant \( k \) can be expressed in terms of the activation energy \( E_a \) as follows:
\[ k = A e^{-\frac{E_a}{RT}} \]
where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

### Step 3: Substitute the Arrhenius equation into the rate constant relationship
Substituting the Arrhenius equation into the expression for \( k \):
\[ k = A e^{-\frac{E_{a1}}{RT}} \cdot A e^{-\frac{E_{a3}}{RT}} \cdot \frac{1}{A e^{-\frac{E_{a2}}{RT}}} \]

### Step 4: Simplify the equation
This simplifies to:
\[ k = A^2 e^{-\frac{E_{a1}}{RT}} \cdot e^{-\frac{E_{a3}}{RT}} \cdot e^{\frac{E_{a2}}{RT}} \]
Combining the exponents:
\[ k = A^2 e^{-\frac{E_{a1} + E_{a3} - E_{a2}}{RT}} \]

### Step 5: Identify the overall activation energy
From the above equation, we can identify the overall activation energy \( E_a \) for the reaction as:
\[ E_a = E_{a1} + E_{a3} - E_{a2} \]

### Step 6: Substitute the given activation energies
We are given:
- \( E_{a1} = 40 \, \text{kJ/mol} \)
- \( E_{a2} = 30 \, \text{kJ/mol} \)
- \( E_{a3} = 20 \, \text{kJ/mol} \)

Substituting these values into the equation:
\[ E_a = 40 + 20 - 30 \]
\[ E_a = 30 \, \text{kJ/mol} \]

### Final Answer
The overall energy of activation is \( 30 \, \text{kJ/mol} \).

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