A reaction takes place in various steps. The rate constatn for first, second, third and fifth steps are `k_(1),k_(2),k_(3)` and `k_(5)` respectively The overall rate constant is given by
`k=(k_(2))/(k_(3))(k_(1)/(k_(5)))^(1//2)`
If activation energy are 40, 60, 50, and 10 kJ/mol respectively, the overall energy of activation (kJ/mol) is :

Options

A. 10
B. 20
C. 25
D. none of these

Answer

To solve the problem, we need to find the overall energy of activation for the given reaction steps. We will use the Arrhenius equation and the provided activation energies for each step.

### Step-by-Step Solution:

1. **Write the Arrhenius Equation**:
The Arrhenius equation is given by:
\[ k = A \cdot e^{-\frac{E_A}{RT}} \]
where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_A \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature.

2. **Express the Rate Constants**:
For each step, we can express the rate constants as follows:
- For step 1: \( k_1 = A_1 \cdot e^{-\frac{E_1}{RT}} \)
- For step 2: \( k_2 = A_2 \cdot e^{-\frac{E_2}{RT}} \)
- For step 3: \( k_3 = A_3 \cdot e^{-\frac{E_3}{RT}} \)
- For step 5: \( k_5 = A_5 \cdot e^{-\frac{E_5}{RT}} \)

3. **Substitute into Overall Rate Constant**:
The overall rate constant is given by:
\[ k = \frac{k_2}{k_3} \cdot \left(\frac{k_1}{k_5}\right)^{\frac{1}{2}} \]
Substituting the expressions for \( k_1, k_2, k_3, \) and \( k_5 \):
\[ k = \frac{A_2 \cdot e^{-\frac{E_2}{RT}}}{A_3 \cdot e^{-\frac{E_3}{RT}}} \cdot \left(\frac{A_1 \cdot e^{-\frac{E_1}{RT}}}{A_5 \cdot e^{-\frac{E_5}{RT}}}\right)^{\frac{1}{2}} \]

4. **Simplify the Expression**:
This simplifies to:
\[ k = \frac{A_2 \cdot A_1^{\frac{1}{2}}}{A_3 \cdot A_5^{\frac{1}{2}}} \cdot e^{-\left(\frac{E_2}{R} - \frac{E_3}{R} + \frac{1}{2}\left(\frac{E_1}{R} - \frac{E_5}{R}\right)\right)\frac{1}{T}} \]

5. **Combine Activation Energies**:
The overall activation energy \( E_A \) can be expressed as:
\[ E_A = E_2 - E_3 + \frac{1}{2}(E_1 - E_5) \]

6. **Substitute Given Activation Energies**:
The activation energies are given as:
- \( E_1 = 40 \, \text{kJ/mol} \)
- \( E_2 = 60 \, \text{kJ/mol} \)
- \( E_3 = 50 \, \text{kJ/mol} \)
- \( E_5 = 10 \, \text{kJ/mol} \)

Substituting these values into the equation:
\[ E_A = 60 - 50 + \frac{1}{2}(40 - 10) \]
\[ E_A = 10 + \frac{1}{2}(30) = 10 + 15 = 25 \, \text{kJ/mol} \]

7. **Final Answer**:
The overall energy of activation is:
\[ \boxed{25 \, \text{kJ/mol}} \]