A reaction takes place in various steps. The rate constant for first, second, third and fifth steps are k1, k2, k3 and k5 respectively. The overall rate constant is is given by:
`k = k_(2)/k_(1) (k_(1)/k_(2))^(1/2)` activation energy are 40,60, 50 and 10 kJ/mol respectively, find the overall energy of activation (kJ/mol).

Explanation

To find the overall energy of activation for the reaction taking place in various steps, we can follow these steps:

### Step 1: Understand the Arrhenius Equation
The Arrhenius equation relates the rate constant (k) to the activation energy (Ea):
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = pre-exponential factor
- \( E_a \) = activation energy
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 2: Write the Rate Constants for Each Step
Given the activation energies for each step:
- \( E_{a1} = 40 \, \text{kJ/mol} \)
- \( E_{a2} = 60 \, \text{kJ/mol} \)
- \( E_{a3} = 50 \, \text{kJ/mol} \)
- \( E_{a5} = 10 \, \text{kJ/mol} \)

We can express the rate constants \( k_1, k_2, k_3, \) and \( k_5 \) using the Arrhenius equation:
- \( k_1 = A e^{-\frac{E_{a1}}{RT}} \)
- \( k_2 = A e^{-\frac{E_{a2}}{RT}} \)
- \( k_3 = A e^{-\frac{E_{a3}}{RT}} \)
- \( k_5 = A e^{-\frac{E_{a5}}{RT}} \)

### Step 3: Substitute into the Overall Rate Constant Expression
The overall rate constant \( k \) is given by:
\[ k = \frac{k_2}{k_3} \left( \frac{k_1}{k_5} \right)^{1/2} \]

Substituting the expressions for \( k_1, k_2, k_3, \) and \( k_5 \):
\[ k = \frac{A e^{-\frac{E_{a2}}{RT}}}{A e^{-\frac{E_{a3}}{RT}}} \left( \frac{A e^{-\frac{E_{a1}}{RT}}}{A e^{-\frac{E_{a5}}{RT}}} \right)^{1/2} \]

### Step 4: Simplify the Expression
The \( A \) terms cancel out:
\[ k = e^{-\frac{E_{a2}}{RT} + \frac{E_{a3}}{RT}} \left( e^{-\frac{E_{a1}}{RT} + \frac{E_{a5}}{RT}} \right)^{1/2} \]

This simplifies to:
\[ k = e^{-\frac{E_{a2} - E_{a3}}{RT}} \cdot e^{-\frac{1}{2}(E_{a1} - E_{a5})/RT} \]

### Step 5: Combine Exponents
Combining the exponents gives:
\[ k = e^{-\frac{E_a}{RT}} \]
where:
\[ E_a = E_{a2} - E_{a3} + \frac{1}{2}(E_{a1} - E_{a5}) \]

### Step 6: Substitute the Activation Energies
Now substituting the given activation energies:
\[ E_a = 60 - 50 + \frac{1}{2}(40 - 10) \]
\[ E_a = 10 + \frac{1}{2}(30) \]
\[ E_a = 10 + 15 = 25 \, \text{kJ/mol} \]

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