For a reaction taking place in three steps at same temperature, overall rate constant $$K = \frac{K_1 K_2}{K_3}$$. If $$Ea_1, Ea_2$$ and $$Ea_3$$ are $$40, 50$$ and $$60$$ kJ/mol respectively, the overall $$Ea$$ is _______ kJ/mol.

We need to find the overall activation energy when the overall rate constant $$K = \frac{K_1 K_2}{K_3}$$.

Recall the Arrhenius equation.

The Arrhenius equation relates the rate constant to activation energy:

$$K_i = A_i \, e^{-Ea_i/RT}$$

Express the overall rate constant.

$$K = \frac{K_1 K_2}{K_3} = \frac{A_1 e^{-Ea_1/RT} \cdot A_2 e^{-Ea_2/RT}}{A_3 e^{-Ea_3/RT}}$$

$$= \frac{A_1 A_2}{A_3} \cdot e^{-(Ea_1 + Ea_2 - Ea_3)/RT}$$

Identify the overall activation energy.

Comparing with $$K = A \, e^{-Ea/RT}$$, the overall activation energy is:

$$Ea = Ea_1 + Ea_2 - Ea_3$$

Substitute the values.

$$Ea = 40 + 50 - 60 = 30 \text{ kJ/mol}$$

The answer is 30 kJ/mol.