Question

Consider a complex reaction taking place in three steps with rate constants \(k_1\), \(k_2\), and \(k_3\) respectively. The overall rate constant \(k\) is given by the expression \( k = \sqrt{\frac{k_1 k_3}{k_2}} \). If the activation energies of the three steps are 60, 30, and 10 kJ mol\(^{-1}\) respectively, then the overall energy of activation in kJ mol\(^{-1}\) is ________________(Nearest integer).

Hint:

To find the overall activation energy in multi-step reactions, apply the Arrhenius equation to each step, then combine the activation energies weighted by their rate constants.

Answer 1

Step 1: Understand the Given Information

The reaction takes place in three steps with rate constants \(k_1\), \(k_2\), and \(k_3\). The overall rate constant \(k\) is given by the expression:

\[ k = \sqrt{\frac{k_1 k_3}{k_2}} \]

The activation energies for the three steps are given as:

• Activation energy for step 1, \(E_1 = 60 \, \text{kJ/mol}\)
• Activation energy for step 2, \(E_2 = 30 \, \text{kJ/mol}\)
• Activation energy for step 3, \(E_3 = 10 \, \text{kJ/mol}\)

Step 2: Apply the Arrhenius Equation

The Arrhenius equation relates the rate constant and activation energy of a reaction as:

\[ k = A \cdot e^{-E/RT} \]

where:
- \(A\) is the pre-exponential factor,
- \(E\) is the activation energy,
- \(R\) is the gas constant (8.314 J/mol·K),
- \(T\) is the temperature in Kelvin.

Step 3: Determine the Overall Activation Energy

The overall rate constant \(k\) is a combination of the three rate constants \(k_1\), \(k_2\), and \(k_3\). Given the expression for \(k\), we can use the activation energies of the individual steps to calculate the overall activation energy \(E\). The overall activation energy for a reaction involving multiple steps can be determined by the following relationship:

\[ E = E_1 + E_3 - E_2 \]

Substituting the values for \(E_1\), \(E_2\), and \(E_3\):

\[ E = 60 + 10 - 30 = 40 \, \text{kJ/mol} \]

Step 4: Calculate the Overall Activation Energy

The overall activation energy can also be related to the rate constants and activation energies of the individual steps by the following formula:

\[ E_{\text{overall}} = \frac{E_1 + E_3}{2} \approx 20 \, \text{kJ/mol} \]

Conclusion

The overall activation energy of the reaction is approximately 20 kJ/mol.