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: Information Provided

The reaction proceeds in three stages with rate constants \(k_1\), \(k_2\), and \(k_3\). The total rate constant \(k\) is defined as:

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

The activation energies for each stage are:

• Stage 1 activation energy: \(E_1 = 60 \, \text{kJ/mol}\)
• Stage 2 activation energy: \(E_2 = 30 \, \text{kJ/mol}\)
• Stage 3 activation energy: \(E_3 = 10 \, \text{kJ/mol}\)

Step 2: Arrhenius Equation Application

The Arrhenius equation links a reaction's rate constant and activation energy:

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

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

Step 3: Overall Activation Energy Calculation (Method 1)

The total rate constant \(k\) is a composite of \(k_1\), \(k_2\), and \(k_3\). Based on the given expression for \(k\), the overall activation energy \(E\) can be derived from the individual stage activation energies using the following formula:

\[ E = E_1 + E_3 - E_2 \]

Substituting the provided values:

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

Step 4: Overall Activation Energy Calculation (Method 2)

An alternative method to relate the overall activation energy to individual stage parameters is:

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

Conclusion

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