Question

The rate constant for a first order reaction is $0.58 \text{ s}^{-1}$ at $300 \text{ K}$ and $0.026 \text{ s}^{-1}$ at $290 \text{ K}$ . What is the energy of activation? $\left(\text{R} = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}\right)$}

• A. 124.48 kJ
• B. 224.55 kJ
• C. 348.18 kJ
• D. 513.21 kJ

Hint:

When temperature increases by 10K, the rate constant often increases significantly; use the log form of Arrhenius for $E_a$ calculations.

Answer 1

The Correct Option is B

Step 1: Concept
The temperature dependence of the rate constant is given by the Arrhenius equation: $\log \frac{k_2}{k_1} = \frac{E_a}{2.303R} \left( \frac{T_2 - T_1}{T_1 T_2} \right)$.

Step 2: Meaning
$E_a$ is the activation energy, the minimum energy required for reactants to transform into products.

Step 3: Analysis
Given: $k_1 = 0.026$, $T_1 = 290$, $k_2 = 0.58$, $T_2 = 300$, $R = 8.314$. $\log \frac{0.58}{0.026} = \frac{E_a}{2.303 \times 8.314} \left( \frac{300 - 290}{300 \times 290} \right)$ $\log(22.3) \approx 1.348$ $1.348 = \frac{E_a}{19.147} \times \frac{10}{87000}$ $E_a = \frac{1.348 \times 19.147 \times 87000}{10} \approx 224550 \text{ J/mol}$.

Step 4: Conclusion
Converting to kJ: $E_a \approx 224.55 \text{ kJ/mol}$. Final Answer: (B)