Question

For the reaction $\text{A} \to \text{B}$, the rate constant $k$ (in $\text{s}^{-1}$) is given by $\log_{10} k = 20.35 - \frac{2.47 \times 10^3}{\text{T}}$. The energy of activation in $\text{kJ mol}^{-1}$ is ________. (Nearest integer) [Given : R = 8.314 J K$^{-1}$ mol$^{-1}$]

Hint:

In the log form of the Arrhenius equation, the coefficient of $1/T$ is always equal to $E_a / (2.303R)$. Always ensure you multiply by 2.303 when the equation is in $\log_{10}$ instead of $\ln$.

Answer 1

Correct Answer: 47

Step 1: Understanding the Concept:
The temperature dependence of the rate constant is described by the Arrhenius equation: $k = A e^{-E_a/RT}$. Taking the base-10 logarithm allows us to relate the slope of the equation to the activation energy.
Step 2: Key Formula or Approach:
1. Arrhenius equation in log form: $\log_{10} k = \log_{10} A - \frac{E_a}{2.303 RT}$
2. Compare this with the given equation: $\log_{10} k = 20.35 - \frac{2.47 \times 10^3}{T}$
Step 3: Detailed Explanation:
1. Identify the term corresponding to the slope:
\[ \frac{E_a}{2.303 R} = 2.47 \times 10^3 \]
2. Substitute $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$ and solve for $E_a$:
\[ E_a = 2.47 \times 10^3 \times 2.303 \times 8.314 \]
\[ E_a = 2470 \times 19.147 \approx 47293.4 \text{ J mol}^{-1} \]
3. Convert to kJ mol$^{-1}$:
\[ E_a \approx 47.29 \text{ kJ mol}^{-1} \]
The nearest integer is 47.
Step 4: Final Answer:
The activation energy is 47 kJ mol$^{-1}$.