To determine the activation energy (Ea) of a chemical reaction, we can utilize the Arrhenius equation, which relates the rate constants of a reaction at two different temperatures. Given the rate constants k₁ = 4.6 × 10^-2 s^-1 at 0°C (273.15 K) and k₂ = 8.1 × 10^-2 s^-1 at 20°C (293.15 K), we can apply the following formula:

\[ \ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]

Here, R is the universal gas constant, approximately 8.314 J/(mol·K). We first calculate the natural logarithm of the ratio of the rate constants:

\[ \ln\left(\frac{8.1 \times 10^{-2}}{4.6 \times 10^{-2}}\right) \approx 0.5658 \]

Next, we substitute the temperatures into the equation:

\[ \frac{1}{T_2} - \frac{1}{T_1} = \frac{1}{293.15} - \frac{1}{273.15} \approx 0.0005 \, \text{K}^{-1} \]

Now, substituting these values into the Arrhenius equation gives:

\[ 0.5658 = -\frac{E_a}{8.314} \times 0.0005 \]

Rearranging to solve for E_a yields:

\[ E_a = \frac{0.5658 \times 8.314}{0.0005} \approx 18,833.91 \, \text{J/mol} \]

For significant figures, we round this value to two significant figures, resulting in:

\[ E_a \approx 1.9 \times 10^{4} \, \text{J/mol} \]

This calculated activation energy indicates the minimum energy required for the reaction to proceed, reflecting the energy barrier that must be overcome for the reactants to transform into products.

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