To determine the activation energy (Ea) of a reaction using the rate constants at two different temperatures, you can apply the Arrhenius equation:

Arrhenius Equation
The Arrhenius equation is given by:
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = pre-exponential factor
- \( E_a \) = activation energy
- \( R \) = universal gas constant (8.314 J/mol·K)
- \( T \) = temperature in Kelvin

Using Two Temperatures
Taking the natural logarithm of the Arrhenius equation for two temperatures, \( T_1 \) and \( T_2 \):
\[ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]

Given Data
- \( k_1 = 0.02 \, \text{s}^{-1} \) at \( T_1 = 200 \, K \)
- \( k_2 = 0.04 \, \text{s}^{-1} \) at \( T_2 = 300 \, K \)

Calculation Steps
1. **Calculate the ratio of rate constants:**
\[ \frac{k_2}{k_1} = \frac{0.04}{0.02} = 2 \]
2. **Calculate the temperature difference:**
\[ \frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{200} - \frac{1}{300} = \frac{3 - 2}{600} = \frac{1}{600} \]
3. **Substituting into the equation:**
\[ \ln(2) = \frac{E_a}{8.314} \left(\frac{1}{600}\right) \]
Rearranging gives:
\[ E_a = 8.314 \cdot 600 \cdot \ln(2) \]
4. **Final Calculation:**
\[ E_a \approx 8.314 \cdot 600 \cdot 0.693 \approx 3450.8 \, J/mol \]
Thus, the activation energy \( E_a \) for the reaction is approximately **3450.8 J/mol**.