According to the Arrhenius equation, the relationship between the rate constant ($k$) and temperature ($T$) at two different temperatures is given by $\ln \frac{k_2}{k_1} = \frac{E_a}{R} \left[ \frac{1}{T_1} - \frac{1}{T_2} \right]$ . Given that the activation energy $E_a = 0$, the right side of the equation becomes zero. Therefore, $\ln \frac{k_2}{k_1} = 0$, which implies $\frac{k_2}{k_1} = e^0 = 1$, or $k_2 = k_1$ . This means the rate constant is completely independent of temperature when the activation energy is zero. Thus, the rate constant at $400 \text{ K}$ will remain exactly the same as at $200 \text{ K}$, which is $1.6 \times 10^6 \text{ s}^{-1}$.