In the rigorous formulation of the collision theory rate equation for a bimolecular reaction between — Chemical Kinetics Chemistry Question
Question
In the rigorous formulation of the collision theory rate equation $\text{Rate} = Z_{AA} \cdot e^{-E_a/RT}$ for a bimolecular reaction between identical gas molecules ($A + A \rightarrow P$), the collision frequency $Z_{AA}$ is fundamentally proportional to $\sqrt{T}$ based on the kinetic theory of gases. If the absolute temperature of the system is artificially increased by a massive factor of $4$ (while assuming $E_a \approx 0$ for an activationless radical recombination), the collision frequency $Z_{AA}$ will purely increase by an integer factor of what value?
💡 Solution & Explanation
According to the kinetic theory of gases, the collision frequency $Z_{AA}$ is directly proportional to the average relative velocity of the colliding molecules, which in turn is strictly proportional to the square root of the absolute temperature ($\sqrt{T}$). If the absolute temperature $T$ becomes $4T$, the new collision frequency $Z_{AA}' \propto \sqrt{4T} = 2\sqrt{T}$. Therefore, $Z_{AA}$ purely increases by a strict factor of 2.