Collision Theory was first introduced in the 1910s by Max Trautz (Trautz, 1916) and William Lewis (Lewis, 1918) to try to account for the magnitudes of rate constants in terms of the frequency of molecular collisions, the collisional energy, and the relative orientations of the molecules involved in the collision. The rate of a reaction, according to collision theory, can be expressed as \[ \text{rate} = Z_{ab} F \label{majorrate} \] where \(Z_{AB}\) is the frequency of collisions between the molecules \(A\) and \(B\) involved in the reaction, and \(F\) is the fraction of those collisions that will lead to a reaction. The first term, \(Z_{AB}\), can be taken from the kinetic molecular theory of gases (for a derivation see LibreTexts (Fleming) 2.6: Collisions with Other Molecules ). \[ Z_{AB} = \sqrt{ \dfrac{8k_BT}{\pi \mu}} \sigma_{AB} [A][B] \label{collision} \] Where the first factor is the average relative velocity of the two particles within which \(\mu ={m_A m_B}/(m_A + m_B)\) is the reduced mass of the A-B collisional system, k B = Boltzmann constant and T = temperature in Kelvin. \(\sigma_{AB}\) is the collisional cross section. [A] and [B] are the concentrations of A and B. The collision cross section \(\sigma_{AB}\) can be understood as the area of a circle that sweeps out the cylindrical volume within which spherical molecules with a center-to-center distance less than or equal to the sum of their radii collide. Those with centers farther apart do not collide (see figure \(\PageIndex{1}\)). If A and B are hard spheres \(\sigma_{AB} = \pi(r_A + r_B)^2\), where 'r' refers to the radius of a sphere. 0 1 2 NaN NaN NaN A B C Figure \(\PageIndex{1}\): (A) no collision; (B) collision; (C) volume swept out by collision cross-section imagining only one particle moving. The factor \(F\) has two important contributors, the energy of the collision and the orientation of the molecules when they collide. The factor \(F\) depends on the activation energy. Assuming a Boltzmann (or Boltzmann-like) distribution of energies, the fraction of molecular collisions that will have enough energy to overcome the activation barrier is given by \[ F= e^{-E_a/RT} \label{arr} \] Combining Equations \ref{collision} and \ref{arr}, the rate of the reaction (Equation \ref{majorrate}) is predicted by \[ \text{rate} = \left( \dfrac{8k_BT}{\pi \mu} \right) ^{1/2} \sigma_{AB} e^{-E_a/RT} [A][B] \nonumber \] So if the rate law can be expressed as a second order rate law \[\text{rate} = k [A][B] \nonumber \] it is clear that the rate constant \(k\) is given by \[k=\left( \dfrac{8k_BT}{\pi \mu} \right)^{1/2} \sigma_{AB} e^{-E_a/RT} \nonumber \] To account for the orientation dependence of collisions an ad-hoc steric factor (P, for the proportion of molecules with E a that react) is often added to the pre-exponential factors yielding: \[k=P\left( \dfrac{8k_BT}{\pi \mu} \right)^{1/2} \sigma_{AB} e^{-E_a/RT} \label{Collfinal}\] By comparison, the theory predicts the form of the Arrhenius pre-expoential factor to be \[A= P\left( \dfrac{8k_BT}{\pi \mu} \right)^{1/2} \sigma_{AB} \nonumber \] Note this implies that A has a temperature dependence, (\(\sqrt{T}\)), so the Arrhenius equation is a simplification. Given equation \(\ref{Collfinal}\), if k and E a are measured for molecules with a known collision cross section, \(\sigma_{AB}\), P can be found. Generally 0 < P < 1. Occasionally, when charged species are involved P is found to be larger than 1. Keep in mind that this model applies only to bimolecular steps in a mechanism. References Atkins, Peter and Julio de Paula. Physical Chemistry for the Life Sciences. 2006. New York, NY: W.H. Freeman and Company. pp.282-288, 290 Atkins, Peter. Concepts in Physical Chemistry. 1995. New York, NY: W.H. Freeman and Company. pp.193 Chang, Raymond. Physical Chemistry for the Biosciences. 2005. Sausalito, CA: University Science Books. pp.28