Editing Collision theory (section)

===Derivation===
Consider the bimolecular elementary reaction:

:A + B → C

In collision theory it is considered that two particles A and B will collide if their nuclei get closer than a certain distance. The area around a molecule A in which it can collide with an approaching B molecule is called the [[Cross section (physics)|cross section]] (σ<sub>AB</sub>) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius (<math>r_{AB}</math>) is the sum of the radii of both reacting molecules, which are supposed to be spherical. 
A moving molecule will therefore sweep a volume <math>\pi r^{2}_{AB} c_A</math> per second as it moves, where <math>c_A</math> is the average velocity of the particle. (This solely represents the classical notion of a collision of solid balls. As molecules are quantum-mechanical many-particle systems of electrons and nuclei based upon the Coulomb and exchange interactions, generally they neither obey rotational symmetry nor do they have a box potential. Therefore, more generally the cross section is defined as the reaction probability of a ray of A particles per areal density of B targets, which makes the definition independent from the nature of the interaction between A and B. Consequently, the radius <math>r_{AB}</math> is related to the length scale of their interaction potential.)

From [[kinetic theory of gases|kinetic theory]] it is known that a molecule of A has an [[Maxwell–Boltzmann distribution|average velocity]] (different from [[root mean square]] velocity) of <math>c_A = \sqrt \frac{8 k_\text{B} T}{\pi m_A}</math>, where <math>k_\text{B}</math> is the [[Boltzmann constant]], and <math>m_A</math> is the mass of the molecule.

The solution of the [[two-body problem]] states that two different moving bodies can be treated as one body which has the [[reduced mass]] of both and moves with the velocity of the [[center of mass]], so, in this system <math>\mu_{AB}</math> must be used instead of <math>m_A</math>.
Thus, for a given molecule A, it travels <math>t=l/c_A=1/(n_B\sigma_{AB}c_A)</math> before hitting a molecule B if all B is fixed with no movement, where <math>l</math> is the average traveling distance. Since B also moves, the relative velocity can be calculated using the reduced mass of A and B.

Therefore, the total '''collision frequency''',<ref name="frequency">{{GoldBookRef | file = C01166| title = collision frequency}}</ref> of all A molecules, with all B molecules, is

:<math> Z = n_\text{A} n_\text{B} \sigma_{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_{AB}} = 10^6N_A^2[A][B] \sigma_{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_{AB}} = z[A][B],</math>

From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is <math>e^{\frac{-E_\text{a}}{RT}}</math>. Therefore, the rate of a bimolecular reaction for ideal gases will be

:<math>r = z \rho [A][B] \exp\left( \frac{-E_\text{a}}{RT} \right),</math> in unit number of molecular reactions s<sup>−1</sup>⋅m<sup>−3</sup>,

where:
* ''Z'' is the collision frequency with unit s<sup>−1</sup>⋅m<sup>−3</sup>. The ''z'' is ''Z'' without [A][B].
* <math>\rho</math> is the [[steric factor]], which will be discussed in detail in the next section,
* ''E<sub>a</sub>'' is the [[activation energy]] (per mole) of the reaction in unit J/mol,
*''T'' is the absolute temperature in unit K,
* ''R'' is the [[gas constant]] in unit J/mol/K.
* [A] is molar concentration of A in unit mol/L,
* [B] is molar concentration of B in unit mol/L.

The product ''zρ'' is equivalent to the [[preexponential factor]] of the [[Arrhenius equation]].