Collision Theory of Chemical Reactions

Collision theory, put forward by Max Trautz and William Lewis around 1916–18, digs deeper than the Arrhenius equation into why a reaction proceeds at the rate it does. It borrows its picture of molecules from the kinetic theory of gases: reactant molecules are treated as hard spheres, constantly moving and colliding, and a reaction is imagined to occur only when two such spheres strike each other.

Collision frequency

The starting idea is simple — more collisions per second should mean a faster reaction. This rate of collisions is captured by a single quantity.

Collision frequency ($Z$) — the number of collisions occurring per second, per unit volume, of the reaction mixture.

For a simple bimolecular elementary step

collision theory writes the rate as

where $Z_{AB}$ is the collision frequency of A and B, and $e^{-E_a/RT}$ is the Boltzmann factor — the fraction of those collisions in which the colliding pair together carries kinetic energy at least equal to the activation energy $E_a$. Matched against the Arrhenius equation, this tells us that the pre-exponential factor $A$ is essentially a measure of how often molecules collide.

This expression works well for reactions between atoms or fairly simple molecules, but for reactions involving more complex molecules the predicted rate constants can be far off the observed ones. The reason is that colliding with enough energy is necessary but not, by itself, sufficient — not every energetic collision actually turns reactants into products.

Two conditions for an effective collision

Collision theory therefore tightens its criterion. A collision only counts as productive — an effective collision — when both of the following hold at once:

1. The colliding molecules must carry kinetic energy equal to or above a certain minimum, called the threshold energy, so that the necessary bonds can break and new bonds can form.
2. The molecules must collide in the correct spatial orientation, so that the reactive parts of each molecule actually meet.

Energy alone getting molecules to crash into one another is not enough — they also have to be pointed the right way when they do.

Why orientation matters

The reaction of bromomethane with hydroxide ion to give methanol,

illustrates this neatly (fig-3-12).

The hydroxide ion needs to approach and bond at the carbon atom, on the side opposite to the C–Br bond, for the old bond to break as the new one forms. If the two species collide with this proper orientation — the negative oxygen end of $^-\text{OH}$ swinging in towards the carbon from the back side — an intermediate stage forms and the collision goes on to give methanol and bromide ion. But if $^-\text{OH}$ instead collides on the wrong side of the molecule — an improper orientation, for instance approaching near the bromine or hydrogen atoms rather than lining up opposite the C–Br bond — the two species simply bounce apart unchanged, and no product is formed at all. So a "good" collision is not just forceful, it is also aimed correctly at the reactive site.

Building orientation into the rate expression

To account for this, collision theory introduces a second factor alongside energy.

Steric factor / probability factor ($P$) — a factor that reflects the probability that colliding molecules are oriented correctly for a reaction to occur.

Bringing $P$ into the earlier expression gives the more complete rate law of collision theory:

Here:

• $P$ — the steric (probability) factor, accounting for proper orientation
• $Z_{AB}$ — the collision frequency between reactants A and B
• $E_a$ — the activation energy of the reaction
• $R$ — the gas constant
• $T$ — the absolute temperature …