Collision theory

Collision theory, proposed by Max Trautz and William Lewis in 1916 and 1918, qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. It assumes that, for a reaction to occur, the reactant particles must collide, but only a certain fraction of the total collisions, the effective collisions, cause the transformation of reactant molecules into products. This is due to the fact that only a fraction of the molecules have sufficient energy and the right orientation at the moment of impact to break the existing bonds and form new bonds. The minimal amount of energy needed so that the molecule is transformed is called activation energy. Collision theory is closely related to chemical kinetics.

Rate constant
The rate constant for a bimolecular gas phase reaction, as predicted by collision theory is:

$$k(T) = Z \rho \exp \left( \frac{-E_{a}}{RT} \right)$$.


 * Z is the collision frequency.
 * $$\scriptstyle \rho$$ is the steric factor.
 * Ea is the activation energy of the reaction.
 * T is the temperature.
 * R is gas constant.

And the collision frequency is:

$$Z = N_A \sigma_{AB} \sqrt \frac{8 k_B T}{\pi \mu_{AB}}$$


 * NA is Avogadro's number
 * σAB is the reaction cross section
 * kB is Boltzmann's constant
 * μAB is the reduced mass of the reactants

Qualitative overview
Fundamentally collision theory is based on kinetic theory and therefore it can only be applied strictly to ideal gases, otherwise approximations are used. Qualitatively, it assumes that the molecules of the reactants are rigid, uncharged spheres that physically collide prior to reacting. Moreover, it postulates that the majority of collisions do not lead to a reaction, but only those in which the colliding species have:


 * A kinetic energy greater than a certain minimum, called the activation energy, Ea
 * The correct spatial orientation (steric factor) with respect to each other.

These collisions which lead to reaction are called effective collisions. The reaction rate, may be defined as the number of effective collisions per unit time.

According to collision theory, two significant factors determine reaction rates:


 * Concentration: Increase in concentration of reactants increases the collision frequency between the reactants. Thus the effective collision frequency also increases.
 * Temperature: The kinetic energy of particles follows the Maxwell-Boltzmann distribution. An increase in temperature not only increases the average speed of the reactant particles and the number of collisions, but also the fraction of particles having kinetic energy higher than the activation energy. Thus, the effective collision frequency increases.

If a heterogeneous reaction takes place, then the surface area of the solid is also important: the more reactive centers exposed on the surface (due to the porosity of the solid and how finely divided it is), the more collisions with reacting molecules.

Derivation
Collision theory can only be applied quantitatively to bimolecular reactions, of the kind:


 * 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 (σAB) of the reaction and is, in principle, the area corresponding to a circle whose radius (rAB) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume $$\scriptstyle \pi r^{2}_{AB} c_A$$ per second as it moves, where $$\scriptstyle c_A$$ is the average velocity of the particle.

From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of $$c_A = \sqrt \frac{8 k_B T}{\pi m_A}$$. Where $$\scriptstyle k_B$$ is Boltzmann constant and $$\scriptstyle m_A$$ 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 $$\mu_{AB}$$ must be used instead of $$m_A$$.

Therefore, the total collision frequency, of all A molecules, with all B molecules, is:

$$N_A^{2} \sigma_{AB} \sqrt \frac{8 k_B T}{\pi \mu_{AB}}[A][B] =N_A^{2} r^{2}_{AB} \sqrt \frac{8 \pi k_B T}{ \mu_{AB}}[A][B] = Z [A][B] $$

From Maxwell Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is $$e^{\frac{-E_a}{k_BT}}$$. Therefore the rate of a bimolecular reaction for ideal gases will be:

$$r = Z \rho [A][B] \exp \left( \frac{-E_{a}}{RT} \right)$$

Where:
 * Z is the collision frequency.
 * $$\scriptstyle \rho$$ is the steric factor, which will be discussed in detail in the next section.
 * Ea is the activation energy of the reaction.
 * T is the temperature.
 * R is gas constant.

The product Zρ is equivalent to the preexponential factor of the Arrhenius equation.

Validity of the theory and steric factor
Once a theory is formulated, its validity must be tested, that is, compare its predictions with the results of the experiments.

When the expression form of the rate constant is compared with the rate equation for an elementary bimolecular reaction, $$\scriptstyle r =k(T) [A][B]$$, it is noticed that $$k(T) = N_A \sigma_{AB} \sqrt \frac{8 k_B T}{\pi m_A} \exp \left( \frac{-E_{a}}{RT} \right)$$.

That expression is similar to the Arrhenius equation, and gives the first theoretical explanation for the Arrhenius equation on a molecular basis. The weak temperature dependence of the preexponential factor is so small compared to the exponential factor that it cannot be measured experimentally, that is, "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T½ dependence of the preexponential factor is observed experimentally"

Steric factor
If the values of the predicted rate constants are compared with the values of known rate constants it is noticed that collision theory fails to estimate the constants correctly and the more complex the molecules are, the more it fails. The reason for this is that particles have been supposed to be spherical and able to react in all directions; that is not true, as the orientation of the collisions is not always the right one. For example in the hydrogenation reaction of ethylene the H2 molecule must approach the bonding zone between the atoms, and only a few of all the possible collisions fulfill this requirement.

A new concept must be introduced: the steric factor, $$\rho$$. It is defined as the ratio between the experimental value and the predicted one (or the ratio between the frequency factor and the collision frequency, and it is most often less than unity(one). $$\rho = \frac{A_{observed}}{Z_{calculated}}$$

Usually, the more complex the reactant molecules, the lower the steric factor. Nevertheless, some reactions exhibit steric factors greater than unity: the harpoon reactions, which involve atoms that exchange electrons, producing ions. The deviation from unity can have different causes: the molecules are not spherical, so different geometries are possible; not all the kinetic energy is delivered into the right spot; the presence of a solvent (when applied to solutions), etc.

Collision theory can be applied to reactions in solution; in that case, the solvent cage has an effect on the reactant molecules and several collisions can take place in a single encounter, which leads to predicted preexponential factors being too large. ρ values greater than unity can be attributed to favorable entropic contributions.