# Collision theory

Collision theory is a theory that was proposed independently by Max Trautz in 1916 and William Lewis in 1918. The collision theory states that when suitable particles of the reactant hit each other, only a certain fraction of the collisions cause any noticeable or significant chemical change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant particles or raising the temperature - which brings about more collisions and hence more successful collisions - therefore increases the rate of a reaction.

When a catalyst is involved in the collision between the reactant molecules, less energy is required for the chemical change to take place, and hence more collisions have sufficient energy for reaction to occur. The reaction rate therefore increases.

Collision theory is closely related to chemical kinetics.

## Rate constant

The rate for a bimolecular gas-phase reaction, as predicted by collision theory is

$r(T)=kn_{A}n_{B}=Z\rho \exp \left({\frac {-E_{\text{a}}}{RT}}\right),$ unit is number of molecular reactions $s^{-1}m^{-3}$ , which can be converted to $molL^{-1}s^{-1}$ by divided by $10^{3}N_{A}$ , where $N_{A}$ is the Avogadro constant.

where:

k is the rate constant in unit (number of molecules) $s^{-1}m^{3}$ ,
nA is the number density of A in the gas (unit $m^{-3}$ ),
nB is the number density of B in the gas (unit $m^{-3}$ ), E.g. for a gas with gas A concentration 0.1 mol/L and B concentration 0.2 mol/L, the number of density of A is $0.1*6.02*10^{23}/10^{-3}=6.02*10^{25}m^{-3}$ , the number of density of B is $0.2*6.02*10^{23}/10^{-3}=1.20*10^{26}m^{-3}$ .
Z is the collision frequency (unit $m^{-3}s^{-1}$ ),
$\rho$ is the steric factor,
Ea is the activation energy of the reaction, (unit J/mol),
T is the temperature (unit K),
R is the gas constant (unit J/mol/K).

For a reaction between A and B, the collision frequency is number of A times number of B times the collision area which is inverse the average distance of A, B times the average relative speed between A and B:

$Z=n_{\text{A}}n_{\text{B}}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}=10^{6}N_{A}^{2}[A][B]\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}},$ SI unit of Z is number of collision per $m^{3}$ per second.

where:

• σAB is the reaction cross section (unit $m^{2}$ ), the area when two molecules collide with each other, simplified to $\sigma _{AB}=\pi (r_{A}+r_{B})^{2}$ , where $r_{A}$ the radius of A and $r_{B}$ the radius of B. This is not the true reaction picture but over simplified. The effective area should be smaller whose effect can be combined into the reaction steric factor in the rate equation.
• kB is the Boltzmann's constant unit J/K = m2⋅kg/(s2⋅K),
• T is the temperature (unit K),
• μAB is the reduced mass of the reactants A and B, $\mu _{AB}={\frac {{m_{A}}{m_{B}}}{{m_{A}}+{m_{B}}}}$ (unit kg)
• [A] is molar concentration of A in unit mol/L,
• [B] is molar concentraiton of B in unit mol/L.

if all units related to dimension are converted to dm, i.e. mol dm−3 for [A],[B], dm2 for $\sigma _{AB}$ , use dm in Boltzmann's constant, then

$Z=N_{\text{A}}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}[A][B],$ (unit mol·dm−3 ·s−1).

### Collision in diluted solution

Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion.

For a diluted solution in the gas or the liquid phase, the above equation is not suitable when diffusion takes control of the collision frequency, i.e., The direct collision between the two molecules no longer dominate. For any given molecule A, it has to collide with a lot of solvent molecules, let's say molecule C, before finding the B molecule.

## Quantitative insights

### 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 sectionAB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius ($r_{AB}$ ) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume $\pi r_{AB}^{2}c_{A}$ per second as it moves, where $c_{A}$ 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 $r_{AB}$ is related to the length scale of their interaction potential.)

From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of $c_{A}={\sqrt {\frac {8k_{\text{B}}T}{\pi m_{A}}}}$ , where $k_{\text{B}}$ is Boltzmann constant, and $m_{A}$ 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 $m_{A}$ . Thus, for a given molecule A, it travels $t=l/c_{A}=1/(n_{B}\sigma _{AB}c_{A})$ before hitting a molecule B if all B is fixed with no movement, where $l$ 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, of all A molecules, with all B molecules, is

$Z=n_{\text{A}}n_{\text{B}}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}=10^{6}N_{A}^{2}[A][B]\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}=z[A][B],$ From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is $e^{\frac {-E_{\text{a}}}{RT}}$ . Therefore, the rate of a bimolecular reaction for ideal gases will be

$r=z\rho [A][B]\exp \left({\frac {-E_{\text{a}}}{RT}}\right),$ in unit number of molecular reactions $s^{-1}m^{-3}$ Where:

Z is the collision frequency with unit $s^{-1}m^{-3}$ ,
$\rho$ is the steric factor, which will be discussed in detail in the next section,
Ea 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 concentraiton of B in unit mol/L.

The product 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, $r=k(T)[A][B]$ , it is noticed that

$k(T)=N_{A}\sigma _{AB}\rho {\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}\exp \left({\frac {-E_{\text{a}}}{RT}}\right)$ , unit $M^{-1}s^{-1}(=dm^{3}mol^{-1}s^{-1})$ , with all dimension unit dm.

This 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, which is not true, as the orientation of the collisions is not always proper for the reaction. 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.

To alleviate this problem, a new concept must be introduced: the steric factor ρ. 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):

$\rho ={\frac {A_{\text{observed}}}{Z_{\text{calculated}}}},$ and it is most often less than unity.

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.

Experimental rate constants compared to the ones predicted by collision theory for gas phase reactions
Reaction A, s−1M−1 Z, s−1M−1 Steric factor
2ClNO → 2Cl + 2NO 9.4×109 5.9×1010 0.16
2ClO → Cl2 + O2 6.3×107 2.5×1010 2.3×10−3
H2 + C2H4 → C2H6 1.24×106 7.3×1011 1.7×10−6
Br2 + K → KBr + Br 1.0×1012 2.1×1011 4.3

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.

Experimental rate constants compared to the ones predicted by collision theory for reactions in solution
Reaction Solvent A, 1011 s−1M−1 Z, 1011 s−1M−1 Steric factor
C2H5Br + OH ethanol 4.30 3.86 1.11
C2H5O + CH3I ethanol 2.42 1.93 1.25
ClCH2CO2 + OH water 4.55 2.86 1.59
C3H6Br2 + I methanol 1.07 1.39 0.77
HOCH2CH2Cl + OH water 25.5 2.78 9.17
4-CH3C6H4O + CH3I ethanol 8.49 1.99 4.27
CH3(CH2)2Cl + I acetone 0.085 1.57 0.054
C5H5N + CH3I C2H2Cl4 2.0 10×10−6