Collision theory is a principle ofchemistry used to predict the rates ofchemical reactions. It states that when suitable particles of thereactant hit each other with the correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known asactivation 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. Theactivation energy is often predicted using thetransition state theory. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently byMax Trautz in 1916[1] andWilliam Lewis in 1918.[2][3]
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 the reaction to occur. The reaction rate therefore increases.
Collision theory is closely related tochemical kinetics.
Collision theory was initially developed for the gas reaction system with no dilution. But most reactions involve solutions, for example, gas reactions in a carrying inert gas, and almost all reactions in solutions. The collision frequency of the solute molecules in these solutions is now controlled bydiffusion orBrownian motion of individual molecules. The flux of the diffusive molecules followsFick's laws of diffusion. For particles in a solution, an example model to calculate the collision frequency and associated coagulation rate is theSmoluchowski coagulation equation proposed byMarian Smoluchowski in a seminal 1916 publication.[4] In this model, Fick's flux at the infinite time limit is used to mimic the particle speed of the collision theory. Jixin Chen proposed a finite-time solution to the diffusion flux in 2022 which significantly changes the estimated collision frequency of two particles in a solution.[5]
The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is[6]
where:
The unit ofr(T) can be converted to mol⋅L−1⋅s−1, after divided by (1000×NA), whereNA is theAvogadro constant.
For a reaction between A and B, thecollision frequency calculated with the hard-sphere model with the unit number of collisions per m3 per second is:
where:
If all the units that are related to dimension are converted to dm, i.e. mol⋅dm−3 for [A] and [B], dm2 forσAB, dm2⋅kg⋅s−2⋅K−1 for theBoltzmann constant, then
unit mol⋅dm−3⋅s−1.
Consider the bimolecular elementary reaction:
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 thecross section (σAB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius () is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume per second as it moves, where 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 is related to the length scale of their interaction potential.)
Fromkinetic theory it is known that a molecule of A has anaverage velocity (different fromroot mean square velocity) of, where is theBoltzmann constant, and is the mass of the molecule.
The solution of thetwo-body problem states that two different moving bodies can be treated as one body which has thereduced mass of both and moves with the velocity of thecenter of mass, so, in this system must be used instead of.Thus, for a given molecule A, it travels before hitting a molecule B if all B is fixed with no movement, where 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 totalcollision frequency,[8] of all A molecules, with all B molecules, is
From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is. Therefore, the rate of a bimolecular reaction for ideal gases will be
where:
The productzρ is equivalent to thepreexponential factor of theArrhenius equation.
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 therate equation for an elementary bimolecular reaction,, it is noticed that
unit M−1⋅s−1 (= dm3⋅mol−1⋅s−1), with all dimension unit dm includingkB.
This expression is similar to theArrhenius 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 predictedT1/2 dependence of the preexponential factor is observed experimentally".[9]
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 thehydrogenation reaction ofethylene 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: thesteric factorρ. It is defined as the ratio between the experimental value and the predicted one (or the ratio between thefrequency factor and the collision frequency):
and it is most often less than unity.[7]
Usually, the more complex the reactant molecules, the lower the steric factor. Nevertheless, some reactions exhibit steric factors greater than unity: theharpoon reactions, which involve atoms that exchangeelectrons, producingions. 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.
| 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, thesolvent 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 favorableentropic contributions.
| Reaction | Solvent | A, 1011 s−1⋅M−1 | Z, 1011 s−1⋅M−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 |
Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated fromFick's laws of diffusion. Theoretical models to calculate the collision frequency in solutions have been proposed byMarian Smoluchowski in a seminal 1916 publication at the infinite time limit,[4] and Jixin Chen in 2022 at a finite-time approximation.[5] A scheme of comparing the rate equations in pure gas and solution is shown in the right figure.
For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable whendiffusion takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. 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 to react with. Thus the probability of collision should be calculated using theBrownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski model and the JChen Model.
For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from theFick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution:[4]
where:
or
where:
There have been a lot of extensions and modifications to the Smoluchowski model since it was proposed in 1916.
In 2022, Chen rationales that because the diffusive flux is evolving over time and the distance between the molecules has a finite value at a given concentration, there should be a critical time to cut off the evolution of the flux that will give a value much larger than the infinite solution Smoluchowski has proposed.[5] So he proposes to use the average time for two molecules to switch places in the solution as the critical cut-off time, i.e., first neighbor visiting time. Although an alternative time could be themean free path time or the average first passenger time, it overestimates the concentration gradient between the original location of the first passenger to the target. This hypothesis yields afractal reaction kinetic rate equation of diffusive collision in a diluted solution:[5]
where: