Collision theory

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 section (σ_AB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius (${\displaystyle r_{AB}}$) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume ${\displaystyle \pi r_{AB}^2{c}_A}$ per second as it moves, where ${\displaystyle c_A}$ 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 ${\displaystyle r_{AB}}$ 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 ${\displaystyle c_A=\sqrt{\frac{8k_{\text{B}}T}{\pi m_A}}}$, where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle m_A}$ 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 ${\displaystyle {\mu }_{AB}}$ must be used instead of ${\displaystyle m_A}$. Thus, for a given molecule A, it travels ${\displaystyle t=l/c_A=1/(n_B{\sigma }_{AB}{c}_A)}$ before hitting a molecule B if all B is fixed with no movement, where ${\displaystyle l}$ 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

${\displaystyle Z=n_{\text{A}}{n}_{\text{B}}{\sigma }_{AB}\sqrt{\frac{8k_{\text{B}}T}{\pi {\mu }_{AB}}}={10}^6{N}_A^2[A][B]{\sigma }_{AB}\sqrt{\frac{8k_{\text{B}}T}{\pi {\mu }_{AB}}}=z[A][B],}$

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

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

where:

• Z is the collision frequency with unit s⁻¹⋅m⁻³. The z is Z without [A][B].
• ${\displaystyle \rho }$ is the steric factor, which will be discussed in detail in the next section,
• E_a 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.

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, ${\displaystyle r=k(T)[A][B]}$, it is noticed that

${\displaystyle k(T)=N_A{\sigma }_{AB}\rho \sqrt{\frac{8k_{\text{B}}T}{\pi {\mu }_{AB}}}\exp \left(\frac{-E_{\text{a}}}{RT}\right)}$

unit M⁻¹⋅s⁻¹ (= dm³⋅mol⁻¹⋅s⁻¹), with all dimension unit dm including k_B.

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 T^⁠1/2⁠ 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):

${\displaystyle \rho =\frac{A_{\text{observed}}}{Z_{\text{calculated}}},}$

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.

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.

Alternative collision models for diluted solutions

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. Theoretical models to calculate the collision frequency in solutions have been proposed by Marian Smoluchowski in a seminal 1916 publication at the infinite time limit.

For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable when diffusion 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 the Brownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski.

For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from the Fick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution:

${\displaystyle Z_{AB}=4\pi R{D}_r{C}_A{C}_B}$

where:

• ${\displaystyle Z_{AB}}$ is the collision frequency, unit #collisions/s in 1 m³ of solution.
• ${\displaystyle R}$ is the radius of the collision cross-section, unit m.
• ${\displaystyle D_r}$ is the relative diffusion constant between A and B, unit m²/s, and ${\displaystyle D_r=D_A+D_B}$.
• ${\displaystyle C_A}$ and ${\displaystyle C_B}$ are the number concentrations of molecules A and B in the solution respectively, unit #molecule/m³.

or

${\displaystyle Z_{AB}=1000N_A\ast 4\pi R{D}_r[A][B]=k[A][B]}$

where:

• ${\displaystyle Z_{AB}}$ is in unit mole collisions/s in 1 L of solution.
• ${\displaystyle N_{\text{A}}}$ is the Avogadro constant.
• ${\displaystyle R}$ is the radius of the collision cross-section, unit m.
• ${\displaystyle D_r}$ is the relative diffusion constant between A and B, unit m²/s.
• ${\displaystyle [A]}$ and ${\displaystyle [B]}$ are the molar concentrations of A and B respectively, unit mol/L.
• ${\displaystyle k}$ is the diffusive collision rate constant, unit L mol⁻¹ s⁻¹.