Reaction rate tends to increase with concentration phenomenon explained by collision theory

Collision theory is a principle of chemistry used to predict the rates of chemical reactions. It states that when suitable particles of the reactant 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 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. The activation energy is often predicted using the Transition 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 by Max Trautz in 1916^[1] and William 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 to chemical 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 by diffusion or Brownian motion of individual molecules. The flux of the diffusive molecules follows Fick's laws of diffusion. For particles in a solution, an example model to calculate the collision frequency and associated coagulation rate is the Smoluchowski coagulation equation proposed by Marian 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]

Rate equations

The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is^[6]

${\displaystyle r(T)=kn_{\text{A}}n_{\text{B}}=Z\rho \exp \left({\frac {-E_{\text{a}}}{RT}}\right)}$

where:

• k is the rate constant in units of (number of molecules)⁻¹⋅s⁻¹⋅m³.
• n_A is the number density of A in the gas in units of m⁻³.
• n_B is the number density of B in the gas in units of m⁻³. E.g. for a gas mixture with gas A concentration 0.1 mol⋅L⁻¹ and B concentration 0.2 mol⋅L⁻¹, the number of density of A is 0.1×6.02×10²³÷10⁻³ = 6.02×10²⁵ m⁻³, the number of density of B is 0.2×6.02×10²³÷10⁻³ = 1.2×10²⁶ m⁻³.
• Z is the collision frequency in units of m⁻³⋅s⁻¹.
• ${\displaystyle \rho }$ is the steric factor.^[7]
• E_a is the activation energy of the reaction, in units of J⋅mol⁻¹.
• T is the temperature in units of K.
• R is the gas constant in units of J mol⁻¹K⁻¹.

The unit of r(T) can be converted to mol⋅L⁻¹⋅s⁻¹, after divided by (1000×N_A), where N_A is the Avogadro constant.

For a reaction between A and B, the collision frequency calculated with the hard-sphere model with the unit number of collisions per m³ per second is:

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

where:

• σ_AB is the reaction cross section (unit m²), the area when two molecules collide with each other, simplified to ${\displaystyle \sigma _{\text{AB}}=\pi (r_{\text{A}}+r_{\text{B}})^{2}}$, where r_A the radius of A and r_B the radius of B in unit m.
• k_B is the Boltzmann constant unit J⋅K⁻¹.
• T is the absolute temperature (unit K).
• μ_AB is the reduced mass of the reactants A and B, ${\displaystyle \mu _{\text{AB}}={\frac {{m_{\text{A}}}{m_{\text{B}}}}{{m_{\text{A}}}+{m_{\text{B}}}}}}$ (unit kg).
• N_A is the Avogadro constant.
• [A] is molar concentration of A in unit mol⋅L⁻¹.
• [B] is molar concentration of B in unit mol⋅L⁻¹.

If all the units that are related to dimension are converted to dm, i.e. mol⋅dm⁻³ for [A] and [B], dm² for σ_AB, dm²⋅kg⋅s⁻²⋅K⁻¹ for the Boltzmann constant, then

${\displaystyle Z=N_{\text{A}}\sigma _{\text{AB}}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{\text{AB}}}}}[{\text{A}}][{\text{B}}]=k[A][B]}$

unit mol⋅dm⁻³⋅s⁻¹.

Quantitative insights

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,^[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.

A scheme comparing direct collision and diffusive collision, with corresponding rate equations.

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 model and the JChen Model.

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:^[4]

${\displaystyle Z_{AB}=4\pi RD_{r}C_{A}C_{B}}$

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

${\displaystyle Z_{AB}=1000N_{A}*4\pi RD_{r}[A][B]=k[A][B]}$

where:

• ${\displaystyle Z_{AB}}$ is in unit mole collisions/s in 1 L of solution.
• ${\displaystyle N_{\text{A}}}$ is the Avogadro constant.
• ${\displaystyle D_{r}}$ 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.

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 the mean free path time or the average first passenger time. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution:^[5]

${\displaystyle Z_{AB}=(1000N_{A})^{4/3}*8\pi ^{-1}A\beta D_{r}([A]+[B])^{1/3}[A][B]=k([A]+[B])^{1/3}[A][B]}$

where:

• ${\displaystyle Z_{AB}}$ is in unit mole collisions/s in 1 L of solution.
• ${\displaystyle N_{\text{A}}}$ is the Avogadro constant.
• ${\displaystyle A}$ is the area of the collision cross-section in unit m².
• ${\displaystyle \beta }$ is the product of the unitless fractions of reactive surface area on A and B. ${\displaystyle \beta A}$ is the effective adsorption cross-section area.
• ${\displaystyle D_{r}}$ is the relative diffusion constant between A and B, unit m²/s, and ${\displaystyle D_{r}=D_{A}+D_{B}}$.
• ${\displaystyle [A]}$ 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^4/3 mol^-4/3 s⁻¹.

See also

• Two-dimensional gas
• Rate equation

References

External links

• Introduction to Collision Theory