text: 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, and Jixin Chen in 2022 at a finite-time approximation. 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 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:
  where:
  *  is the collision frequency, unit #collisions/s in 1 m of solution.
  *  is the radius of the collision cross-section, unit m.
  *  is the relative diffusion constant between A and B, unit m/s, and .
  *  and  are the number concentrations of molecules A and B in the solution respectively, unit #molecule/m.
  or
  where:
  *  is in unit mole collisions/s in 1 L of solution.
  *  is the Avogadro constant.
  *  is the radius of the collision cross-section, unit m.
  *  is the relative diffusion constant between A and B, unit m/s. 
  *  and  are the molar concentrations of A and B respectively, unit mol/L.
  *  is the diffusive collision rate constant, unit L mol s.
  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. 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, it overestimates the concentration gradient between the original location of the first passenger to the target. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution:  
  where:
  *  is in unit mole collisions/s in 1 L of solution.
  *  is the Avogadro constant.
  *  is the area of the collision cross-section in unit m.
  *  is the product of the unitless fractions of reactive surface area on A and B.  is the effective adsorption cross-section area.
  *  is the relative diffusion constant between A and B, unit m/s, and .
  *  and  are the molar concentrations of A and B respectively, unit mol/L.
  *  is the diffusive collision rate constant, unit L mol s.