Collision frequency at surfaces

Question

Collision frequency for particles in gases is well known, and collision theory is used to derive chemical reaction rates in gases, (and particles in liquid solutions as well). Using the mean velocity as a function of temperature, one arrives at $Z = N_A \sigma_{AB}\sqrt{\frac{8k_BT}{\pi\mu_{AB}}}$.

I need something similar where the collisions/interactions can only occur at the surface between two areas. I understand phenomenologically the collision frequency will be proportional to the area, but I am looking for something a little more rigorous. I have two types of particles, A and B. They occupy separate volumes of space and do not mix. One can assume no mixing because the time window I am interested in is too small compared to the average velocities or because of surface tension. I haven't thought it out completely yet.

The only thing I've thought of yet is to define some sort of mixing depth, and then just use the traditional collision frequency, multiplying by the mixing depth to arrive at collision frequency per unit surface area. But then the mixing depth should itself be derived from the average velocities of the particles and it should not really have sharp boundaries.

What is the canonical kind of model used for similar problems, such as reactions on liquid/liquid or liquid/gas boundaries?

Answer 1

Collision theory will not help for calculating chemical rates in the liquid phase since the mass transport there is governed by diffusion. It is not perfectly clear from your question, so I will have to assume that you are interested in a two-phase gas/liquid system.

First, liquid phase is much denser, molecules there are always in contact and don't have to wait for collision so reaction rates in the liquid phase are much higher. This means that the major reaction pathway has these steps

1. Molecule A in the gas phase collides with the surface (defined by the collision theory)
2. The molecule has a certain chance $\alpha$ to be absorbed by the liquid and transferred to the bulk phase.
3. Molecule A diffuses through the solution, reacting with molecule B.

Depending on your system, any of these steps could be rate-limiting. Very often it is diffusion-limited at step 3: reaction depletes molecules B close to the surface so that molecules have to diffuse further before they have a chance to react.

Accommodation coefficient $\alpha$ is can be just 0.001 - most of the gas phase molecules will bounce back from the surface, no matter how reactive they are.