Concentration gradient

Sorption rate and collision frequency of diluted solute

A more problematic result of the above equations is they predict the lower limit of adsorption under ideal situations but is very difficult to predict the actual adsorption rates. The equations are derived at the long-time-limit condition when a stable concentration gradient has been formed near the surface. But real adsorption is often done much faster than this infinite time limit i.e. the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have predicted for low or none energy barrier adsorption (unless there is a significant adsorption energy barrier that slows down the absorption significantly), for example, thousands to millions time faster in the self-assembly of monolayers at the water-air or water-substrate interfaces. As such, it is necessary to calculate the evolution of the concentration gradient near the surface and find out a proper time to stop the imagined infinite evolution for practical applications. While it is hard to predict when to stop but it is reasonably easy to calculate the shortest time that matters, the critical time when the first nearest neighbor from the substrate surface feels the building-up of the concentration gradient. This yields the upper limit of the adsorption rate under an ideal situation when there are no other factors than diffusion that affect the absorber dynamics: $${\displaystyle \langle r\rangle ={\frac {4}{\pi }}AC_{b}^{4/3}D,}$$where: