Editing Adsorption (section)

===Single-molecule explanation===
The adsorption of ensemble molecules on a surface or interface can be divided into two processes: adsorption and desorption. If the adsorption rate wins the desorption rate, the molecules will accumulate over time giving the adsorption curve over time. If the desorption rate is larger, the number of molecules on the surface will decrease over time. The adsorption rate is dependent on the temperature, the diffusion rate of the solute (related to mean free path for pure gas), and the [[activation energy|energy barrier]] between the molecule and the surface. The diffusion and key elements of the adsorption rate can be calculated using [[Fick's laws of diffusion]] and [[Einstein relation (kinetic theory)]].
Under ideal conditions, when there is no energy barrier and all molecules that diffuse and collide with the surface get adsorbed, the number of molecules adsorbed <math>\Gamma</math> at a surface of area <math>A</math> on an infinite area surface can be directly integrated from [[Fick's second law]] differential equation to be:<ref>{{Cite journal| author1 = Langmuir, I. | author2 = Schaefer, V.J.| date = 1937 | title = The Effect of Dissolved Salts on Insoluble Monolayers| journal = Journal of the American Chemical Society | volume = 29 | issue = 11 | pages = 2400–2414 | doi = 10.1021/ja01290a091| bibcode = 1937JAChS..59.2400L}}</ref>

:<math> \Gamma= 2AC\sqrt{\frac{Dt}{\pi}}</math>

where <math>A</math> is the surface area (unit m<sup>2</sup>), <math>C</math> is the number concentration of the molecule in the bulk solution (unit #/m<sup>3</sup>), <math>D</math> is the diffusion constant (unit m<sup>2</sup>/s), and <math>t</math> is time (unit s). Further simulations and analysis of this equation<ref name="ReferenceA">{{Cite journal| author1 = Chen, Jixin | date = 2020 | title = Stochastic Adsorption of Diluted Solute Molecules at Interfaces | journal = ChemRxiv | doi = 10.26434/chemrxiv.12402404| s2cid = 242860958 }}</ref> show that the square root dependence on the time is originated from the decrease of the concentrations near the surface under ideal adsorption conditions. Also, this equation only works for the beginning of the adsorption when a well-behaved concentration gradient forms near the surface. Correction on the reduction of the adsorption area and slowing down of the concentration gradient evolution have to be considered over a longer time.<ref>{{Cite journal| author1 = Ward, A.F.H. | author2 = Tordai, L.|date=1946| title = Time-dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time-effects| journal = Journal of Chemical Physics | volume = 14 | issue = 7| pages = 453–461 | doi = 10.1063/1.1724167| bibcode = 1946JChPh..14..453W}}</ref>
Under real experimental conditions, the flow and the small adsorption area always make the adsorption rate faster than what this equation predicted, and the energy barrier will either accelerate this rate by surface attraction or slow it down by surface repulsion. Thus, the prediction from this equation is often a few to several orders of magnitude away from the experimental results. Under special cases, such as a very small adsorption area on a large surface, and under [[chemical equilibrium]] when there is no concentration gradience near the surface, this equation becomes useful to predict the adsorption rate with debatable special care to determine a specific value of <math>t</math> in a particular measurement.<ref name="ReferenceA"/>

The desorption of a molecule from the surface depends on the binding energy of the molecule to the surface and the temperature. The typical overall adsorption rate is thus often a combined result of the adsorption and desorption.