Editing Adsorption (section)

===Kisliuk===
[[File:Wiki kisliuk n2-tungsten.JPG|thumb|Two adsorbate nitrogen molecules adsorbing onto a tungsten adsorbent from the precursor state around an island of previously adsorbed adsorbate (left) and via random adsorption (right)]]

In other instances, molecular interactions between gas molecules previously adsorbed on a solid surface form significant interactions with gas molecules in the gaseous phases. Hence, adsorption of gas molecules to the surface is more likely to occur around gas molecules that are already present on the solid surface, rendering the Langmuir adsorption isotherm ineffective for the purposes of modelling. This effect was studied in a system where nitrogen was the adsorbate and tungsten was the adsorbent by Paul Kisliuk (1922–2008) in 1957.<ref>{{cite journal |last1=Kisliuk |first1=P. |title=The sticking probabilities of gases chemisorbed on the surfaces of solids |journal=Journal of Physics and Chemistry of Solids |date=January 1957 |volume=3 |issue=1–2 |pages=95–101 |doi=10.1016/0022-3697(57)90054-9 |bibcode=1957JPCS....3...95K }}</ref> To compensate for the increased probability of adsorption occurring around molecules present on the substrate surface, Kisliuk developed the precursor state theory, whereby molecules would enter a precursor state at the interface between the solid adsorbent and adsorbate in the gaseous phase. From here, adsorbate molecules would either adsorb to the adsorbent or desorb into the gaseous phase. The probability of adsorption occurring from the precursor state is dependent on the adsorbate's proximity to other adsorbate molecules that have already been adsorbed. If the adsorbate molecule in the precursor state is in close proximity to an adsorbate molecule that has already formed on the surface, it has a sticking probability reflected by the size of the S<sub>E</sub> constant and will either be adsorbed from the precursor state at a rate of ''k''<sub>EC</sub> or will desorb into the gaseous phase at a rate of ''k''<sub>ES</sub>. If an adsorbate molecule enters the precursor state at a location that is remote from any other previously adsorbed adsorbate molecules, the sticking probability is reflected by the size of the S<sub>D</sub> constant.

These factors were included as part of a single constant termed a "sticking coefficient", ''k''<sub>E</sub>, described below:

:<math>k_\text{E} = \frac{S_\text{E}}{k_\text{ES} S_\text{D}}.</math>

As S<sub>D</sub> is dictated by factors that are taken into account by the Langmuir model, S<sub>D</sub> can be assumed to be the adsorption rate constant. However, the rate constant for the Kisliuk model (''R''’) is different from that of the Langmuir model, as ''R''’ is used to represent the impact of diffusion on monolayer formation and is proportional to the square root of the system's diffusion coefficient. The Kisliuk adsorption isotherm is written as follows, where θ<sub>(''t'')</sub> is fractional coverage of the adsorbent with adsorbate, and ''t'' is immersion time:

:<math>\frac{d\theta_{(t)}}{dt} = R'(1 - \theta)(1 + k_\text{E}\theta).</math>

Solving for θ<sub>(''t'')</sub> yields:

:<math>\theta_{(t)} = \frac{1 - e^{-R'(1 + k_\text{E})t}}{1 + k_\text{E} e^{-R'(1 + k_\text{E})t}}.</math>