Editing Adsorption (section)

==Isotherms==
The adsorption of gases and solutes is usually described through isotherms, that is, the amount of adsorbate on the adsorbent as a function of its pressure (if gas) or concentration (for liquid phase solutes) at constant temperature. The quantity adsorbed is nearly always normalized by the mass of the adsorbent to allow comparison of different materials. To date, 15 different isotherm models have been developed.<ref name="FooHameed2010">{{cite journal |last1=Foo |first1=K. Y. |last2=Hameed |first2=B. H. |title=Insights into the modeling of adsorption isotherm systems |journal=Chemical Engineering Journal |volume=156 |issue=1 |year=2010 |pages=2–10 |issn=1385-8947 |doi=10.1016/j.cej.2009.09.013|s2cid=11760738 }}</ref>

=== Freundlich ===
{{Main|Freundlich equation}}
The first mathematical fit to an isotherm was published by Freundlich and Kuster (1906) and is a purely empirical formula for gaseous adsorbates:

:<math>\frac{x}{m} = kP^{1/n},</math>

where <math>x</math> is the mass of adsorbate adsorbed, <math>m</math> is the mass of the adsorbent, <math>P</math> is the pressure of adsorbate (this can be changed to concentration if investigating solution rather than gas), and <math>k</math> and <math>n</math> are empirical constants for each adsorbent–adsorbate pair at a given temperature. The function is not adequate at very high pressure because in reality <math>x/m</math> has an asymptotic maximum as pressure increases without bound. As the temperature increases, the constants <math>k</math> and <math>n</math> change to reflect the empirical observation that the quantity adsorbed rises more slowly and higher pressures are required to saturate the surface.

===Langmuir===
{{See also|Langmuir equation}}
[[Irving Langmuir]] was the first to derive a scientifically based adsorption isotherm in 1918.<ref name=Langmuir/> The model applies to gases adsorbed on solid surfaces. It is a semi-empirical isotherm with a kinetic basis and was derived based on statistical thermodynamics. It is the most common isotherm equation to use due to its simplicity and its ability to fit a variety of adsorption data. It is based on four assumptions:
# All of the adsorption sites are equivalent, and each site can only accommodate one molecule.
# The surface is energetically homogeneous, and adsorbed molecules do not interact.
# There are no [[phase transition]]s.
# At the maximum adsorption, only a monolayer is formed. Adsorption only occurs on localized sites on the surface, not with other adsorbates.

These four assumptions are seldom all true: there are always imperfections on the surface, adsorbed molecules are not necessarily inert, and the mechanism is clearly not the same for the first molecules to adsorb to a surface as for the last. The fourth condition is the most troublesome, as frequently more molecules will adsorb to the monolayer; this problem is addressed by the [[#BET|BET isotherm]] for relatively flat (non-[[microporous material|microporous]]) surfaces. The Langmuir isotherm is nonetheless the first choice for most models of adsorption and has many applications in surface kinetics (usually called [[Langmuir–Hinshelwood kinetics]]) and [[thermodynamics]].

Langmuir suggested that adsorption takes place through this mechanism: <math>A_\text{g} + S \rightleftharpoons AS</math>, where ''A'' is a gas molecule, and ''S'' is an adsorption site. The direct and inverse rate constants are ''k'' and ''k''<sub>−1</sub>. If we define surface coverage, <math>\theta</math>, as the fraction of the adsorption sites occupied, in the equilibrium we have:

:<math>K = \frac{k}{k_{-1}} = \frac{\theta}{(1 - \theta)P},</math>

or

:<math>\theta = \frac{KP}{1 + KP},</math>

where <math>P</math> is the partial pressure of the gas or the molar concentration of the solution.
For very low pressures <math>\theta \approx KP</math>, and for high pressures <math>\theta \approx 1</math>.

The value of <math>\theta</math> is difficult to measure experimentally; usually, the adsorbate is a gas and the quantity adsorbed is given in moles, grams, or gas volumes at [[standard temperature and pressure]] (STP) per gram of adsorbent. If we call ''v''<sub>mon</sub> the STP volume of adsorbate required to form a monolayer on the adsorbent (per gram of adsorbent), then <math>\theta = \frac{v}{v_\text{mon}}</math>, and we obtain an expression for a straight line:

:<math>\frac{1}{v} = \frac{1}{Kv_\text{mon}}\frac{1}{P} + \frac{1}{v_\text{mon}}.</math>

Through its slope and ''y'' intercept we can obtain ''v''<sub>mon</sub> and ''K'', which are constants for each adsorbent–adsorbate pair at a given temperature. ''v''<sub>mon</sub> is related to the number of adsorption sites through the [[ideal gas law]]. If we assume that the number of sites is just the whole area of the solid divided into the cross section of the adsorbate molecules, we can easily calculate the surface area of the adsorbent.
The surface area of an adsorbent depends on its structure: the more pores it has, the greater the area, which has a big influence on [[reactions on surfaces]].

If more than one gas adsorbs on the surface, we define <math>\theta_E</math> as the fraction of empty sites, and we have:

:<math>\theta_E = \dfrac{1}{1 + \sum_{i=1}^n K_i P_i}.</math>

Also, we can define <math>\theta_j</math> as the fraction of the sites occupied by the ''j''-th gas:

:<math>\theta_j = \dfrac{K_j P_j}{1 + \sum_{i=1}^n K_i P_i},</math>

where ''i'' is each one of the gases that adsorb.

'''Note:'''

1) To choose between the Langmuir and Freundlich equations, the enthalpies of adsorption must be investigated.<ref name="Burke GM p V">Burke GM, Wurster DE, Buraphacheep V, Berg MJ, Veng-Pedersen P, Schottelius DD. Model selection for the adsorption of phenobarbital by activated charcoal. Pharm Res. 1991;8(2):228-231. doi:10.1023/a:1015800322286</ref> While the Langmuir model assumes that the energy of adsorption remains constant with surface occupancy, the Freundlich equation is derived with the assumption that the heat of adsorption continually decrease as the binding sites are occupied.<ref>Physical Chemistry of Surfaces. Arthur W. Adamson. Interscience (Wiley), New York 6th ed</ref> The choice of the model based on best fitting of the data is a common misconception.<ref name="Burke GM p V"/>

2) The use of the linearized form of the Langmuir model is no longer common practice. Advances in computational power allowed for nonlinear regression to be performed quickly and with higher confidence since no data transformation is required.

===BET===<!-- linked in the caption of the image in the lead -->
{{Main|BET theory}}
Often molecules do form multilayers, that is, some are adsorbed on already adsorbed molecules, and the Langmuir isotherm is not valid. In 1938 [[Stephen Brunauer]], [[Paul H. Emmett|Paul Emmett]], and [[Edward Teller]] developed a model isotherm that takes that possibility into account. Their theory is called [[BET theory]], after the initials in their last names. They modified Langmuir's mechanism as follows:

:A<sub>(g)</sub> + S ⇌ AS,

:A<sub>(g)</sub> + AS ⇌ A<sub>2</sub>S,

:A<sub>(g)</sub> + A<sub>2</sub>S ⇌ A<sub>3</sub>S and so on.

[[File:Isothermes.svg|thumb|Langmuir (blue) and BET (red) isotherms]]

The derivation of the formula is more complicated than Langmuir's (see links for complete derivation). We obtain:

:<math>\frac{x}{v(1 - x)} = \frac{1}{v_\text{mon}c} + \frac{x(c - 1)}{v_\text{mon}c},</math>

where ''x'' is the pressure divided by the [[vapor pressure]] for the adsorbate at that temperature (usually denoted <math>P/P_0</math>), ''v'' is the STP volume of adsorbed adsorbate, ''v<sub>mon</sub>'' is the STP volume of the amount of adsorbate required to form a monolayer, and ''c'' is the equilibrium constant ''K'' we used in Langmuir isotherm multiplied by the vapor pressure of the adsorbate. The key assumption used in deriving the BET equation that the successive heats of adsorption for all layers except the first are equal to the heat of condensation of the adsorbate.

The Langmuir isotherm is usually better for chemisorption, and the BET isotherm works better for physisorption for non-microporous surfaces.

===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>

===Adsorption enthalpy===
Adsorption constants are [[equilibrium constant]]s, therefore they obey the [[Van 't Hoff equation]]:

:<math>\left( \frac{\partial \ln K}{\partial \frac{1}{T}} \right)_\theta = -\frac{\Delta H}{R}.</math>

As can be seen in the formula, the variation of ''K'' must be isosteric, that is, at constant coverage.
If we start from the BET isotherm and assume that the entropy change is the same for liquefaction and adsorption, we obtain
:<math>\Delta H_\text{ads} = \Delta H_\text{liq} - RT\ln c,</math>
that is to say, adsorption is more exothermic than liquefaction.

===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.