Adsorption

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

${\displaystyle {\frac {x}{m}}=kP^{1/n},}$

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

Irving Langmuir was the first to derive a scientifically based adsorption isotherm in 1918.[13] 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:

1. All of the adsorption sites are equivalent, and each site can only accommodate one molecule.
2. The surface is energetically homogeneous, and adsorbed molecules do not interact.
3. There are no phase transitions.
4. 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 isotherm for relatively flat (non-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: ${\displaystyle A_{\text{g}}+S\rightleftharpoons AS}$, where A is a gas molecule, and S is an adsorption site. The direct and inverse rate constants are k and k₋₁. If we define surface coverage, ${\displaystyle \theta }$, as the fraction of the adsorption sites occupied, in the equilibrium we have:

${\displaystyle K={\frac {k}{k_{-1}}}={\frac {\theta }{(1-\theta )P}},}$

or

${\displaystyle \theta ={\frac {KP}{1+KP}},}$

where ${\displaystyle P}$ is the partial pressure of the gas or the molar concentration of the solution. For very low pressures ${\displaystyle \theta \approx KP}$, and for high pressures ${\displaystyle \theta \approx 1}$.

The value of ${\displaystyle \theta }$ 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_mon the STP volume of adsorbate required to form a monolayer on the adsorbent (per gram of adsorbent), then ${\displaystyle \theta ={\frac {v}{v_{\text{mon}}}}}$, and we obtain an expression for a straight line:

${\displaystyle {\frac {1}{v}}={\frac {1}{Kv_{\text{mon}}}}{\frac {1}{P}}+{\frac {1}{v_{\text{mon}}}}.}$

Through its slope and y intercept we can obtain v_mon and K, which are constants for each adsorbent–adsorbate pair at a given temperature. v_mon 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 ${\displaystyle \theta _{E}}$ as the fraction of empty sites, and we have:

${\displaystyle \theta _{E}={\dfrac {1}{1+\sum _{i=1}^{n}K_{i}P_{i}}}.}$

Also, we can define ${\displaystyle \theta _{j}}$ as the fraction of the sites occupied by the j-th gas:

${\displaystyle \theta _{j}={\dfrac {K_{j}P_{j}}{1+\sum _{i=1}^{n}K_{i}P_{i}}},}$

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.[14] 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.[15] The choice of the model based on best fitting of the data is a common misconception.[14]

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.

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 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_(g) + S ⇌ AS,

A_(g) + AS ⇌ A₂S,

A_(g) + A₂S ⇌ A₃S and so on.

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

${\displaystyle {\frac {x}{v(1-x)}}={\frac {1}{v_{\text{mon}}c}}+{\frac {x(c-1)}{v_{\text{mon}}c}},}$

where x is the pressure divided by the vapor pressure for the adsorbate at that temperature (usually denoted ${\displaystyle P/P_{0}}$), v is the STP volume of adsorbed adsorbate, v_mon 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.

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.[16] 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_E constant and will either be adsorbed from the precursor state at a rate of k_EC or will desorb into the gaseous phase at a rate of k_ES. 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_D constant.

These factors were included as part of a single constant termed a "sticking coefficient", k_E, described below:

${\displaystyle k_{\text{E}}={\frac {S_{\text{E}}}{k_{\text{ES}}S_{\text{D}}}}.}$

As S_D is dictated by factors that are taken into account by the Langmuir model, S_D 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_(t) is fractional coverage of the adsorbent with adsorbate, and t is immersion time:

${\displaystyle {\frac {d\theta _{(t)}}{dt}}=R'(1-\theta )(1+k_{\text{E}}\theta ).}$

Solving for θ_(t) yields:

${\displaystyle \theta _{(t)}={\frac {1-e^{-R'(1+k_{\text{E}})t}}{1+k_{\text{E}}e^{-R'(1+k_{\text{E}})t}}}.}$

Adsorption constants are equilibrium constants, therefore they obey the Van 't Hoff equation:

${\displaystyle \left({\frac {\partial \ln K}{\partial {\frac {1}{T}}}}\right)_{\theta }=-{\frac {\Delta H}{R}}.}$

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

${\displaystyle \Delta H_{\text{ads}}=\Delta H_{\text{liq}}-RT\ln c,}$

that is to say, adsorption is more exothermic than liquefaction.

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 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 ${\displaystyle \Gamma }$ at a surface of area ${\displaystyle A}$ on an infinite area surface can be directly integrated from Fick's second law differential equation to be:[17]

${\displaystyle \Gamma =2AC{\sqrt {\frac {Dt}{\pi }}}}$

where ${\displaystyle A}$ is the surface area (unit m²), ${\displaystyle C}$ is the number concentration of the molecule in the bulk solution (unit #/m³), ${\displaystyle D}$ is the diffusion constant (unit m²/s), and ${\displaystyle t}$ is time (unit s). Further simulations and analysis of this equation[18] 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.[19] 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 ${\displaystyle t}$ in a particular measurement.[18]

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.

Adsorbents are used usually in the form of spherical pellets, rods, moldings, or monoliths with a hydrodynamic radius between 0.25 and 5 mm. They must have high abrasion resistance, high thermal stability and small pore diameters, which results in higher exposed surface area and hence high capacity for adsorption. The adsorbents must also have a distinct pore structure that enables fast transport of the gaseous vapors.[25] Most industrial adsorbents fall into one of three classes:

• Oxygen-containing compounds – Are typically hydrophilic and polar, including materials such as silica gel, limestone (calcium carbonate)[26] and zeolites.
• Carbon-based compounds – Are typically hydrophobic and non-polar, including materials such as activated carbon and graphite.
• Polymer-based compounds – Are polar or non-polar, depending on the functional groups in the polymer matrix.

Silica gel is a chemically inert, non-toxic, polar and dimensionally stable (< 400 °C or 750 °F) amorphous form of SiO₂. It is prepared by the reaction between sodium silicate and acetic acid, which is followed by a series of after-treatment processes such as aging, pickling, etc. These after-treatment methods results in various pore size distributions.

Silica is used for drying of process air (e.g. oxygen, natural gas) and adsorption of heavy (polar) hydrocarbons from natural gas.

Zeolites are natural or synthetic crystalline aluminosilicates, which have a repeating pore network and release water at high temperature. Zeolites are polar in nature.

They are manufactured by hydrothermal synthesis of sodium aluminosilicate or another silica source in an autoclave followed by ion exchange with certain cations (Na⁺, Li⁺, Ca²⁺, K⁺, NH₄⁺). The channel diameter of zeolite cages usually ranges from 2 to 9 Å. The ion exchange process is followed by drying of the crystals, which can be pelletized with a binder to form macroporous pellets.

Zeolites are applied in drying of process air, CO₂ removal from natural gas, CO removal from reforming gas, air separation, catalytic cracking, and catalytic synthesis and reforming.

Non-polar (siliceous) zeolites are synthesized from aluminum-free silica sources or by dealumination of aluminum-containing zeolites. The dealumination process is done by treating the zeolite with steam at elevated temperatures, typically greater than 500 °C (930 °F). This high temperature heat treatment breaks the aluminum-oxygen bonds and the aluminum atom is expelled from the zeolite framework.

The term "adsorption" itself was coined by Heinrich Kayser in 1881 in the context of uptake of gases by carbons.[27]

Activated carbon is a highly porous, amorphous solid consisting of microcrystallites with a graphite lattice, usually prepared in small pellets or a powder. It is non-polar and cheap. One of its main drawbacks is that it reacts with oxygen at moderate temperatures (over 300 °C).

Activated carbon can be manufactured from carbonaceous material, including coal (bituminous, subbituminous, and lignite), peat, wood, or nutshells (e.g., coconut). The manufacturing process consists of two phases, carbonization and activation.[28][29] The carbonization process includes drying and then heating to separate by-products, including tars and other hydrocarbons from the raw material, as well as to drive off any gases generated. The process is completed by heating the material over 400 °C (750 °F) in an oxygen-free atmosphere that cannot support combustion. The carbonized particles are then "activated" by exposing them to an oxidizing agent, usually steam or carbon dioxide at high temperature. This agent burns off the pore blocking structures created during the carbonization phase and so, they develop a porous, three-dimensional graphite lattice structure. The size of the pores developed during activation is a function of the time that they spend in this stage. Longer exposure times result in larger pore sizes. The most popular aqueous phase carbons are bituminous based because of their hardness, abrasion resistance, pore size distribution, and low cost, but their effectiveness needs to be tested in each application to determine the optimal product.

Activated carbon is used for adsorption of organic substances[30] and non-polar adsorbates and it is also usually used for waste gas (and waste water) treatment. It is the most widely used adsorbent since most of its chemical (e.g. surface groups) and physical properties (e.g. pore size distribution and surface area) can be tuned according to what is needed.[31] Its usefulness also derives from its large micropore (and sometimes mesopore) volume and the resulting high surface area. Recent research works reported activated carbon as an effective agent to adsorb cationic species of toxic metals from multi-pollutant systems and also proposed possible adsorption mechanisms with supporting evidences.[32]