Langmuir adsorption model
: From these basic hypotheses the mathematical formulation of the Langmuir adsorption isotherm can be derived in various independent and complementary ways: by the kinetics, the thermodynamics, and the statistical mechanics approaches respectively (see below for the different demonstrations).
of a gas molecules monolayer covering the whole surface of the solid and completely occupied by the adsorbate.
A continuous monolayer of adsorbate molecules covering a homogeneous flat solid surface is the conceptual basis for this adsorption model.
[1] In 1916, Irving Langmuir presented his model for the adsorption of species onto simple surfaces.
He hypothesized that a given surface has a certain number of equivalent sites to which a species can "stick", either by physisorption or chemisorption.
[2] Langmuir published two papers that confirmed the assumption that adsorbed films do not exceed one molecule in thickness.
The first experiment involved observing electron emission from heated filaments in gases.
[3] The second, a more direct evidence, examined and measured the films of liquid onto an adsorbent surface layer.
[4] Inherent within this model, the following assumptions[5] are valid specifically for the simplest case: the adsorption of a single adsorbate onto a series of equivalent sites onto the surface of the solid.
In case of two competing adsorbed species, the competitive adsorption model is required, while when a sorbed species dissociates into two distinct entities, the dissociative adsorption model need to be used.
The model assumes adsorption and desorption as being elementary processes, where the rate of adsorption rad and the rate of desorption rd are given by where pA is the partial pressure of A over the surface, [S] is the concentration of free sites in number/m2, [Aad] is the surface concentration of A in molecules/m2 (concentration of occupied sites), and kad and kd are constants of forward adsorption reaction and backward desorption reaction in the above reactions.
The thermodynamic derivation allows for the activity coefficients of adsorbates in their bound and free states to be included.
[6][7] This derivation[8][9] based on statistical mechanics was originally provided by Volmer and Mahnert[10] in 1925.
factor accounts for the overcounting arising due to the indistinguishable nature of the adsorbates.
, finally, we have It is plotted in the figure alongside demonstrating that the surface coverage increases quite rapidly with the partial pressure of the adsorbants, but levels off after P reaches P0.
[11] Here, the following assumptions would be held to be valid: Using similar kinetic considerations, we get The 1/2 exponent on pD2 arises because one gas phase molecule produces two adsorbed species.
Applying the site balance as done above, The formation of Langmuir monolayers by adsorption onto a surface dramatically reduces the entropy of the molecular system.
The Langmuir adsorption model deviates significantly in many cases, primarily because it fails to account for the surface roughness of the adsorbent.
Moreover, specific surface area is a scale-dependent quantity, and no single true value exists for this parameter.
[1] Thus, the use of alternative probe molecules can often result in different obtained numerical values for surface area, rendering comparison problematic.
Experimentally, there is clear evidence for adsorbate–adsorbate interactions in heat of adsorption data.
The modifications try to account for the points mentioned in above section like surface roughness, inhomogeneity, and adsorbate–adsorbate interactions.
However, the Freundlich equation is not unique; consequently, a good fit of the data points does not offer sufficient proof that the surface is heterogeneous.
When the adsorbate pressure (or concentration) is low, the fractional occupation is small and as a result, only low-energy sites are occupied, since these are the most stable.
The heat of adsorption ΔHad is defined as He derived a model assuming that as the surface is loaded up with adsorbate, the heat of adsorption of all the molecules in the layer would decrease linearly with coverage due to adsorbate–adsorbate interactions: where αT is a fitting parameter.
Assuming the Langmuir adsorption isotherm still applied to the adsorbed layer,
is expected to vary with coverage as follows: Langmuir's isotherm can be rearranged to Substituting the expression of the equilibrium constant and taking the natural logarithm: Brunauer, Emmett and Teller (BET)[18] derived the first isotherm for multilayer adsorption.
It assumes a random distribution of sites that are empty or that are covered with by one monolayer, two layers and so on, as illustrated alongside.
This section describes the surface coverage when the adsorbate is in liquid phase and is a binary mixture.
[19] For ideal both phases – no lateral interactions, homogeneous surface – the composition of a surface phase for a binary liquid system in contact with solid surface is given by a classic Everett isotherm equation (being a simple analogue of Langmuir equation), where the components are interchangeable (i.e. "1" may be exchanged to "2") without change of equation form: where the normal definition of multi-component system is valid as follows: By simple rearrangement, we get This equation describes competition of components "1" and "2".
