The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressure i.e. at these conditions the adsorbate's partial pressure ${\displaystyle p_{A}}$ is related to its volume V adsorbed onto a solid adsorbent. The adsorbent, as indicated in the figure, is assumed to be an ideal solid surface composed of a series of distinct sites capable of binding the adsorbate. The adsorbate binding is treated as a chemical reaction between the adsorbate gaseous molecule ${\displaystyle A_{\text{g}}}$ and an empty sorption site S. This reaction yields an adsorbed species ${\displaystyle A_{\text{ad}}}$ with an associated equilibrium constant ${\displaystyle K_{\text{eq}}}$:

${\displaystyle {\ce {A_{g}{}+ S <=> A_{ad}}}}$.

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).

The Langmuir adsorption equation is

${\displaystyle \theta _{A}={\frac {V}{V_{\text{m}}}}={\frac {K_{\text{eq}}^{A}\,p_{A}}{1+K_{\text{eq}}^{A}\,p_{A}}},}$

where ${\displaystyle \theta _{A}}$ is the fractional occupancy of the adsorption sites, i.e., the ratio of the volume V of gas adsorbed onto the solid to the volume ${\displaystyle V_{\text{m}}}$ 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]

Background and experiments

In 1916, Irving Langmuir presented his model for the adsorption of species onto simple surfaces. Langmuir was awarded the Nobel Prize in 1932 for his work concerning surface chemistry. He hypothesized that a given surface has a certain number of equivalent sites to which a species can “stick”, either by physisorption or chemisorption. His theory began when he postulated that gaseous molecules do not rebound elastically from a surface, but are held by it in a similar way to groups of molecules in solid bodies.^[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. He also noted that generally the attractive strength between the surface and the first layer of adsorbed substance is much greater than the strength between the first and second layer. However, there are instances where the subsequent layers may condense given the right combination of temperature and pressure.^[4]

Basic assumptions of the model

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.

1. The surface containing the adsorbing sites is a perfectly flat plane with no corrugations (assume the surface is homogeneous). However, chemically heterogeneous surfaces can be considered to be homogeneous if the adsorbate is bound to only one type of functional groups on the surface.
2. The adsorbing gas adsorbs into an immobile state.
3. All sites are energetically equivalent, and the energy of adsorption is equal for all sites.
4. Each site can hold at most one molecule (mono-layer coverage only).
5. No (or ideal) interactions between adsorbate molecules on adjacent sites. When the interactions are ideal, the energy of side-to-side interactions is equal for all sites regardless of the surface occupancy.

Derivations of the Langmuir adsorption isotherm

The mathematical expression of the Langmuir adsorption isotherm involving only one sorbing species can be demonstrated in different ways: the kinetics approach, the thermodynamics approach, and the statistical mechanics approach respectively. 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.

Kinetic derivation

This section^[5] provides a kinetic derivation for a single-adsorbate case. The kinetic derivation applies to gas-phase adsorption. However, it has been mistakenly applied to solutions. The multiple-adsorbate case is covered in the competitive adsorption sub-section. The model assumes adsorption and desorption as being elementary processes, where the rate of adsorption r_ad and the rate of desorption r_d are given by

${\displaystyle r_{\text{ad}}=k_{\text{ad}}p_{A},}$

${\displaystyle r_{\text{d}}=k_{d},}$

where p_A is the partial pressure of A over the surface, is the concentration of free sites in number/m², is the surface concentration of A in molecules/m² (concentration of occupied sites), and k_ad and k_d are constants of forward adsorption reaction and backward desorption reaction in the above reactions.

At equilibrium, the rate of adsorption equals the rate of desorption. Setting r_ad = r_d and rearranging, we obtain

${\displaystyle {\frac {}{p_{A}}}={\frac {k_{\text{ad}}}{k_{\text{d}}}}=K_{\text{eq}}^{A}.}$

The concentration of sites is given by dividing the total number of sites (S₀) covering the whole surface by the area of the adsorbent (a):

${\displaystyle S_{0}=S_{0}/a.}$

We can then calculate the concentration of all sites by summing the concentration of free sites S and occupied sites:

${\displaystyle S_{0}=S+A_{\text{ad}}.}$

Combining this with the equilibrium equation, we get

${\displaystyle S_{0}={\frac {A_{\text{ad}}}{K_{\text{eq}}^{A}p_{A}}}+A_{\text{ad}}={\frac {1+K_{\text{eq}}^{A}p_{A}}{K_{\text{eq}}^{A}p_{A}}}A_{\text{ad}}.}$

We define now the fraction of the surface sites covered with A as

${\displaystyle \theta _{A}={\frac {A_{\text{ad}}}{S_{0}}}.}$

This, applied to the previous equation that combined site balance and equilibrium, yields the Langmuir adsorption isotherm:

${\displaystyle \theta _{A}={\frac {K_{\text{eq}}^{A}p_{A}}{1+K_{\text{eq}}^{A}p_{A}}}.}$

Thermodynamic derivationedit

In condensed phases (solutions), adsorption to a solid surface is a competitive process between the solvent (A) and the solute (B) to occupy the binding site. The thermodynamic equilibrium is described as

Solvent (bound) + Solute (free) ↔ Solvent (free) + Solute (bound).

If we designate the solvent by the subscript "1" and the solute by "2", and the bound state by the superscript "s" (surface/bound) and the free state by the "b" (bulk solution / free), then the equilibrium constant can be written as a ratio between the activities of products over reactants:

${\displaystyle K=\frac{a_1^{\text{b}}\times <!--\; \times \; -->a_2^{\text{s}}}{a_2^{\text{b}}}}$Zdroj:https://en.wikipedia.org?pojem=Langmuir_adsorption_model
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