Langmuir adsorption isotherm approach

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

It is important to recognize that Kp is unitless, and is related to thermodynamic quantities by Eq. 9.93, for example. However, Eq. 11.17 has exactly the same form as the classic Langmuir adsorption isotherem, Eq. 11.11, if we take K = Kp/p°. Thus the two approaches are entirely equivalent. In addition the discussion above shows how the more restrictive form that is usually written for the Langmuir adsorption isotherm can be converted to the extensible mass-action kinetics form to be used, for example, within a more extensive surface reaction mechanism. 

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

Mg2+ influences calcite dissolution rates the same way, but not to the same extent as Ca2+. The inhibition effects of Mg2+ can be described in terms of a Langmuir adsorption isotherm. Sjoberg (1978) found he could model results for the combined influences of Ca2+ and Mg2+ in terms of site competition consistent with ion exchange equilibrium. The inhibition effects of Mg2+ in calcite powder runs increase with increasing Mg2+ concentration and as equilibrium is approached. 

Adsorption isotherms are by no means all of the Langmuir type as to shape, and Brunauer [34] considered that there are five principal forms, as illustrated in Fig. XVII-7. TVpe I is the Langmuir type, roughly characterized by a monotonic approach to a limiting adsorption at presumably corresponds to a complete monolayer. Type II is very common in the case of physical adsorption... 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

We start by noting that the Langmuir isotherm approach does not take into account the electrostatic interaction between the dipole of the adsorbate and the field of the double layer. This interaction however is quite important as already shown in section 4.5.9.2. In order to account explicitly for this interaction one can write the adsorption equilibrium (Eq. 6.24) in the form ... 

Plots of an amount of material adsorbed versus pressure at a fixed temperature are known as adsorption isotherms. They are generally classified in the five main categories described by Brunauer and his co-workers (4). In Figure 6.2 adsorbate partial pressures (P) are normalized by dividing by the saturation pressure at the temperature in question (P0). Type I is referred to as Langmuir-type adsorption and is characterized by a monotonic approach to a limiting amount of adsorption, which presumably corresponds to formation of a monolayer. This type of behavior is that expected for chemisorption. 

Once the transport step is defined, it is necessary to take into account which is the adsorption isotherm preceding the internalisation (the latter is always assumed to be first order). By extending results from Section 2, analytical solutions can be written for steady-state for both linear and Langmuir isotherms, as well as for the transient case with the linear isotherm. Langmuirian isotherms require numerical approaches for the transient case. 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

Temkin (1941) approached the development of an adsorption isotherm by considering a heterogeneous surface (Section 6.8.3) where no molecular interactions exist. He divided the surface into different patches, and since there are no interactions between molecules, in each patch the Langmuir isotherm can be applied [see Eq. (6.196)] (Fig. 6.98). Thus, for the ith patch,... 

It should be apparent — since an adsorption isotherm can be derived from a two-dimensional equation of state —that an isotherm can also be derived from the partition function since the equation of state is implicitly contained in the partition function. The use of partition functions is very general, but it is also rather abstract, and the mathematical difficulties are often formidable (note the cautious in principle in the preceding paragraph). We shall not attempt any comprehensive discussion of the adsorption isotherms that have been derived by the methods of statistical thermodynamics instead, we derive only the Langmuir equation for adsorption from the gas phase by this method. The interested reader will find other examples of this approach discussed by Broeckhoff and van Dongen (1970). 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

An unusual dependence for adsorption on a uniform surface arises when it is assumed that the rate of change of the surface is considerably slower than the rate of adsorption (see figure). If adsorption and desorption occur rapidly, the state of the surface remains practically unaltered and we then get an adsorption isotherm corresponding to a non-uniform surface with a distribution p(E, 0) of the heat of adsorption (curve 1) [3]. However, when the time interval is considerable, slow adsorption accompanies changes in the properties of the surface, and the amount of gas adsorbed approaches that given by the Langmuir isotherm (curve 2, point B), which describes a state of complete equilibrium (see above). 

When the equilibrium constant b is large (highly favorable adsorption) the Langmuir isotherm approaches irreversible or rectangular form,... 

Ravikovitch PI, Vishnyakov A, Russo R, and Neimark AV. Unified approach to pore size characterization of microporous carbonaceous materials from N2, Ar, and C02 adsorption isotherms. Langmuir, 2000 16(5) 2311-2320. 

The more desirable approach is to determine f(Q) from an assumed 0(P,T,Q) and the experimental adsorption isotherm. Sips (16) showed that Equation 1 could be treated by a Stieltjes transform, so that in principle an explicit function could be written for f(Q), provided the experimental isotherm function, 0, could be expressed in analytical form. Subsequently, Honig and coworkers (10, 11, 12) investigated this approach further. The difficulty is that only for certain types of assumed functions 0 and 0 is the approach practical. As a consequence the procedure has been first to restrict the choice of 0 to the Langmuir equation, and second to assume certain simple functions for 0 such as the Freundlich and Temkin isotherm equations. The system is thus forced into an arbitrary mold and again it is not certain how much reliance should be placed on the site energy distributions obtained. 

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