The Langmuir isotherm, type

The asymptotic approach of the quantity adsorbed toward a limiting value indicates that type I isotherms are limited to, at most, a few molecular layers. In the case of chemisorption only one layer can be bonded to the surface and therefore, chemisorption always exhibits a type I isotherm.t Although it is possible to calculate the number of molecules in the monolayer from the type I chemisorption isotherm, some serious 

Using a kinetic approach, Langmuir was able to describe the type I isotherm with the assumption that adsorption was limited to a monolayer. According to the kinetic theory of gases, the number of molecules striking each square centimeter of surface per second is given by 

The constant k is N j InMRTY. Ths number of molecules striking and adhering to each square centimeter of surface is 

At equilibrium the rates of adsorption and desorption are equal. From equations (4.3) and (4.4) one obtains 

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Figure 5 shows the adsorbent loading, Q, as a function of C°, based on data at various pHs. The result suggests that the neutral form of berberine was adsorbed on XAD-7. This result was consistent with the result that adsorption onto XAD-7 was due to hydrophobic interaction. In Fig. 5, the adsorption isotherm for berberine was of the Langmuir isotherm type. This result suggested that berberine might be adsorbed onto XAD-7 form a monolayer. 

Type I isotherm is the Langmuir isotherm type (monolayer coverage), typical of adsorption in microporous solids, such as adsorption of oxygen in charcoal. Type II typifies the BET adsorption mechanism. Type III is the type typical of water adsorption on charcoal where the adsorption is not favorable at low pressure because of the nonpolar (hydrophobic) nature of the charcoal surface. At sufficiently high pressures, the adsorption is due to the capillary condensation in mesopores. Type IV and type V are the same as types II and III with the exception that they have finite limit as P Pq due to the finite pore volume of porous solids. 

The BET isotherm was developed primarily to describe the commonly encountered type II isotherm shown in Figure 9.7, such as is found for the adsorption of relatively inert gases (N2, Ar, He, etc.) on polar surfaces (c 100). However, it reduces to the Langmuir isotherm (type 1) when restricted to monolayer coverage (A//a A//l), and describes type 111 isotherms in the unusual situation where the adsorption of the first monolayer is less exothermic that that of the subsequent layers (e.g., c < 1) resulting in low adsorption at low values of p/po. 

Eig. 4. The Bmnaner classification of isotherms (I V). Langmuir Isotherm. Type I isotherms are commonly represented by the ideal Langmuir model ... 

Figure 5.19 shows an idealized form of the adsorption isotherm for physisorption on a nonporous or macroporous solid. At low pressures the surface is only partially occupied by the gas, until at higher pressures (point B on the curve) the monolayer is filled and the isotherm reaches a plateau. This part of the isotherm, from zero pressures to the point B, is equivalent to the Langmuir isotherm. At higher pressures a second layer starts to form, followed by unrestricted multilayer formation, which is in fact equivalent to condensation, i.e. formation of a liquid layer. In the jargon of physisorption (approved by lUPAC) this is a Type II adsorption isotherm. If a system contains predominantly micropores, i.e. a zeolite or an ultrahigh surface area carbon (>1000 m g ), multilayer formation is limited by the size of the pores. 

The saturation of the SHG response at high cation concentrations suggests that the process of complex formation at the membrane surface may be treated by a Langmuir-isotherm type analysis [24,27]. At constant temperature, the Langmuir equation is given by... 

Sorption and desorption are usually modeled as one fully reversible process, although hystersis is sometimes observed. Four types of equations are commonly used to describe sorption/desorption processes Langmuir, Freundlich, overall and ion or cation exchange. The Langmuir isotherm model was developed for single layer adsorption and is based on the assumption that maximum adsorption corresponds to a saturated monolayer of solute molecules on the adsorbent surface, that the energy of adsorption is constant, and that there is no transmigration of adsorbate on the surface phase. 

Virtually all theoretical treatments of adsorption phenomena are based on or can be readily related to the analysis developed by Langmuir (5,6). The Langmuir isotherm corresponds to a highly idealized type 6f adsorption and the analysis is predicated on the following key assumptions. 

Ferrocene-based Linear Polymers. The first derivative that was studied from the electrochemical point of view was polyvinylferrocene (PVF). As illustrated in Figure 25, it displays a single oxidation process, which in some solvents is affected by problems of adsorption of the oxidation product (though not of the ideal Langmuir isotherm type discussed in Chapter 2, Section 1.6). 

It should be stressed here that although each type of molecule by itself produces a Langmuir isotherm, the combination in Eq. (2.5.2) has a form different fi om the typical hyperbolic form of the Langmuir isotherm. Figure 2.3 shows the BI for a mixture of two types of molecules with k The Bis are quite different fi om the... 

This model is directly derived from the Langmuir isotherm. It assumes that the adsorbent surface consists of two different types of independent adsorption sites. Under this assumption, the adsorption energy distribution can be modeled by a bimodal discrete probability density function, where two spikes (delta-Dirac functions) are located at the average adsorption energy of the two kinds of sites, respectively. The equation of the Bilangmuir isotherm is... 

Treatment of the above SHG curves for membranes 80, 83, 85, and 86 by a Langmuir-isotherm type analysis indicates Langmuir-type saturation occurring at the membrane surface at high cation concentrations, which may be interpreted in terms of a tightly packed monolayer of the SHG active cation complexes at the membrane surface. However, the SHG active layer may actually have a thickness of several monolayers if the electrical field aligns the complexes with respect to the interface. 

In the case of adsorbed species, a unique standard state—one that makes the bottom part of the logarithm equal to zero—is not defined because it will depend on the type of isotherm chosen to describe the process, i.e., it will depend on the function /(0) in Eq. (6.184) characteristic of each isotherm. Thus, it would be impossible to compare the adsorption energies (i.e., AG° s) of different adsorbates if they were represented by different adsorption isotherms. However, instead of a unique standard state, it is possible to define convenient standard states common to any isotherm. The way to choose this convenient standard state is explained in the following analysis of the Langmuir isotherm. 

With this in mind, some important adsorption isotherms were introduced, and we found that each of them describes important characteristics of the adsorption process (Table 6.10). Thus, the Langmuir isotherm considers the basic step in the adsorption process the Frumkin isotherm was one of the first isotherms involving lateral interactions the Temkin is a surface heterogeneity isotherm and the Flory-Huggins-type isotherms include the substitution step of replacing adsorbed water molecules by the adsorbed entities (Fig. 6.98). 

The Type I isotherm is reminiscent of Figure 7.16, the Langmuir isotherm. The plateau is interpreted as indicating monolayer coverage. We see that this type of behavior implies a sufficiently specific interaction between adsorbate and adsorbent to be more typical of chemisorption than physical adsorption. 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

The Langmuir isotherm is based on the simplest model that involves the following assumptions (1) the adsorption energy of all sites is the same and is unaffected by adsorption on neighboring sites, (2) the adsorption is immobile, (3) each site accommodates only one adsorbed particle, and (4) adsorbed atoms (molecules) do not interact with each other. Figure 10.2a shows that the Langmuir-type isotherm for... 

Figure 4.15 The six types of adsorption isotherms (V = volume adsorbed) Type I shows a monolayer (Langmuir isotherm) types II and III show multilayer adsorption type IV shows first a monolayer, followed by filling of mesopores. The knee in isotherms I, II, and IV, indicated bya blackdot, indicates the point of monolayer formation (point B ).

This means that the decomposition occurs uniformly across the surface, where the products are weakly bound and rapidly desorbed consequently, the rate-determining step is the surface decomposition step. This type of reaction shows two rate-limiting laws corresponding to the two extreme behaviors of the Langmuir isotherm. That is, at low pressure, 0A is small and proportional to the pressure, and the rate becomes first order in A(g) ... 

Isotherms that follow the Langmuir isotherm (i.e, form only a monolayer) are termed type I isotherms. 

In general, adsorption isotherms obtained with this and other zeolitic substrates are of Brunauer s Type I, the simple hyperbolic form also known as the Langmuir isotherm. Consequently, the asymptotic limit of adsorption is used instead of the value of Vm normally derived from the BET evaluation of specific surface area. It is, of course, not possible to define exact monolayer or multilayer adsorption in these three-dimensional interconnected pore systems. 

The AGf, AGAa AGAs. and AGss values, and, correspondingly, In fi and a values depend on the electric state of the surface, i.e., on the electrode potential or charge. This isotherm was deduced by Frumkin [i] (and named after him soon) as a general case of the -> Langmuir isotherm, which corresponds to a = 0. A statistical derivation of the Frumkin isotherm is available [ii] various model considerations and relations to other types of isotherms are discussed in [iii]. Another typical form of the Frumkin isotherm is... 

Type I is concave and V approaches a limiting value. This type is characteristic for microporous solids ) with negligible external surface areas (e.g. activated carbons, molecular sieve zeolites and certain porous oxides). Although mathematically type 1 isotherms are very well described by the Langmuir isotherm equation, calling them "Langmuir isotherms" is not recommended because the limited uptake is governed by the accessible pore volume rather than by the Internal surface area. 

We obtained adsorption kinetic data and isotherm curves by determining the amount of protein adsorbed onto polymers by internal reflection IR spectroscopy which gave good reproducibility (19). The adsorption character was consistent with the Langmuir adsorption type and adsorption vs. time curves showed the expected plateau usually found in macro-molecular adsorption. A competitive adsorption study is being carried... 

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