Langmuir isotherm assumption

The assumptions made to derive the Langmuir isotherm (Eq. 2.7) are well known Energetic equivalence of all adsorption sites, and no lateral (attractive or repulsive) interactions between the adsorbate molecules on the surface. This is equivalent to a constant, coverage independent, heat (-AH) of adsorption. 

There are several reasons for deviations from the LHHW kinetics Surface heterogeneity, surface reconstruction, adsorbate island formation and, most important, lateral coadsorbate interactions.18,19 All these factors lead to significant deviations from the fundamental assumption of the Langmuir isotherm, i.e. constancy of AHa (and AHB) with varying coverage. 

Give the essential assumptions made in deriving the BET isotherm. Ho v does the BET isotherm differ from the Langmuir isotherm ... 

Often, mnltistep reactions are enconntered where a reactant j first becomes adsorbed on the electrode, then is converted electrochemically (or chemically) to a desorbing prodnct. We shall consider the case where the electrochemical step involving adsorbed particles is rate determining. With a homogeneons electrode surface and without interaction forces between the adsorbed particles [i.e., in conditions when the Langmuir isotherm (10.14) can be apphed], the assumption can be made that the rate of this step is proportional not to the bulk concentration Cy j but to the surface concentration Aj or to the degree of surface coverage 0 hence. 

One of the simplest nonlinear isotherm models is the Langmuir model. Its basic assumption is that adsorbate deposits on the adsorbent surface in the form of the monomolecular layer, owing to the delocalized interactions with the adsorbent snrface. The Langmuir isotherm can be given by the following relationship ... 

The assumptions mentioned above for the Langmuir isotherm generally do not hold true in a complex heterogeneous medium such as soil. The deep-well environment is similarly complex and... 

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. 

In addition to the assumptions implicit in the use of the Langmuir isotherm the following assumption is applicable to all Hougen-Watson models the reaction involves at least one species chemisorbed on the catalyst surface. If reaction takes place between two adsorbed species, they must be adsorbed on neighboring sites in order for reaction to occur. The probability of reaction between adsorbed A and adsorbed B is assumed to be proportional to the product of the fractions of the sites occupied by each species (0A9B). Similar considerations apply to termolecular reactions occurring on the surface. 

Figure3.4 (a) Experimental l-E curve obtained at 0 C and curves computed from the assumption of a Langmuir isotherm, a = experimental curve b = partial curve for the strongly bound hydrogen, c = partial curve Tor the weakly bound hydrogen, d = computed net curve, (b) Experimental l - E curve obtained at 70 C and curves computed from the assumption of a Frumkin isotherm, a = experimental curve b = partial curve for the strongly bound hydrogen, c = partial curve for the weakly bound hydrogen, d = computed net curve. After Breiter(1964).

This term is analogous to relative volatility or its reciprocal (or to an equilibrium selectivity). Similarly, the assumption of a constant separation factor is a useful assumption in many sorptive operations. [It is constant for the Langmuir isotherm, as described below, and for mass-action equilibrium with za = zh in Eq. (16-24).] This gives the constant separation factor isotherm... 

If the Gibbs energy of adsorption AG-M is considered as independent of the coverage the resulting formula is known as the Langmuir isotherm this assumption is reasonable when the interaction between the adsorbed particles is small. 

The Langmuir Isotherm. The most simple assumption in adsorption is that the adsorption sites, S, on the surface of a solid (adsorbent) become occupied by an adsorbate from the solution, A. Implying a 1 1 stoichiometry... 

We present here a few straightforward generalizations of the Langmuir isotherm. The essential assumptions made in Section 2.4 are retained. 

If a straight hne is obtained when is plotted against x/m at low concentrations, the Langmuir equation is applicable. However, due to the restrictive assumptions, Langmuir isotherms usually are of miuor importance in heterogeneous media... 

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... 

Pankow (1987) showed that this assumption of a linear Langmuir isotherm is consistent with the pioneering work of Junge (1977) on the adsorption of SOCs on particles. The fraction of the total SOC in the atmosphere adsorbed on aerosol particles, denoted , was hypothesized by Junge to be related to the surface area (ST, cm2 per cm3 of air) of the TSP and the saturation vapor pressure of the SOC by... 

The following equations constitute the approximate solutions of the fixed-bed model under the constant pattern and plug-flow assumption for the favorable Langmuir isotherm and linear driving forces (Perry and Green, 1999) ... 

The Langmuir isotherm is based on the assumption of localized adsorption. This means that an adsorbed molecule has such a high statistical preference for a certain surface site as to possess a negligible translational entropy in the adsorbed state. Localized adsorption is thus... 

Probably the reason why the (1 — 0) concept has received such widespread credence is that Langmuir was able to derive his famous adsorption isotherm on the basis of this concept. Since the Langmuir isotherm equation has been experimentally verified in many cases, it was felt that the (1 — 0) concept must be essentially correct. This again is fallacious reasoning, since in the derivation two other assumptions are necessarily made which are not in accord with recent experiments. These are ( ) the rate of evaporation is proportional to 0 and (2) one can treat the experimental data as if the surface were homogeneous. Because of this situation, it is desirable that someone derive the Langmuir isotherm equation on more realistic assumptions. (See ref. 10a.)... 

If a given isotherm cannot be described by Eq. 9-2, then some assumptions behind the Freundlich multi-site conceptualization are not valid. For example, if there are limited total sorption sites that become saturated (case shown in Fig. 9.3c), then C,s cannot increase indefinitely with increasing Ciw. In this case, the Langmuir isotherm may be a more appropriate model ... 

The model involves four basic assumptions. First, it is postulated that an adsorbed phase of silicic acid exists on the particle surface, the equilibrium of which can be expressed by a Langmuir isotherm (Equation 1, symbols are defined in table of nomenclature). 

There are several explanations for these deviations. As Figure 11 indicates, it takes some time to establish the adsorption equilibrium. Thus, the theoretical assumption concerning the adsorption equilibrium and resulting in Equation 3 may not be satisfied adequately. Here a higher initial silicic acid concentration would result. On the other hand, at high silicic acid concentrations the Langmuir isotherm is an idealization which will not properly represent experimental data of extensive... 

Assuming that the second process is rapid, we obtain the following standard picture of adsorption on a uniform surface the equilibrium concentration q, which depends on the pressure of the gas, is determined by the Langmuir isotherm. The only difference from the standard picture is that the statistical sum for all states of the adsorbed molecule in a potential hole must be replaced by a combination of two statistical sums for all states of the adsorbed molecule and for all possible states of the surface element. This, of course, has no effect on the form of the Langmuir equation. Under very simple assumptions the kinetics of establishment of equilibrium will also not differ from those on a uniform surface. Thus, the initial velocity is proportional to the pressure and approaches equilibrium exponentially. 

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... 

There is an interesting parallel between substrate binding and adsorption. Since each enzyme molecule has one active site, and since these active sites all have the same structure, we can think of enzyme molecules in solution as a surface with many equivalent adive sites. In this case, k2 in the Michaelis-Menten kinetics (Eq. 5.1 see Chapter 2 for a detailed discussion) represents the rate of adsorption, k x the rate of desorption, and k2 the rate of the surface readion followed by fast product desorption. Moreover, this system fits the assumptions of the Langmuir isotherm (all sites identical, one molecule per site, no lateral interadions) even better than the adive sites on some real solid catalysts ... 

Langmuir (1918) laid down a systematic and theoretically grounded adsorption isotherm, the well-known Langmuir isotherm, which has served as a basis for a number of future isotherms, such as the BET isotherm, described in 1.4.1.3. The basic assumptions of the Langmuir adsorption model include monolayer coverage, localized adsorption (i.e., the specific adsorption sites exist and the interactions are between a specific molecule and the site), and homogeneous materials. In addition, the heat of adsorption is independent of coverage. He proposed that the rate of adsorption is proportional to the dimensionless pressure of the adsorbate, p/po (= p ), where po is the saturated vapor pressure, and the... 

The assumption of monolayer adsorption in the Langmuir isotherm model is unrealistic in most cases, and a modification to multilayer adsorption should be considered. In 1938, Brunauer, Emmett, and Teller modified the Langmuir approach of balancing the rates of adsorption and desorption for the various molecular layers [Brunauer et al., 1938], This approach is known as the BET method. The BET isotherm assumes that the adsorption of the first layer has a characteristic heat of adsorption A Ha and the adsorption and desorption on subsequent layers are controlled by the heat of condensation of the vapor, A Hc. The derivation of the BET equation is beyond the scope of this book however, a common form of the BET equation is given as... 

Physical understanding of the assumptions underlying the various isotherms can be increased by deriving them by statistical mechanics. This will be done only for the Langmuir isotherm. At equilibrium, the chemical potential of the adsorbate in the gas phase and on the surface must be equal ... 

The Langmuir isotherm, regarded as the starting point for any discussion of isotherms because of its simplicity and applicability, was derived by Irving Langmuir in 1918. Langmuir made four assumptions ... 

Most of the assumptions in deriving the Langmuir isotherm are not accurate. It works best for cases of chemisorption where only one layer is formed. A straightforward check on the validity of the Langmuir isotherm is to plot the inverse of the surface coverage, 0, versus the inverse of the equilibrium pressure ... 

Many different adsorption isotherms may be derived where all or some of the basic assumptions going into the derivation have been released or relaxed [101-105], It should be stated, however, that the general form of the Langmuir isotherm, which is shown for two temperatures in Fig. 13, may be used for a phenomenological description of many processes. It is clear, from the adsorption isotherm, the sticking probability. v may also be determined given that all other parameters are known [1,7]. 

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