Isotherm Langmuir equation

The Langmuir isotherm equation is the first theoretically developed adsorption isotherm. Many of the equations proposed later and which fit the experimental results over a wide range are either based on this equation, or these equations have been developed using the Langmuir concept. Thus, the Langmuir equation still retains an important position in physisorption as well as chemisorption theories. The equation has also been derived using thermodynamic and statistical approaches but we shall discuss the commonly used kinetic approach for its derivation. 

The American scientist I. Langmuir derived this equation based on certain assumptions. More important of these assumptions are 

In the original kinetic approach, the equilibrium is assumed to be dynamic in that the rate at which the molecules lirom the gas phase strike the solid surface and condense on the bare sites is equal to the rate at which molecules evaporate from the occupied sites. In other words, the rate of adsorption is equal to the rate of desorption. If at any pressure p of the gas the fraction of the sites occupied is 9, and the fraction of the bare sites is (so l at 6+ 6q= ) then the rate of condensation (adsorption) is 

The evaporation (desorption) of an adsorbed molecule from the surface is essential an activated process in which the energy of activation, E, for desorption may be identified with -AH, the differential (isosteric) heat of desorption. The rate of evaporation can be written as 

The fraction 9 of the surface covered can also be written as the ratio of volume V of the gas or vapor adsorbed at pressure p and the volume of the adsorbate required to form a monomolecular layer. Thus, Equation 2.32 can also written as 

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Markham andBenton. This model (34) is known as the extended Langmuir isotherm equation for two components, i and j. 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

Langmuir isotherm equation phys chem An equation, useful chiefly for gaseous systems, for the amount of material adsorbed on a surface as a function of pressure, while the temperature Is held constant, assuming that a single layer of molecules is adsorbed it is / = ap/ 1 + ap). where / is the fraction of surface covered, p is the pressure, and a is a constant. lar),myur Ts-3,th3rm i,kwa zh3n ) lanthana See lanthanum oxide. lan tha-na ... 

Selecting the isotherm model and initial estimations for the values of its parameters. For instance, in the case of the Langmuir isotherm (Equation 10.39), an estimation for the a parameter can be achieved by an injection made under linear conditions (through Equation 10.50). At this point, the least-squares estimation of b—starting from an initial guess— does not present any difficulty. 

An adsorption isotherm may be written for these ions that allows for surface saturation and thus introduces the idea of finite ionic size. The Langmuir isotherm (Equation (7.67)) is one expression that can be used for this purpose ... 

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.)... 

This equation, with a constant value for D, has been shown to provide a satisfactory correlation of experimental diffusivity data for several zeolitic systems (6-8). For a system which obeys the Langmuir isotherm, Equation 5 becomes... 

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). 

The Michaelis-Menten equation is analogous to the Langmuir isotherm equation... 

At low pressures the Langmuir isotherm equation reduces to V— Vmap - i.e. the volume of gas adsorbed varies linearly with pressure. At high pressures a limiting monolayer coverage, V=Vm, is reached. The curvature of the isotherm at intermediate pressures depends on the value of the constant a and, hence, on the temperature. 

The Langmuir isotherm equation for the correlation of the liquid-phase adsorption equilibrium of a single component, can, in principle, as was previously stated, be extended to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [2,87] ... 

Equation 4.27 has the form of the generalized Freundlich isotherm equation.11 If the product A[C]cb < < 1, it reduces to the van Bemmelen-Freundlich isotherm equation and, ifb 1, it becomes the Langmuir isotherm equation (cf. Eq. 3.50b).M Thus Eq. 4.18 and a gamma distribution of the two rate coefficients it contains are sufficient to generate three very common adsorption... 

If, in an ideal case, the probability of desorption of an adsorbed molecule from the surface is independent of the surface coverage (i.e. there are no lateral interactions between the adsorbed molecules), then the value of E is constant for a particular adsorption system. Equation (4.10) is then applicable over the complete range of monolayer coverage. By rearrangement and simplification of Equation (4.10), we arrive at the familiar Langmuir isotherm equation,... 

The absorption of vinyl chloride(VC) on surface-treated light-grade and nanoscale calcium carbonate was shown to obey the Langmuir isothermal equation in VC/calcium carbonate/water system. The absorption of VC on calcium carbonate was shown to increase with increase of the partial pressure of VC up to the saturation absorption and the absorption of VC on nanoscale calcium carbonate was greater than that of light-grade calcium carbonate at the same temp, and partial pressure of VC. The presence of calcium carbonate in VC suspension polymerisation system was found to influence the pressure/temp./ conversion(PTC) relationship of the reaction system. Based on the absorption of VC on calcium carbonate and VC distribution in vapour, water and polymer phases, a modified model to represent the PTC relationship of VC suspension polymerisation in the presence of calcium carbonate was proposed. 10 refs. 

Figure 21-18. Total local derivatives of the isotherms belongiag to different equilibrium states. Experimental data (squares) and theoretical results (solid lines) using the modified competitive Langmuir isotherm equation (21-8). (From reference 124, with permission.)...

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. 

The Langmuir Isotherm equation [1.1.3) is obtained either from [1.5.4) by equating ju° to or grand canonically from [1.5.2] using [I.A6.24). For the Langmuir constant... 

Figure 2 shows the effect of NaCl on the adsorption isotherm for citric acid on Chitopeari CS. K and were calculated by the Langmuir isotherm equation. 

In obtaining the Langmuir isotherm equations, several aspects of the adsorption system were presupposed in the derivations. The most important of these,and the one that has been subject to the greatest doubt, is that a uniform surface is assumed. In other words, any active site has the same attraction for an impinging molecule as does any other site. Isotherms different from the Langmuir types may be derived based on various assumptions concerning the adsorption system, including different types of nommiform smfaces. 

In the model [57], the coumarin coverage is defined by a Langmuir isotherm (Equation 2.2) with electrochemical reduction of the adsorbed coumarin (Equation 2.9) being balanced by diffusion of coumarin to the interface (Equation 2.3). Incorporation of the coumarin in the solid was considered to be negligible. The results of the model have been summarized by three characteristic dimensionless groups, X, y and p ... 

The last point on the concentration plateau for the first component is also the point at concentration Cj on a continuous profile, where it moves at the velocity 1, as given by Eq. 8.6a. Using the competitive Langmuir isotherm equations, the directional derivatives (Eq. 8.7a) become... 

The solid and liquid film linear driving force models can be written under the same general form of a second order Langmuir kinetic model [1]. We can insert the Langmuir isotherm equation q = qsbC)/ l bC)) in the partial differential equation of tire solid fihn linear driving force model (Eq. 14.3)... 

Write the Freundlich and Langmuir isotherm equations and explain how one determines their applicability. Discuss important applications and limitations of the isotherm equations, particularly in their use to describe the adsorption of multivalent cations and organic contaminants, such as pesticides, and the effects of ionic strength, pH, competitive adsorption, and complexalion. 

According to the Langmuir isotherm equation, the degree of surfactant adsorption increases with concentration. Note that there are some special cases. Figure 7.47 shows the adsorption isotherm of petroleum sulfonate on kaolinite with or without Ca + present there is a maximum adsorption in the plot of... 

The general shape of the Langmuir isotherm (equation 10,4) is shown in Figure 10.3 for several selected values of b, the bonding constant. This constant can be shown from the statistical mechanical derivation to be a function of the bonding energy, Q, as follows ... 

Dependent upon temperature and additive concentrations, any of the above can be the rate-limiting step controlling film formation. The issue is further complicated by the temperature-variable influence of interactions of molecules in solution. These can influence the equilibrium between surface and solution by influencing the flux of molecules/unit time impinging upon the surface and the probability of reaction at the surface. To a reasonable approximation, the kinetics of these processes can be described by the Langmuir isotherm. Equation (3.4) ... 

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