Kinetics Langmuir theory

The Langmuir theory is based on a kinetic principle, that is the rate of adsorption (which is the striking rate at the surface multiplied by a sticking coefficient, sometimes called the accommodation coefficient) is equal to the rate of desorption from the surface. 

The last section dealt with the basic Langmuir theory, one of the earliest theories in the literature to describe adsorption equilibria. One should note that the Langmuir approach is kinetic by nature. Adsorption equilibria can be described quite readily by the thermodynamic approach. What to follow in this section is the approach due to Gibbs. More details can be found in Yang (1987) and Rudzinski and Everett (1992). 

In the last chapter, we discussed the description of pure component adsorption equilibrium from the fundamental point of view, for example Langmuir isotherm equation derived from the kinetic approach, and Volmer equation from the Gibbs thermodynamic equation. Practical solids, due to their complex pore and surface structure, rarely conform to the fundamental description, that is very often than not fundamental adsorption isotherm equations such as the classical Langmuir equation do not describe the data well because the basic assumptions made in the Langmuir theory are not readily satisfied. To this end, many semi-empirical approaches have been proposed and the resulting adsorption equations are used with success in describing equilibrium data. This chapter will particularly deal with these approaches. We first present a number of commonly used empirical equations, and will discuss some of these equations in more detail in Chapter 6. 

At this low pressure the density of the vapour is small and thus the distillate rate. The relationship can be deduced from the kinetic gas theory and is named according to Langmuir [1] ... 

Although the Langmuir theory depicts a highly idealized situation that many real systems do not follow, it is nevertheless of great value in developing the kinetic interpretation of the heterogeneous reactions on solid surfaces. The concepts of the Langmuir theory have been extended to multilayer physical adsorption by Brunauer et al. [55]. The BET isotherm takes the form... 

Til. Hilczer, M., Brzykin, A.V., Tachiya, M. Theory of the stopped-flow method for studying micelle exchange kinetics. Langmuir 2001, 77(14), 4196-4201. 

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

Surface Area and Permeability or Porosity. Gas or solute adsorption is typicaUy used to evaluate surface area (74,75), and mercury porosimetry is used, ia coajuactioa with at least oae other particle-size analysis, eg, electron microscopy, to assess permeabUity (76). Experimental techniques and theoretical models have been developed to elucidate the nature and quantity of pores (74,77). These iaclude the kinetic approach to gas adsorptioa of Bmaauer, Emmett, and TeUer (78), known as the BET method and which is based on Langmuir s adsorption model (79), the potential theory of Polanyi (25,80) for gas adsorption, the experimental aspects of solute adsorption (25,81), and the principles of mercury porosimetry, based on the Young-Duprn expression (24,25). 

Its main features are given by the use of a stream of inert carrier gas which percolates through a bed of an adsorbent covered with adsorbate and heated in a defined way. The desorbed gas is carried off to a detector under conditions of no appreciable back-diffusion. This means that the actual concentration of the desorbed species in the bed is reproduced in the detector after a time lag which depends on the flow velocity and the distance. The theory of this method has been developed for a linear heating schedule, first-order desorption kinetics, no adsorbable component in the entering carrier gas (Pa = 0), and the Langmuir concept, and has already been reviewed (48, 49) so that it will not be dealt with here. An analysis of how closely the actual experimental conditions meet the idealized model is not available. 

Langmuir s research on how oxygen gas deteriorated the tungsten filaments of light bulbs led to a theory of adsorption that relates the surface concentration of a gas to its pressure above the surface (1915). This, together with Taylor s concept of active sites on the surface of a catalyst, enabled Hinshelwood in around 1927 to formulate the Langmuir-Hinshelwood kinetics that we still use today to describe catalytic reactions. Indeed, research in catalysis was synonymous with kinetic analysis... 

The BET surface area equation is based on Langmuir s kinetic theory of monolayer gas adsorption on surfaces [6], Langmuir theorized that the collision... 

In classical kinetic theory the activity of a catalyst is explained by the reduction in the energy barrier of the intermediate, formed on the surface of the catalyst. The rate constant of the formation of that complex is written as k = k0 cxp(-AG/RT). Photocatalysts can also be used in order to selectively promote one of many possible parallel reactions. One example of photocatalysis is the photochemical synthesis in which a semiconductor surface mediates the photoinduced electron transfer. The surface of the semiconductor is restored to the initial state, provided it resists decomposition. Nanoparticles have been successfully used as photocatalysts, and the selectivity of these reactions can be further influenced by the applied electrical potential. Absorption chemistry and the current flow play an important role as well. The kinetics of photocatalysis are dominated by the Langmuir-Hinshelwood adsorption curve [4], where the surface coverage PHY = KC/( 1 + PC) (K is the adsorption coefficient and C the initial reactant concentration). Diffusion and mass transfer to and from the photocatalyst are important and are influenced by the substrate surface preparation. 

In 1938, Brunauer, Emmett and Teller(12) and Emmett and de Witt(13) developed what is now known as the BET theory. As in the case in Langmuir s isotherm, the theory is based on the concept of an adsorbed molecule which is not free to move over the surface, and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theory does, however, allow different numbers of adsorbed layers to build up on different parts of the surface, although it assumes that the net amount of surface which is empty or which is associated with a monolayer, bilayer and so on is constant for any particular equilibrium condition. Monolayers are created by adsorption on to empty surface and by desorption from bilayers. Monolayers are lost both through desorption and through the adsorption of additional layers. The rate of adsorption is proportional to the frequency with which molecules strike the surface and the area of that surface. From the kinetic theory of gases, the frequency is proportional to the pressure of the molecules and hence ... 

The Freundlich equation proved to be applicable to the adsorption of liquids with only limited ranges of concentration. It was replaced by the Langmuir equation (see later on) and others which had a theoretical basis in the kinetic theory of gases. It is clear that neither the Freundlich nor the Langmuir equation can describe isotherms of the shape shown in Figure 10.5. 

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... 

Brunauer, Emmett and Teller, in 1938, extended Langmuir s kinetic theory to multilayer adsorption. The BET theory assumes that the uppermost molecules in adsorbed stacks are in dynamic equilibrium with the vapor. This means that, where the surface is covered with only one layer of adsorbate, an equilibrium exists between that layer and the vapor, and where two layers are adsorbed, the upper layer is in equilibrium with the vapor, and so forth. Since the equilibrium is dynamic, the actual location of the surface sites covered by one, two or more layers may vary but the number of molecules in each layer will remain constant. 

Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (/ ), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium. 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

This vapor pressure can be deduced from kinetic theory and the rate of arrival of Cs atoms per square centimeter per second (3,4). This arrival rate. A, can be calculated from the measured value of the saturation positive ion current. Langmuir first showed that if the tungsten is hot enough every cesium atom that strikes the surface evaporates off as a positive ion of cesium. This saturated positive ion current, i.p, can be measured easily with a medium sensitive galvanometer when the collector is negative. It is related to A by the equation... 

A simple but instructive kinetic theory model due to Langmuir (1918) is often used as a starting point for an atomic scale view of adsorption. It is assumed that the sorbent surface, of total area A, is comprised of Ns sites each capable of binding only one sorbate atom (or molecule). If, at a given instant, a fraction of 6 of these sites are occupied by sorbate atoms, the mean life against desorption is t, and r is independent of 6 (i.e., no interaction among sorbate atoms), the flux (j> of desorbing atoms is... 

The BET theory, developed by Brunauer, Emmett, and Teller [26], is based on the kinetic model of adsorption proposed by Langmuir [24,25] and was extended to describe the multilayer adsorption by the introduction of some assumptions listed below ... 

The permeation of a gas through a porous polymer is generally described by equations based on the kinetic theory of gases. The sorption isotherm described by Eq. 1 is concave to the pressure axis and is commonly observed for a penetrant gas in a glassy polymer. It is composed of Henry s law and Langmuir-terms [20] ... 

The theory of equilibrium adsorption and kinetics of adsorption upon the surface of adsorbents have been thoroughly treated in the Russian literature since experimental results were obtained which did not agree with Langmuir s initial assumptions (1) that the surface of the adsorbent is energetically uniform (2) that the adsorbed particles do not interact with one another (3) that the surface is covered by only a monolayer of the adsorbate. If the third assumption is retained then three alternative possibilities are offered (a) assumption (1) is rejected and assumption (2)... 

For a reaction of such complexity as methanation (or FTS) an exact kinetic theory is actually out of the question. One has to introduce one or more approximations. The usual assumption made is that one reaction step is rate determining (r.d.s.) and other steps are in equilibrium or steady state. Adsorption equilibria are described by Langmuir formulas (Langmuir-Hinshelwood, Hougen-Watson... 

Laplace (Marquis), P. S. (1805) Traite de Mecanique Celeste, fourth volume, first section (theorie de Taction capillaire) of the supplement to Book 10 (sur divers points relatifs au systeme du monde), published by Chez Courier, Paris Marmur, A. (1996) Langmuir, 12, 5704 Marmur, A. (1997) 7. Colloid and Interface Science, 186, 462 McNutt, J. E. and Andes, G. M. (1959)7. Chem. Phys., 30,1300 Merlin, V. (1992) Ph.D. Thesis, INP Grenoble, France Mullins, W. W. (1957)7. Appl. Phys., 28, 33 Mullins, W. W. (1960) Trans. Met. Soc. AIME, 218, 354 Mullins, W. W. (1963) Metal Surfaces Structure, Energetics and Kinetics, ASM... 

By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller (1938) were able to extend the Langmuir mechanism to multilayer adsorption and obtain an isotherm equation (the BET equation), which has Type II character. The original BET treatment involved an extension of the Langmuir kinetic theory of monomolecular adsorption to the formation of an infinite number of adsorbed layers. 

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