Langmuir desorption equation

Replacing the fugacity by coverage p = (1/T/0 )c ewe arrive at the Langmuir desorption equation, which shows the dependence of the desorption rate on coverage... 

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,... 

Since 1916 when Langmuir published his fundamental paper on adsorption, the Theory of Activated Adsorption/Desorption Kinetics (TAAD) has, almost exclusively, been used for the interpretation of adsorption/desorption kinetics. However, contraiy to the success of Langmuir equation to represent the adsorption equilibria, a dramatic failure of TAAD was observed to represent by the Langmuir kinetic equation and its further modifications, [1-3] the monitored adsorption/desorption kinetics. 

The Langmuir isotherm equation can also be derived from the formal adsorption and desorption rate equations derived from chemical reaction kinetics. In Section 3.2.2, we see that the mass of molecules that strikes 1 m2 in one second can be calculated using Equation (186), by applying the kinetic theory of gases as [dmldt = P2 (MJ2nRT)m], where P2 is the vapor pressure of the gas in (Pa), Mw is the molecular mass in (kg mol ), T is the absolute temperature in Kelvin, R is the gas constant 8.3144 (nT3 Pa mol-K-1). If we consider the mass of a single molecule, mw (kg molecule-1), (m = Nmw), where N is the number of molecules, by considering the fact that (R = kNA), where k is the Boltzmann constant, and (Mw = NAmw), we can calculate the molecular collision rate per unit area (lm2) from Equation (186) so that... 

Some molecules are dissociated upon adsorption, and this type of adsorption is called dissociative adsorption. The adsorption reaction can be shown as [(XY)gas + 2 Surface <-> Xads-Surface + Yads-Surface], and the Langmuir adsorption expression must be modified because two sites on the adsorbent are consumed per adsorbate molecule. The probability of desorption is also different. When these differences are considered, the Langmuir adsorption equation for the dissociative adsorption becomes... 

Christiansen s formula cannot be used because two adsorbed species react with one another. If the rate is controlled by the surface reaction, reactants and products are at adsorption quasi-equilibrium, that is, for each the adsorption and desorption rates are practically equal. With Langmuir s equations for these rates ... 

At higher partial pressures, the behavior becones nonlinear, and more complex models are required to describe the observed equilibrium data. A frequently used model for monomolecular layer adsorption is the Langmuir isotherm equation. This equation is derived from simple mass-action kinetics. It assumes that the surface of the pores of the adsorbent is homogeneous and that the forces of interaction between the adsorbed molecules are negligible. Let/be the fraction of the surface covered by adsorbed molecules. Therefore, 1 -/ is the fraction of the bare surface. Then, the net rate of adsorption is the difference between the rate of adsorption on the bare surface and desorption from the covered surface ... 

This is the simplest kinetic case in which the overall adsorption process is governed by the rates of adsorption and desorption of the solute molecules onto and from the surface. For this case, the adsorption kinetics is usually described by the well-known Langmuir kinetic equation ... 

This equation, the Langmuir-Hinshelwood equation, was first proposed by Langmuir and Hinshelwood in the 1920-30s for solid-catalyzed gas-phase reactions under the assumption that adsorption and desorption rates are high compared with rates of other chemical transformations on the catalyst surface. In this model, adsorption-desorption steps are considered to be at equilibrium. Later, Hougen, and Watson proposed a similar equation, the Hougen-Watson equation, for a reversible catalytic reaction, again under the assumption that the adsorption-desorption steps are at equilibrium. 

In this book we use models of the first three types, with emphasis on the first one. For the more complicated models of the other two types die reader is referred to more specialized litterature. In modelling chemical reactors we usually proceed stepwise, starting with the smallest scale. On that scale we observe the chemical kinetics of homogeneous or heterogeneous reactions, including adsorption and desorption effects. The well known Langmuir-Hinshelwood equations for reactions at solid surfaces are examples of this sort of model. We call these molecular scale models. 

Langmuir s equation [24] is regularly derived in a variety of ways, of which the most inmitive is based on chemical equilibrium (Chapter 12) rather than phase equilibrium. He originally based it only on the external surface of crystalline solids, not on the internal surface of the pores in crystalline or amorphous solids - by far the most industrially important adsorption application it is most often applied to these internal surfaces. The discussion here is a significant simplification of that he presented. He assumed that at equilibrium the rates of adsorption and desorption were equal and that... 

Detailed Modeling Results. The results of a series of detailed calculations for an ideal isothermal plug-flow Langmuir system are summarized in Figure 15. The soHd lines show the form of the theoretical breakthrough curves for adsorption and desorption, calculated from the following set of model equations and expressed in terms of the dimensionless variables T, and P ... 

At low values of the bulk concentration Bcy surface coverage is proportional to this concentration, but at high values it tends toward a limit of unity. This equation was derived by Irving Langmuir in 1918 with four basic assumptions (1) the adsorption is reversible (2) the number of adsorption sites is limited, and the value of adsorption cannot exceed A° (3) the surface is homogeneous aU adsorption sites have the same heat of adsorption and hence, the same coefficient B and (4) no interaction forces exist between the adsorbed particles. The rate of adsorption is proportional to the bulk concentration and to the fraction 1-9 of vacant sites on the surface = kjil - 9), while the rate of desorption is proportional to the fraction of sites occupied Vj = kjd. In the steady state these two rates are equal. With the notation kjk = B, we obtain Eq. (10.14). 

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. 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

The binding energy in field adsorption can be derived from consideration of the kinetics of field adsorption. Specifically, it can be determined from a temperature dependence of the probability of field adsorption on an adsorption site, or the degree of coverage of field adsorption on a plane. As will be shown, a consideration of the probability of field adsorption based on adsorption time and desorption time leads to an equation equivalent to the Langmuir adsorption isotherm, but specific to the problem of field adsorption.112115 Let us focus on one surface atom. The average time it takes to have an image gas atom field adsorbed on the surface atom, ra, is... 

Most solid sorbents rely on vapors being sorbed by a physical adsorption mechanism the substance enters the internal pores of the sorbent and is held there by attractive forces considerably weaker and less specific than those of chemical bonds. These weakly attractive forces facilitate desorption for subsequent analysis. The mechanisms for physical adsorption have been studied extensively and are described mathematically by equations such as the Langmuir isotherm. 

The rate of desorption can be described by a Langmuir equation, so that the adsorption rate in the general case will be non-linear. We will have ... 

Equations (20) and (21) are Langmuir expressions for adsorption and desorption rates. If the equilibrium is reached with respect to the process (19), then, in accordance with mass action law,... 

If, on the other hand, surface reaction determined the overall chemical rate, equation 3.68 (or 3.69 if an Eley-Rideal mechanism operates) would represent the rate. If it is assumed that a pseudo-equilibrium state is reached for each of the adsorption-desorption processes then, by a similar method to that already discussed for reactions where adsorption is rate determining, it can be shown that the rate of chemical reaction is (for a Langmuir-Hinshelwood mechanism) ... 

To illustrate the analogy more clearly, it is necessary to consider the derivation of the Langmuir adsorption isotherm. We can incorporate the above assumptions into an equilibrium expression which equates the rate of adsorption racis to that of desorption rdes of gas molecules of type J. The desorption rate is directly proportional to the fraction of monolayer sites occupied /, and is expressed as... 

The Langmuir adsorption isotherm is based on the characteristic assumptions that (a) only monomolecular adsorption takes place, (b) adsorption is localised and (c) the heat of adsorption is independent of surface coverage. A kinetic derivation follows in which the velocities of adsorption and desorption are equated with each other to give an expression representing adsorption equilibrium. 

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... 

Derive the Langmuir isotherm by equating the rate of desorption of adsorbate from the surface (proportional to the fraction of surface sites occupied) to the rate of adsorption (proportion to the pressure of adsorbate in the gas phase). 

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