Langmuir adsorption rate equation

The rate of formation of the affinity complex is often described by the second-order Langmuir adsorption rate equation... 

Chase [32] used the adsorption rate-limited model [Eqs. (7 —(11)[ to analyze the experimental breakthrough curves in affinity chromatography. This empirical approach assumes that all the rate-limiting processes can be represented by an apparent single second-order Langmuir adsorption rate equation in which k is considered a lumped" parameter. 

Langmuir Adsorption Isotherm Equation The Langmuir adsorption isotherm equation was developed originally to describe gas adsorption onto clean surfaces and can be derived theoretically based on rates of evaporation and condensation (Adamson, 1976). The Langmuir adsorption isotherm equation is... 

All the balances have accumulation, convection, axial dispersion, and reaction terms. The equations include liquid holdup, Bi, and superficial liquid velocity, w. Langmuir-type rate equation, for the main reaction, Equation 15.4, included also an activity correction term a. Kst and in Equations 15.5-15.7 indicate the adsorption parameters for stearic acid and heptadecene, respectively. Equation 15.4 corresponds to a monomolecular transformation of stearic acid via the adsorption of the reactant to the main product. Adsorption terms for stearic acid and heptadecene were used, since both of these compounds contain functional groups enabling adsorption on the active sites of the catalyst Reaction rates were assumed not to be limited by heptadecane adsorp-UoiL Thus, the adsorption term of heptadecane was n ected. In line with the experimental observations indicating catalyst deactivation. Equation 15.4 (Table 15.2) was modified to incorporate the decrease in catalyst activity. In particular, the activity was assumed... 

Combined usage of Langmuir adsorption kinetics equation (2) and equation (1) allows us to calculate characteristics of adsorption-desorption process of asphaltenes onto surface of searched surfaces relative fraction of occupied surface by molecules (a), the maximal adsorbed mass density the equilibrium constant of adsorption (K), the rate... 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

Kinetic orders in CO oxidation on M/A1203 can be explained by the classical Langmuir-Hinshelwood expression for the rate equation, as a function of the rate constant k, the adsorption constants K and the partial pressures P ... 

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. 

In the steady-state approach (equations (35) and (36)), no attempt is made to isolate the adsorption step from the internalisation of solutes. In this case, a Langmuir adsorption via membrane carriers is coupled to an irreversible and rate-limiting internalisation of the solute carrier complex [186], The process can be described by the Michaelis-Menten equation ... 

This identity of formulation means that in normal cases correct results for the velocity equation will be obtained by assuming that adsorption equilibrium is established and the adsorbed molecules react at a rate proportional to their surface concentration when they possess the critical energy. Then, for the surface concentration 07 of every reactant the Langmuir adsorption isotherm holds ... 

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. 

The monomolecular conversion of three components has also been considered in some detail by Kallo (30). Rate equations based on Langmuir adsorption were developed assuming a number of different mechanistic schemes including steps in which surface adsorption was not at equilibrium. Since the rate equations developed became complicated, practical application was devoted to cases in which only initial reaction rates were observed, so... 

The kinetics of the ethylene oxidation are rather complicated as they depend not only on ethylene and oxygen pressure but also on the concentration of the reaction products. These influence the rate by adsorption competition with the reactants. Moreover, different forms of adsorbed oxygen may occur on the catalyst surface. Consequently, the rate equations proposed in the literature consist of either Langmuir—Hinshelwood and Eley—Rideal types or power rate models with non-integer coefficients. Power rate models are less appropriate as their coefficients inevitably depend on the reaction conditions. 

The next problem of the Langmuir-Hinshelwood kinetics, the validity of the rate-determining step approximation, has not been rigourously examined. However, as has been shown (e.g. refs. 57 and 63), the mathematical forms of the rate equations for the Langmuir-Hinshelwood model and for the steady-state models are very similar and sometimes indistinguishable. This makes the meaning of the constants in the denominators of the rate equations somewhat doubtful in the Langmuir—Hinshelwood model, they stand for adsorption equilibrium constants and in the steady-state models, for rate coefficients or products and quotients of several rate coefficients. 

The kinetics of the ammoxidation of xylenes over a vanadium catalyst and mixed vanadium catalysts were studied. The reaction rate data obtained were correlated with the parallel consecutive reaction scheme by the rate equations based upon the Langmuir-Hinshelwood mechanism where the adsorption of xylenes was strong. The reaction rates of each path are remarkably affected by the kind of xylene and catalyst. The results of the physical measurement of catalysts indicated that the activity and the selectivity of reaction were affected by the nature and the distribution of metal ions and oxygen ion on catalyst surface. 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

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 D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

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) ... 

In principle it is possible to write down the rate equation for any rate determining chemical step assuming any particular mechanism. To take a specific example, the overall rate may be controlled by the adsorption of A and the reaction may involve the dissociative adsorption of A, only half of which then reacts with adsorbed B by a Langmuir-Hinshelwood mechanism. The basic rate equation which represents such a process can be transposed into an equivalent expression in terms of partial... 

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... 

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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