Langmuir equation mechanism

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Although the preceding derivation is the easier to follow, the BET equation also may be derived from statistical mechanics by a procedure similar to that described in the case of the Langmuir equation [41,42]. 

Similar equations were written by Eley [204] for the exchange of N2 with N2 catalyzed by Fe or W, and mechanisms such as Eq. XVIII-33 have come to be known as Eley-Rideal mechanisms. Mechanisms such as that of Eq. XVIII-32 are now most commonly called Langmuir-Hinshelwood mechanisms (see... 

Conformity to the Langmuir equation, where it occurs, does not constitute proof of the correctness of the mechanism, since both B (or c) and n are disposable constants the quantities v, and , (cf. Equation... 

The provenance of expressions like that in Equation 3 has never been shown to be uniquely an adsorption mechanism. On the contrary, it is possible to derive special cases of Equation 3, such as the classical Langmuir equation... 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

This form of an isotherm is characteristic for a chemisorption process on an n-type catalyst forming a boundary layer with a space charge. It is evident that this mechanism is quite different from the mechanism of physical or van der Waals adsorption, which is represented by the Langmuir equation. 

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. 

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... 

The adsorption process in the micropores occurs by a volume-filling mechanism rather than by a surface-coverage mechanism. Thus, nm values obtained either by BET or Langmuir equation are similar to the limited uptake of the type I isotherms. Therefore, the amount adsorbed for different adsorptives (expressed as volume of liquid) at a relative pressure near unity is very similar, which implies that micropore filling occurs rather than the surface area coverage of the adsorbent. 

The mechanism of the CO oxidation is well established on extended surfaces of the Pt group metals since the work performed in the Ertl group in the late seventies [3, 124]. The reaction occurs via the Langmuir-Hinshelwood mechanism. CO adsorbs molecularly and 02 dissociatively, the two adsorbed species react to form C02 that readily desorbs. The rate of production of C02 is given by the following equation ... 

Murase et al.,9 by fitting their adsorption curves to a Langmuir equation, deduced that the aromatic molecule investigated (benzyl alcohol) was adsorbed on the PNIPA chains - as shown in the left cartoon of Figure 4 - and the collapse of the gel occurred in the vicinity of the solubility limit of the aromatic molecule in water. The suggested mechanism of phenol sorption is site-binding... 

These equations are consistent with a Langmuir Hinshelwood mechanism involving the rate-determining step... 

Although the simple Langmuir equation is more applicable to some forms of chemisorption, the underlying theory is of great historical importance and has provided a starting point for the development of the BET treatment and of other more refined physisorption isotherm equations. It is therefore appropriate to consider briefly die mechanism of gas adsorption originally proposed by Langmuir (1916, 1918). 

From the Langmuir-Hinshelwood mechanism a formula similar to Equation 2.11 can be derived. This formula, however, is rather complex and bulky. Therefore, only the limit is discussed here, where adsorption and desorption rates are again much higher than the rate of the surface reaction (i.e. k, k, kb, k4, kd kr, kr). For this situation,... 

Various Langmuir-Hinshelwood mechanisms were assumed. CO and CO2 were assumed to adsorb on one kind of active site, si, and H2 and IT2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

Over the years there has been a lot of debate concerning the applicability of the Dubinin-Radushkevich equation on the very low pressure region of isotherms of microporous solids. The experimental downward deviation of the DR-plot for very low pressures is generally attributed to kinetic barriers, especially in the case of nitrogen adsorption at 77K. This low pressure region of isotherms of various adsorbents can be fitted with the Langmuir equation. Hence it is shown that the downward deviation is not due to experimental factors but reflects a different adsorption mechanism. 

Raoulfs laW 371 Rates, chemical reactions, 549 basic equations, 554 constant pressure, 554 constant volume, 554 integrals of equations, 556 Langmuir-Hinshelwood mechanism, 554... 

Bandyopadhyay, D. Ghosh, A. (1996). Validity of rate equation based on Langmuir-Hinshelwood mechanism for gasification of carbon - a reappraisal. 

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