Langmuir equation treatment

Last three figures designate the heat-treatment temperature for th, Calculated by BET equation for the samples unless otherwise specified, and by Langmuir equation for the samples designated with, Pore volume ratio of the mesoporous material to its precursory synthetic hectorites. 

The Langmuir-Hinshelwood treatment of the kinetics of surface catalyzed reactions affords a useful representation of some of the characteristics of catalytic hydrogenation. It is a limiting form of more exact equations which recognize that, even though the elementary steps are reversible, few if any will be at equilibrium (ref. 15). Not surprisingly, alternative assumptions regarding the relative rates of the forward and reverse elementary reactions can lead to approximate equations of the same form. 

Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surfece potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surfece conditions, since it does not imply any of the above restrictions. 

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). 

If the adsorption at saturation is restricted to a finite number of layers, N, the BET treatment leads to a modified equation which includes this additional parameter (cf. Chapter 6). Naturally, in the special case when N = 1, the extended BET equation corresponds to the Langmuir equation. 

The original derivation of the BET equation was an extension and generalization of Langmuir s treatment of monolayer adsorption. This derivation is based on kinetic considerations—in particular on the fact that at equilibrium the rate of condensation of... 

It is instructive to compare the Langmuir equation, Eq. (8-1), with Eq. (9-4). If the latter is correct, then k in Eq. (8-1) is a function of surface coverage. However, the reason a and E are functions of 9 is that the first two postulates of the Langmuir treatment (see Sec. 8-4) are not satisfied experimentally that is, in real surfaces all sites do not have the same activity, and interactions do exist. 

Physisorption arises from the van der Waals forces, and these forces also condense gas molecules into their liquid state. Thus, in principle, there is no reason to stop upon completion of a monolayer during physisorption. Indeed, the formation of multi-layers, which are basically liquid in nature, is very common in physisorption experiments. Brunauer, Emmett and Teller developed a theory in 1938 to describe physisorption, where the adsorbate thickness exceeds a monolayer, and this isotherm equation is known by the initials of the authors (B.E.T.). The original derivation of the B.E.T. equation is an extension of Langmuir s treatment of monolayer adsorption from kinetic arguments. Later, in 1946, Hill derived this equation from statistical mechanics. In the B.E.T. isotherm, it is assumed that ... 

In the theoretical derivation of the Langmuir equation (11.22) which is usually presented in some detail in physical chemistry textbooks, the solid surface is modeled as a chessboard (Fig. 11-14), each site of which is able with equal probability to host the adsorbed molecules (no more than one molecule per site is allowed). The treatment is restricted to the case of localized adsorption, i.e. when the exchange between molecules of the gas phase and those adsorbed on the surface is considered, while the possibility of migration of molecules from one site to another is not taken into account. The rates of adsorption and desorption are functions of the fraction of sites occupied, 0a = T / Tmax. If the molecules in the adsorption layer are not interacting with each other, the rate of adsorption, ua, is proportional to the fraction of unoccupied sites, (1 - 0a), and the vapor pressure p ... 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

The assumption of surface homogeneity is essential in the Langmuir equation otherwise a different value of K would apply to Equations 4.37 through 4.45 at various places, and eventually at every site, on the surface. For a few different values of K, the treatment of Section 4.4.3 can be applied. Attempts to deal with surface heterogeneity have been undertaken, and some of these will be detailed in Section 4.4.7. 

On the assumption that the forces that produce condensation are chiefly responsible for the binding energy of multilayer adsorption, ey proceeded to derive an isotherm equation for multilayer adsorption by a method that was a generalization of Langmuir s treatment of the unimolecular layer. The generalization of the ideal localized monolayer treatment is effected by assuming that each first layer adsorbed molecule serves as a site for the adsorption of a molecule into the second layer and so on. Hence, the concept of localization prevails at all layers and forces of mutual interaction are neglected. 

Due to the inherent uncertainty of the Langmuir model and difficulties in solving the transcendental equation (41), probably the most accurate treatment in the near-equilibrium cases is a numerical or graphical integration of the expression... 

Fig. 5 (a) shows the nitrogen adsorption isotherms of aluminum hydroxy pillared clays after heat-treatment at 300-500°C. These are of the typical Langmuir type isotherm for microporous crystals. Fig, 5 (b) shows the water adsorption isotherms on the same Al-hydroxy pillared clays [27]. Unlike the water adsorption isotherms for hydrophilic zeolites, such as zeolites X and A, apparently these isotherms cannot be explained by Langmuir nor BET adsorption equations the water adsorption in the early stages is greatly suppressed, and shows hydrophobicity. Water adsorption isotherms for several microporous crystals [20] are compared with that of the alumina pillared clay in Fig. 6. Zeolites NaX and 4A have very steep Langmuir type adsorption isotherms, while new microporous crystals such as silicalite and AlPO -S having no cations in the...

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. 

In deriving the kinetic equations of heterogeneous catalytic reactions, the surface concentrations are assumed to be steady state (or stationary), as has been done by Langmuir in the previously mentioned study of the reactions of CO and H2 with 02 on platinum (22). The treatment of surface reactions as including adsorption equilibria widely used by Hinshelwood and other authors is a particular case of this more general approach of Langmuir. 

An equation of the form of Eq. (2.32) was given by Langmuir (Carberry, 1976) for the treatment of data from the adsorption of gas on a solid surface. If the Michaelis-Menten equation is applicable, the Langmuir plot will result in a straight line, and the slope will be equal to l/rmax. The intercept will be KM/rmax, as shown in Figure 2.5. 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

Two expressions for the pre-equilibrium step in the reaction as in the above equation can be adopted one is the Michaelis-Menten equation (44) and the other is the Langmuir isotherm (45). Philosophically, the former is treated for a continuous reaction and the latter is done for a reaction whose steps can be analyzed independently. For the reaction system in which quantities of the adsorptive site and the catalytic site vary, the kinetical treatment by the latter is convenient. In this article,... 

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. 

The adsorption capacity of activated carbon may be determined by the use of an adsorption isotherm. The adsorption isotherm is an equation relating the amount of solute adsorbed onto the solid and the equilibrium concentration of the solute in solution at a given temperature. The following are isotherms that have been developed Freundlich Langmuir and Brunauer, Emmet, and Teller (BET). The most commonly used isotherm for the application of activated carbon in water and wastewater treatment are the Ereundlich and Langmuir isotherms. The Freundlich isotherm is an empirical equation the Langmuir isotherm has a rational basis as will be shown below. The respective isotherms are ... 

Adsorbates can become non-ideal for two reasons non-negligible molecular cross-section and non-negligible lateral interaction. The former non-ideality is automatically accounted for in the Langmuir treatment and this is also the case in our derivation of the Volmer equation. Lateral interaction will now be considered. It is still assumed that the solid surface is ideally flat and homogeneous. 

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