Uniform-Langmuir isotherm

Constant solution / I I. A constant and uniform Langmuir isotherm means that pR is the same for all simulated domains and does not change with time. 

Steepness of the isotherm beyond the monolayer region, that is, point B or where the knee is located, is a direct reflection of the surface energy heterogeneity. Mathematically, the factor s in the Unilam (i.e., uniform Langmuir) isotherm is an indicator of the steepness of this portion of the isotherm, and is hence referred to as the heterogeneity parameter. The Unilam isotherm has been discussed by Valenzuela and Myers (1989) and by Do (1998). It is given by ... 

Besides having the capital property of being able to imitate any kinetics—see Eq. 61—Astarita s uniform kinetics are often patient of a mechanistic derivation. Thus, for what Astarita and Ocone call uniform generalized Langmuir isotherm kinetic mechanism (UGLIKM), we might take 6(x) dx to be the fraction of catalytic surface occupied by the species A(x) with index in... 

Langmuir equation for a uniform surface, but by the Zel dovich and Rogin-skii equation or by the Bangham equation. Adsorption equilibrium is described not by the hyperbolic Langmuir isotherm, but by the Freundlich isotherm or the logarithmic isotherm (40). 

Assuming that the second process is rapid, we obtain the following standard picture of adsorption on a uniform surface the equilibrium concentration q, which depends on the pressure of the gas, is determined by the Langmuir isotherm. The only difference from the standard picture is that the statistical sum for all states of the adsorbed molecule in a potential hole must be replaced by a combination of two statistical sums for all states of the adsorbed molecule and for all possible states of the surface element. This, of course, has no effect on the form of the Langmuir equation. Under very simple assumptions the kinetics of establishment of equilibrium will also not differ from those on a uniform surface. Thus, the initial velocity is proportional to the pressure and approaches equilibrium exponentially. 

An unusual dependence for adsorption on a uniform surface arises when it is assumed that the rate of change of the surface is considerably slower than the rate of adsorption (see figure). If adsorption and desorption occur rapidly, the state of the surface remains practically unaltered and we then get an adsorption isotherm corresponding to a non-uniform surface with a distribution p(E, 0) of the heat of adsorption (curve 1) [3]. However, when the time interval is considerable, slow adsorption accompanies changes in the properties of the surface, and the amount of gas adsorbed approaches that given by the Langmuir isotherm (curve 2, point B), which describes a state of complete equilibrium (see above). 

This means that the decomposition occurs uniformly across the surface, where the products are weakly bound and rapidly desorbed consequently, the rate-determining step is the surface decomposition step. This type of reaction shows two rate-limiting laws corresponding to the two extreme behaviors of the Langmuir isotherm. That is, at low pressure, 0A is small and proportional to the pressure, and the rate becomes first order in A(g) ... 

Various attempts have been made to modify the Langmuir model. One of the best known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-adsorbate interactions in a localized monolayer on a uniform surface. However, on an empirical basis the Fowler-Guggenheim equation turns out to be no more successful than the original Langmuir isotherm. The highly complex problem of localized adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everett (1992). 

The assumption of uniformity is in fact justified for some realistic kinetic schemes, such as Langmuir isotherm catalyzed reactions, Michaelis-Menten kinetics, and others (Aris, 1989 Cicarelli et al, 1992). The assumption bears a more than superficial analogy with those systems termed pseudo-monomolecular by Wei and Prater (1962). Mathematically, it is a very powerful assumption By crossing out the dependence of F[ ] on x, its value has been reduced from an infinite-dimensional vector (a function of x) to a scalar. This simplification makes Eq. (102) a quasilinear one, and it can be integrated explicitly by introducing a warped time scale t(0. (0) = 0. The solution, as can be verified by inspection, is... 

In obtaining the Langmuir isotherm equations, several aspects of the adsorption system were presupposed in the derivations. The most important of these,and the one that has been subject to the greatest doubt, is that a uniform surface is assumed. In other words, any active site has the same attraction for an impinging molecule as does any other site. Isotherms different from the Langmuir types may be derived based on various assumptions concerning the adsorption system, including different types of nommiform smfaces. 

The isotherm we have developed in equation 3.29 is the Langmuir Isotherm which, in principle, applies only to sets of sites of uniform strength. However, the surfaces of catalysts usually contain sites with a distribution of strengths of adsorption, a fact that would be reflected in the activation energy of the constant k., and in the heat of adsorption in KA. It has been found, however, that real catalytic rate equations based on the equilibrium isotherm are generally satisfactory in fitting the rates of catalytic reactions without taking this complication into account. Nevertheless, as in all such cases, we should bear this approximation in mind so we can be alert to the appearance of a counter-example. 

The Langmuir isotherm is widely used for reaetants that reversibly chemisorb on the catalyst surface. The surface is assumed to be completely uniform, with all sites having equal reactivity. Adsorption occurs when molecules with sufficient energy strike vacant sites or uncovered parts of the surface. The process is described as a reaction between a molecule from the gas phase and an unoccupied site, s. 

Unilan equation is another empirical relation obtained by assuming a patchwise topography on the surface and each patch is ideal such that the local Langmuir isotherm is applicable on each patch. The distribution of energy is assumed uniform. Integrating eq. (3.2-17a) with the following uniform energy distribution... 

The isotherm equation of BLK presented above is in fact identical to the Unilan equation, derived in Example 6.3-2 (eq. 6.3-9) using a local Langmuir isotherm and a uniform energy distribution. To prove this, we note from the equation for the heat of adsorption (6.5-1) that the minimum and maximum heats of adsorption corresponding to maximum and minimum coverages are... 

Because all sites on a uniform surface are identical, it is sufficient to insert in the Langmuir isotherm the condition that 6 varies linearly according to an equation... 

This isotherm characterized by linear variation of In p with 0 is known as the Temkin isotherm. The Temkin isotherm for nonuniform surfaces can be derived by dividing the surface into a number of uniform elanents, on each of which the heat of these elements obeys the Langmuir isotherm. The isotherm equation obtained after certain assumptions... 

Adsorption Equilibrium on Uniform Surfaces - Langmuir Isotherms 91... 

With this introduction, the derivation of an adsorption isotherm, which represents an equilibrium adsorption process, can be conducted [5]. The Langmuir isotherm is obtained based on the assumptions associated with an ideal uniform surface on which all sites are identical [5], i.e.,... 

If a nonuniform surface is now considered, it is again neeessary to divide the surface into domains, di, each of which has uniform sites giving a constant heat of adsorption [1]. Within each of these domains, a Langmuir isotherm is... 

Two ideas have been considered to derive valid lumped models for systems with nonlinear kinetics. The first is the use of cooperative kinetics,which is applicable to catalytic reactions governed by Langmuir isotherm adsorption, and can be extended to other models if there is uniform catalytic surface coverage. The second is a transformation of the coordinates (such as nonlinear stretching of the time coordinate). ... 

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