Other adsorption isotherms

This is one of the oldest empirical adsorption isotherms, developed by Freundlich in 1907. It is useful for adsorption from liquid solutions and also for chemisorption isotherms. 

A plot of In V versus In P2 gives a line with slope 1/n of the intensity of the adsorption, and the intercept (V, Kr) gives a measure of adsorbent capacity. 

This empirical adsorption isotherm is useful for the chemisorption experiments where a monolayer forms, and it considers that all sites are not energetically equivalent. The enthalpy of adsorption is assumed to vary linearly with the fractional coverage 

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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. 

The preceding set of equation is valid only for Langmuir adsorption isotherms, and numerical simulation must be used to obtain the flow rates for other adsorption isotherm shapes or for multicomponent mixtures. 

The use of other adsorption isotherms such as the Freimdlich, T6th, or Langmuir-Freundlich isotherms results in different thermodynamic correction factors [53]. 

Other adsorption isotherms have been used in the literature to analyze experimental data. However, it can usually be shown that they are limiting forms of the general isotherm derived here (equation (10.8.38)). For example, in early work the importance of the term in the diffuse layer potential (zj f ([) ) was not recognized. By using equation (10.8.7) for this contribution, the exact dependence of this term on Qni and Qad is obtained. Equation (10.8.7) can be simplified in limiting cases and the form of the isotherm without an explicit dependence on ([) obtained. 

There are several other derivations of the Langmuir adsorption isotherm from statistical mechanics and thermodynamics. Although the model is physically unrealistic for describing the adsorption of gases on real surfaces, its successes, just like the success of other adsorption isotherms also based on different simple adsorption models, is due to the relative insensitivity of macroscopic adsorption measurements to the atomic details of the adsorption process. Thus the adsorption isotherm... 

The principle of microscopic reversibility applies at equilibrium, which states that the rate of chemisorption equals the rate of desorption. It is important to emphasize that individual rates of the forward and backward steps are not zero at equilibrium but that these rates are nonzero and equal to each other. Adsorption isotherms predict surface coverage at equilibrium where iRadsoipaon = 0. Hence,... 

A somewhat subtle point of difficulty is the following. Adsorption isotherms are quite often entirely reversible in that adsorption and desorption curves are identical. On the other hand, the solid will not generally be an equilibrium crystal and, in fact, will often have quite a heterogeneous surface. The quantities ys and ysv are therefore not very well defined as separate quantities. It seems preferable to regard t, which is well defined in the case of reversible adsorption, as simply the change in interfacial free energy and to leave its further identification to treatments accepted as modelistic. 

Equation XI-27 shows that F can be viewed as related to the difference between the individual adsorption isotherms of components 1 and 2. Figure XI-9 [140] shows the composite isotherms resulting from various combinations of individual ones. Note in particular Fig. XI-9a, which shows that even in the absence of adsorption of component 1, that of component 2 must go through a maximum (due to the N[ factor in Eq. XI-27), and that in all other cases the apparent adsorption of component 2 will be negative in concentrated solution. 

One application of the grand canonical Monte Carlo simulation method is in the study ol adsorption and transport of fluids through porous solids. Mixtures of gases or liquids ca separated by the selective adsorption of one component in an appropriate porous mate The efficacy of the separation depends to a large extent upon the ability of the materit adsorb one component in the mixture much more strongly than the other component, separation may be performed over a range of temperatures and so it is useful to be to predict the adsorption isotherms of the mixtures. 

A factor militating against the use of other adsorptives for pore size determination at the present time is the lack of reliable r-curves. The number of published isotherms of vapours such as benzene, carbon tetrachloride or the lower alkanes, or even such simple inorganic substances as carbon dioxide, on a reasonable number of well-defined non-porous adsorbents, is very small. 

The channels in zeoHtes are only a few molecular diameters in size, and overlapping potential fields from opposite walls result in a flat adsorption isotherm, which is characterized by a long horizontal section as the relative pressure approaches unity (Fig. 6). The adsorption isotherms do not exhibit hysteresis as do those in many other microporous adsorbents. Adsorption and desorption are reversible, and the contour of the desorption isotherm foUows that of adsorption. 

Adsorption Isotherms. EquiUbrium dialysis studies indicate around 10 repeat VP units (base moles) are required to form favorable complexes (89,90). This figure can rise to several hundred for methyl orange and other anions depending on stmcture (91,92). 

The major surfactant in the foam may usually be considered to be present at the bubble surfaces in the form of an adsorbed monolayer with a substantially constant F, often of the order of 3 X 10" (g mol)/ cm", for a molecular weight of several hundred. On the other hand, trace materials follow the linear-adsorption isotherm Tj = KiCj if their concentration is low enough. For a wider range of concentration a Langmuir or other type of isotherm may be applicable (Davies and Rideal, loc. cit.). 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

FIGURE 3.5 Major components of classical receptor theory. Stimulus is the product of intrinsic efficacy (s), receptor number [R], and fractional occupancy as given by the Langmuir adsorption isotherm. A stimulus-response transduction function f translates this stimulus into tissue response. The curves defining receptor occupancy and response are translocated from each other by the stimulus-response function and intrinsic efficacy. 

As noted in Chapter 1, the most simple and theoretically sound model for drug-receptor interaction is the Langmuir adsorption isotherm. Other models, based on receptor behavior (see Chapter 3), are available. One feature of all of these models (with the exception of some instances of the... 

In order to discuss the nature of the interaction between an adsorbed molecule and a surface it is important that the surface coverage be less than one monolayer since in multimolecular adsorption and capillary condensation the spectrum of the adsorbate molecule perturbed by interaction with other adsorbate molecules may mask the spectrum of the adsorbate molecule perturbed by interaction with the adsorbent. Surface coverages may be determined by obtaining an adsorption isotherm with the adsorbate... 

Static Involving Use of Adsorption Isotherms BRUNAUER, EMMETT, AND TELLER (B.E.T.). In this method tire surface area is not measured directly, but the number of molecules of the adsorbed substance required to give a monolayer (N) is determined. If the mean area per molecule (a) of the adsorbed substance is known by other means, the area of the solid may... 

Such a model should be as simple as possible, without however missing any of the underlying thermodynamic and physicochemical factors which cause electrochemical promotion. In particular it will be shown that even the use of Langmuir-type adsorption isotherms, appropriately modified due to the application of potential (or equivalently by the presense of promoters) suffice to describe all the experimentally observed rules G1 to G7 as well as practically all other observations regarding electrochemical promotion including the effect of potential on heats of adsorption as well as on kinetics and reaction orders. 

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