Isotherm statistical derivation

F. Fowler. A Statistical Derivation of Langmuir s Adsorption Isotherm. Proc. Camb. Phil. Soc., 31 260-280,1935. 

The AGf, AGAa AGAs. and AGss values, and, correspondingly, In fi and a values depend on the electric state of the surface, i.e., on the electrode potential or charge. This isotherm was deduced by Frumkin [i] (and named after him soon) as a general case of the -> Langmuir isotherm, which corresponds to a = 0. A statistical derivation of the Frumkin isotherm is available [ii] various model considerations and relations to other types of isotherms are discussed in [iii]. Another typical form of the Frumkin isotherm is... 

Other ions in the solution. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager s theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechaiucs from which the adsorption isotherm is derived. 

The DA equation (4.2-5) is obtained by assuming the temperature invariance of the adsorption potential at constant loading and a choice of the Weibull s distribution to describe the filling of micropore over the differential molar work of adsorption. It can be shown to be a special case of an isotherm equation derived from the statistical mechanical principles when the loading is appreciable (Chen and Yang, 1994). They derived the following isotherm... 

Comparisons between the binary isotherm predictions derived from the varous theoretical approaches have been presented by Danner and Choi for C2H6-C2H4-I3X sieve, by KauF for mixtures of Oj, CO, CH4, CjH, etc., on activated carbon, and by Sorial, Granville, and Daly for O2-N2-5A sieve (see Section 11.3). When the molecular volumes of both components are similar, there is little difference between the predictions of the ideal adsorbed solution theory and the simple statistical model as is to be expected from Eqs. (4.17)-(4.26). Both approaches generally give good predictions for sorption of mixtures of saturated hydrocarbons and other nonpolar species. However, the... 

The statistical derivation shows that the Freundlich isotherm is expected to be valid at low surface coverages in fact, the isotherm successfully predicts that 0a —> 0 when Pa O but fails to predict 9a- -I when Pa — oo. The Freundlich isotherm can handle multicomponent adsorption to some extent, and in some cases, the Langmuir isotherm can be reduced to the power function form of the Freundlich isotherm. 

The original derivation of the BET isotherm was based on kinetic arguments [12], although statistical derivations are known [13]. The hypotheses upon which the BET theory is built are the following ... 

It should be noted the result mentioned earlier holds only for the van der Waals (or Volmer) isotherm. Instead, if the Frumkin (or Langmuir) isotherm is used, the value of a obtained from the surface tension fits is about 33% greater than that obtained from molecular size [44], A possible explanation of this difference could be the fact that the Frumkin (and Langmuir) isotherm is statistically derived for localized adsorption and is more appropriate to describe adsorption at solid interfaces. In contrast, the van der Waals (and Volmer) isotherm is derived for nonlocalized adsorption, and they provide a more adequate theoretical desaiption of the surfactant adsorption at liqnid-flnid interfaces. This conclnsion refers also to the calculation of the surface (Gibbs) elasticity by means of the two types of isotherms [44]. 

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. 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

While there are several books that deal with the subject matter of this volume, the only one that develops the statistical mechanical approach is T. L. Hill s monograph (1985), which includes equilibrium as well as nonequilibrium aspects of cooperativity. Its style is quite condensed, formal, and not always easy to read. The emphasis is on the effect of cooperativity on the form of the PF and on the derived binding isotherm (BI). Less attention is paid to the sources of cooperativity and to the mechanism of communication between ligands, which is the main subject of the present volume. 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

Adsorption isotherms may be derived from a consideration of two-dimensional equations of state, from partition functions by statistical thermodynamics, or from kinetic arguments. Even though these methods are not fundamentally different, they differ in ease of visualization. We consider examples of each method in Sections 9.3 and 9.4. 

Our approach until now has been to discuss adsorption isotherms on the basis of the equation of state of the corresponding two-dimensional matter. This procedure is easy to visualize and establishes a parallel with adsorption on liquid surfaces (Chapter 7) however, it is not the only way to proceed. In the following section we consider the use of statistical thermodynamics in the derivation of adsorption isotherms and examine some other approaches in subsequent sections. 

It should be apparent — since an adsorption isotherm can be derived from a two-dimensional equation of state —that an isotherm can also be derived from the partition function since the equation of state is implicitly contained in the partition function. The use of partition functions is very general, but it is also rather abstract, and the mathematical difficulties are often formidable (note the cautious in principle in the preceding paragraph). We shall not attempt any comprehensive discussion of the adsorption isotherms that have been derived by the methods of statistical thermodynamics instead, we derive only the Langmuir equation for adsorption from the gas phase by this method. The interested reader will find other examples of this approach discussed by Broeckhoff and van Dongen (1970). 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

How is statistical thermodynamics used for deriving adsorption isotherms What are the similarities and differences between this procedure and the one based on phenomenological thermodynamics How is the kinetic theory of gases used for deriving adsorption isotherms ... 

An isotherm that is not too difficult to derive by the methods of statistical mechanics assumes an adsorbed layer that obeys the two-dimensional analog of the van der Waals equation. The result of such a derivation is the equation (see Table 9.1)... 

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

The Langmuir adsorption isotherm can be derived [134,417] using the statistical thermodynamics techniques discussed in Chapters 8 and 9. The assumptions necessary are basically the same as were used in deriving the Langmuir adsorption isotherm in Section 11.4.1. That is, adsorption is assumed to occur on a fixed array of surface sites there is assumed to be no interaction between adsorbed species the particular sites that are filled are assumed to be random and adsorbed species are immobile, corresponding to a chemisorbed species. 

This paper by Ya.B. was translated and published, with a few changes, in the collection Statistical phenomena in heterogeneous systems, 1 which was devoted especially to the theory of non-uniform surfaces and to statistical phenomena in adsorption and catalysis. In the review article by V. I. Levin in this collection the priority of Ya.B. s article in statistical research on the theory of adsorption and catalysis is emphasized. The article also cites articles by other authors who came to similar conclusions, but later than Ya.B. The significance of Ya.B. s work for the theory of catalysis is elucidated in detail in S. Z. Roginskii s book, Adsorption and Catalysis on a Non-Uniform Surface. 2 After this a summary of this paper by Ya.B. has entered into the majority of monographs and textbooks on catalysis. Thus, in the course of Thomas and Thomas3 the derivation of the adsorption isotherm on a non-uniform surface is given in full and referred to as classical. 

A statistical thermodynamic equation for gas adsorption on synthetic zeolites is derived using solid solution theory. Both adsorbate-adsorbate and adsorbate-adsorbent interactions are calculated and used as parameters in the equation. Adsorption isotherms are calculated for argon, nitrogen, ammonia, and nitrous oxide. The solution equation appears valid for a wide range of gas adsorption on zeolites. 

The isotherm model of Schirmer et al. (T.) for sorption in molecular sieves is based on statistical thermodynamics in which the configuration integrals describing the sorption behaviour are extracted from the available data. The model does not presuppose any specific kind of sorption mechanism. The multi-component form of this isotherm derived by Loughlin and Roberts (8 ) is also not limited to any particular sorption mechanism,... 

Schirmer et al. (7.) indicate that the constants and E j may be derived from physical or statistical thermodynamic considerations but do not advise this procedure since theoretical calculations of molecules occluded in zeolites are, at present, at least only approximate, and it is in practice generally more convenient to determine the constants by matching the theoretical equations to experimental isotherms. We have determined the constants in the model by a method of parameter determination using the measured equilibrium data. Defining the entropy constants and energy constants as vectors... 

The kinetic derivation has the disadvantage that it refers to a certain model. The Langmuir adsorption isotherm, however, applies under more general conditions and it is possible to derive it with the help of statistical thermodynamics [8,373], Necessary and sufficient conditions for the validity of the Langmuir equation (9.21) are ... 

Physical understanding of the assumptions underlying the various isotherms can be increased by deriving them by statistical mechanics. This will be done only for the Langmuir isotherm. At equilibrium, the chemical potential of the adsorbate in the gas phase and on the surface must be equal ... 

Adsorption isotherms represent a relationship between the adsorbed amount at an interface and the equilibrium activity of an adsorbed particle (also the concentration of a dissolved substance or partial gas pressure) at a constant temperature. The analysis of adsorption isotherms can yield thermodynamic data for the given adsorption system. Theoretical adsorption isotherms derived from statistical and kinetic data, and using the described assumptions (see 3.1), are known only for the gas-solid interface or for dilute solutions of surfactants (Gibbs). Those for the system gas-solid are of a few basic types that can be thermodynamically predicted81. From temperature relations it is possible to calculate adsorption and activation energies or rate constants for individual isotherms. Since there are no theoretically founded equations of adsorption isotherms for dissolved surfactants on solids, the adsorption of gases on solides can be used as a starting point for an interpretation. 

The statistical mechanical expressions for other thermodynamic quantities can also be written in terms of g(r) and u(r). The derivation is similar to that leading to the energy equation [25,32]. Let us state here only the equations corresponding to the pressure p and the isothermal compressibility a. The... 

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