Adsorption statistical thermodynamics

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

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. 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

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,... 

Statistical Thermodynamic Isotherm Models. These approaches were pioneered by Fowler and Guggenheim (21) and Hill (22). Examples of the appHcation of this approach to modeling of adsorption in microporous adsorbents are given in references 3, 23—27. Excellent reviews have been written (4,28). 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

D. Nicholson, N. D. Parsonage. Computer Simulation and the Statistical Thermodynamics of Adsorption. New York Academic Press, 1983. 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

By applying the machinery of statistical thermodynamics we have derived expressions for the adsorption, reaction, and desorption of molecules on and from a surface. The rate constants can in each case be described as a ratio between partition functions of the transition state and the reactants. Below, we summarize the most important results for elementary surface reactions. In principle, all the important constants involved (prefactors and activation energies) can be calculated from the partitions functions. These are, however, not easily obtainable and, where possible, experimentally determined values are used. 

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. 

Instead of the classical approaches, a molecular-based statistical thermodynamic theory can be applied to allow a model of adsorption to be related to the microscopic properties of the system in terms of fluid-fluid and fluid-solid interactions, pore size, pore geometry and temperature. Using such theories the whole range of pore sizes measured can be calculated using a single approach. Two simulation... 

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 ... 

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. 

The statistical thermodynamics analysis of -mers adsorption in a one-dimensional lattice provides an intuitive approach to linear molecules confined in quasi-one-dimensional nanotubes. More elaborated analytical solutions that incorporate nearest and next-nearest-neighbors between fc-mer s ends can be obtained by applying the mapping proposed in the present work. 

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 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 ... 

As expected, the total interaction energies depend strongly on the van der Waals radii (of both sorbate and sorbent atoms) and the surface densities. This is true for both HK type models (Saito and Foley, 1991 Cheng and Yang, 1994) and more detailed statistical thermodynamics (or molecular simulation) approaches (such as Monte Carlo and density functional theory). Knowing the interaction potential, molecular simulation techniques enable the calculation of adsorption isotherms (see, for example, Razmus and Hall, (1991) and Cracknell etal. (1995)). 

The mathematical deduction of these isotherms is carried out with the help of the laws of statistical thermodynamics as described elsewhere [2,3,25,56], In this sense, the isotherm equation obtained in the case of immobile adsorption with lateral interactions is [2,3,25,56]... 

The entropy of a mobile adsorption process can be determined from the model given in [4], It is based on the assumption that during the adsorption process a species in the gas phase, where it has three degrees of freedom (translation), is transferred into the adsorbed state with two translational degrees of freedom parallel to the surface and one vibration degree of freedom vertical to the surface. From statistical thermodynamics the following equation for the calculation of the adsorption entropy is derived ... 

The development of the most important macroscopic and statistical thermodynamic models that describe adsorption phenomena on electrodes... 

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