Langmuir adsorption isotherm statistical derivation

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 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 ... 

Fowler, R. M. 1935. A statistical derivation of the Langmuir adsorption isotherm. Proc. Camb. Phil. Soc. 31 260-264. 

The Langmuir adsorption isotherm was developed by Irving Langmuir in 1916 from kinetic considerations to describe the dependence of the surface fractional coverage of an adsorbed gas on the pressure of the same gas above the adsorbent surface at a constant temperature. The Langmuir isotherm expression was re-derived thermodynamically by Volmer and statistically mechanically by Fowler. In his original treatment, Langmuir made several assumptions for his model ... 

Theoretical modelling of surfactant adsorption uses as a starting point the theoretical isotherms derived from statistical and kinetic data for the gas-solid interface. The most common model is the one based on the Langmuir adsorption isotherm (see Chapter 11,2) ... 

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 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. 

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). 

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. 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

The Langmuir isotherm equation is the first theoretically developed adsorption isotherm. Many of the equations proposed later and which fit the experimental results over a wide range are either based on this equation, or these equations have been developed using the Langmuir concept. Thus, the Langmuir equation still retains an important position in physisorption as well as chemisorption theories. The equation has also been derived using thermodynamic and statistical approaches but we shall discuss the commonly used kinetic approach for its derivation. 

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. 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

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]. 

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 ... 

The theory of Nitta et al. (1984) assumes a localised monolayer adsorption on surface with an allowance for multi-site adsorption. This is an extension of the Langmuir isotherm for localised monolayer adsorption. Nitta et al. s theory is based on the statistical thermodynamics, and its derivation is given briefly below. 

The Langmuir isotherm can also be derived by other methods including statistical mechanics, thermodynamics, and chemical reaction equilibrium. The last approach is especially straightforward and useful, and it is developed as follows. For nondissociative chemisorption, the adsorption step is represented as a reaction, i.e., for an adsorbing gas-phase molecule. A, which adsorbs on a site, ... 

At K , —> 1, the equation reduces to the Simha and Frisch equation, and at 0 (i.e., 0 —> 0) to the Langmuir equation. The same equation may be obtained on the basis of statistical concept of the behavior of the flexible molecrde in space. There are other approaches to derive the isotherms of adsorption based on the statistical physics of pol5aner. The equations obtained cannot be proven by experiment, due to a great number of unknown parameters. 

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