Langmuir equation/approach

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

With these reservations in mind, we will next consider three approaches that have been used in the past to measure the efficacy of a partial agonist acting on an intact tissue. Each will be analyzed in two ways with the details given in Appendix 1.4C (Section 1.4.9.3). The first is of historical interest only and is based on Stephenson s original formulation, as expressed in Eq. (1.27) (Section 1.4.2) and with receptor occupancy given by the Hill-Langmuir equation in its simplest form, which we have already seen to be inadequate for agonists. The second analysis defines receptor occupancy as all the receptors that are occupied, active plus inactive. 

Derivation of the Langmuir Equation— Adsorption of a Single Species. The kinetic approach to deriving a mathematical expression for the Langmuir isotherm assumes that the rate of adsorption on the surface is proportional to the product of the partial pressure of the adsorbate in the gas phase and the fraction of the surface that is bare. (Adsorption may occur only when a gas phase molecule strikes an uncovered site.) If the fraction of the surface covered by an adsorbed gas A is denoted by 0Ay the fraction that is bare will be 1 — 0A if no other species are adsorbed. If the partial pressure of A in the gas phase is PA, the rate of adsorption is given by... 

In this case the kinetic approach to the derivation of the Langmuir equation requires that the process be regarded as a reaction between the gas molecule and Wo surface sites. Thus the... 

Inasmuch as the Langmuir equation does not allow for nonuniform surfaces, interactions between neighboring adsorbed species, or multilayer adsorption, a variety of theoretical approaches that attempt to take one or more of these factors into account have been pursued by different investigators. The best-known alternative is the BET isotherm, which derives its name from the initials of the three individuals responsible for its formulation, Brunauer, Em-... 

The three isotherms discussed, BET, (H-J based on Gibbs equation) and Polanyi s potential theory involve fundamentally different approaches to the problem. All have been developed for gas-solid systems and none is satisfactory in all cases. Many workers have attempted to improve these and have succeeded for particular systems. Adsorption from gas mixtures may often be represented by a modified form of the single adsorbate equation. The Langmuir equation, for example, has been applied to a mixture of n" components 11). 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

The isotherm also reveals that for m p > 1,0 approaches unity as an upper limit. Thus at both the upper and lower limits, the isotherm gives the same results as the Langmuir equation. At intermediate values the two functions differ slightly, but it would probably be difficult to distinguish between them in fitting experimental data. ... 

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). 

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. 

Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surfece potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surfece conditions, since it does not imply any of the above restrictions. 

The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used. 

The more desirable approach is to determine f(Q) from an assumed 0(P,T,Q) and the experimental adsorption isotherm. Sips (16) showed that Equation 1 could be treated by a Stieltjes transform, so that in principle an explicit function could be written for f(Q), provided the experimental isotherm function, 0, could be expressed in analytical form. Subsequently, Honig and coworkers (10, 11, 12) investigated this approach further. The difficulty is that only for certain types of assumed functions 0 and 0 is the approach practical. As a consequence the procedure has been first to restrict the choice of 0 to the Langmuir equation, and second to assume certain simple functions for 0 such as the Freundlich and Temkin isotherm equations. The system is thus forced into an arbitrary mold and again it is not certain how much reliance should be placed on the site energy distributions obtained. 

In geological surfaces, the solid-gas and solid-liquid interfaces are important, so the correct thermodynamic adsorption equation (Gibbs isotherm) cannot be used. Instead, other adsorption equations are applied, some of them containing thermodynamic approaches, and others being empirical or semiempirical. One of the most widespread isotherms is the Langmuir equation, which was derived for the adsorption of gas molecules on planar surfaces (Langmuir 1918). It has four basic assumptions for adsorption (Fowler 1935) ... 

An extension of the Langmuir approach to multilayers adsorption was made by Brunauer, Emmett, and Teller, BET [17]. They assumed that the Langmuir equation applies to each layer. The heat of adsorption of the first layer was assumed to have a special value, but for the subsequent layers, the heat of adsorption was assumed to be equal to the heat of condensation of the gas. The volume adsorbed is then a summation of the adsorbed volumes of each layer. Upon evaluation of the summation, the BET equation results ... 

This equation is known as the Fnimkin isotherm. It is clear that the Langmuir isotherm is a special case of the Prumkin isotherm, which can be derived from it by setting r = 0. It can also be seen that, for reasonable values of the parameter r, the exponential term in this equation approaches unity for very small values of 0 and becomes constant when 0 is close to unity. Thus, at extreme values of 0, the Prumkin and the Langmuir isotherms lead to the same dependence of coverage on potential, hence to the same rate equations in electrode kinetics. 

It is seen that regarding the derivation of the Langmuir equation, "many roads lead to Rome". The fact that very different physical models all lead to the same result is one of the reasons for its wide application. Considering these disparate approaches quantitatively, the Interpretation of kJ, requires more attention. To that end, let us take a closer look at the kinetics, thereby following an idea originally due to Langmuir and elaborated by others ). 

In contrast, the liquid interface can be considered homogeneous which is a precondition for a non-localised adsorption. Any free space at the liquid interface is available for ion adsorption. At the transition from a localised to a non-localised adsorption the exponential term of the Stem-Langmuir equation can be preserved and the pre-exponential multiplier must be changed. This was done by Martynov (1979). The basis of this approach is the notation of a homogeneous potential well along the liquid interface as a whole. 

Several mathematical models have been proposed for representing the overall adsorption of pesticides by soils. The most comprehensive one appears to be that of Lambert et al. (9), in which the partition of pesticides between soil water and soil is represented by a linear adsorption equation similar to the Langmuir equation. In this model they have assumed that the active adsorbent for pesticides in soils is the soil organic matter. This approach has been successful in modeling the adsorption of nonionic pesticides on soils (12), Lambert (13) has introduced an index of soil adsorption of pesticides which is intended to indicate the amount of active organic matter in a soil and therefore may be used to compare the adsorption capacity of one soil with that of another. Lambert states that the index is independent of the pesticide being adsorbed. 

There is evidence that certain chemical adsorption processes involve dissociation of the adsorbate to form two bonds with the adsorbent surface. On many metals, hydrogen is adsorbed in atomic form. For such situations the kinetic approach to the derivation of the Langmuir equation requires that the process be regarded as a reaction between the gas molecule and two vacant surface sites. Thus, the adsorption rate is written as... 

This approach do not requires an assumption on model fitting with experimental data even though we have shown that Langmuir equation described well the adsorption process in the studied T, P ranges. Thus, (Qsi) values were extracted from the slops of isosteres (Figure 5)... 

Analyzing the behavior of the inhibition coeificient IC (Figure 14.6), it is shown that it continuously decreases for both mixtures with hydrogen molar fraction i.e. increasing with CO partial pressure), approaching the plateau value according to the Sievert-Langmuir equation (first expression in eqn (14.16)). ... 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

Chapters 2 to 4 deal with pure component adsorption equilibria. Chapter 5 will deal with multicomponent adsorption equilibria. Like Chapter 2 for pure component systems, we start this chapter with the now classical theory of Langmuir for multicomponent systems. This extended Langmuir equation applies only to ideal solids, and therefore in general fails to describe experimental data. To account for this deficiency, the Ideal Adsorption Solution Theory (lAST) put forward by Myers and Prausnitz is one of the practical approaches, and is presented in some details in Chapter 5. Because of the reasonable success of the IAS, various versions have been proposed, such as the FastlAS theory and the Real Adsorption Solution Theory (RAST), the latter of which accounts for the non-ideality of the adsorbed phase. Application of the RAST is still very limited because of the uncertainty in the calculation of activity coefficients of the adsorbed phase. There are other factors such as the geometrical heterogeneity other than the adsorbed phase nonideality that cause the deviation of the IAS theory from experimental data. This is the area which requires more research. 

The Langmuir equation can also be derived from the statistical thermodynamics, based on the lattice statistics. Readers interested in this approach are referred to the book by Rudzinski and Everett (1992) for more detail. 

In the last chapter, we discussed the description of pure component adsorption equilibrium from the fundamental point of view, for example Langmuir isotherm equation derived from the kinetic approach, and Volmer equation from the Gibbs thermodynamic equation. Practical solids, due to their complex pore and surface structure, rarely conform to the fundamental description, that is very often than not fundamental adsorption isotherm equations such as the classical Langmuir equation do not describe the data well because the basic assumptions made in the Langmuir theory are not readily satisfied. To this end, many semi-empirical approaches have been proposed and the resulting adsorption equations are used with success in describing equilibrium data. This chapter will particularly deal with these approaches. We first present a number of commonly used empirical equations, and will discuss some of these equations in more detail in Chapter 6. 

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