Langmuir equation derivation

The Langmuir equation (derived in Chapter 4), which is applicable to the chemisorption of H or OH on oxides, has the general form ... 

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

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. 

This difference looks large enough to be diagnostic of the state of the adsorbed film. However, to be consistent with the kinetic derivation of the Langmuir equation, it was necessary to suppose that the site acted as a potential box and, furthermore, that a weak adsorption bond of ifi corresponding to 1 /tq was present. With these provisions we obtain... 

Although the preceding derivation is the easier to follow, the BET equation also may be derived from statistical mechanics by a procedure similar to that described in the case of the Langmuir equation [41,42]. 

The rate of physical adsorption may be determined by the gas kinetic surface collision frequency as modified by the variation of sticking probability with surface coverage—as in the kinetic derivation of the Langmuir equation (Section XVII-3A)—and should then be very large unless the gas pressure is small. Alternatively, the rate may be governed by boundary layer diffusion, a slower process in general. Such aspects are mentioned in Ref. 146. 

Derive the general form of the Langmuir equation (Eq. XVII-11). 

Since in chemisorption systems it is reasonable to suppose that the strong adsorbent-adsorbate interaction is associated with specific adsorption sites, a situation that may arise is that the adsorbate molecule occupies or blocks the occupancy of a second adjacent site. This means that each molecule effectively requires two adjacent sites. An analysis [106] suggests that in terms of the kinetic derivation of the Langmuir equation, the rate of adsorption should now be... 

Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional iTuilti-component Langmuir equation, obtained via a kinetic derivation, is... 

Appendix 1.2B Step-by-Step Derivation of the Hill-Langmuir Equation... 

Here, KA and KB are the dissociation equilibrium constants for the binding of agonist and antagonist, respectively. This is the Gaddum equation, named after J. H. Gaddum, who was the first to derive it in the context of competitive antagonism. Note that if [B] is set to zero, we have the Hill-Langmuir equation (Section 1.2.1). 

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 number of gas molecules can be measured either directly with a balance (gravimetric method) or calculated from the pressure difference of the gas in a fixed volume upon adsorption (manometric method). The most frequently apphed method to derive the monolayer capacity is a method developed by Brunauer, Emmett, and Teller (BET) [1], Starting from the Langmuir equation (monolayer adsorption) they developed a multilayer adsorption model that allows the calculation of the specific surface area of a sohd. The BET equation is typically expressed in its linear form as... 

This is the result of an action of assumptions that were set at the derivation of BET (as well as Langmuir) equations eneigetically homogeneous surfaces and the absence of lateral interactions of G/G type (see Figure 9.14). But, the surfaces of real PSs are usually energetically heterogeneous, the... 

The Langmuir equation is derived here from application of the mass law, in a similar way as the surface complex formation equilibria were derived in Chapter 2. In principle at a constant pH there is no difference between a Langmuir constant and a surface complex formation constant. 

It is necessary to remind ourselves, that the adsorption of humic acids or fulvic acids correspond to the adsorption of a mixture of adsorbates. Adsorption equations derived for the adsorption of a single adsorbate (Langmuir, Frumkin or Gibbs Equation) cannot be used for mechanistic interpretation of the data even if these data can be fitted to such equations (Tomaic and Zutic, 1988). 

The provenance of expressions like that in Equation 3 has never been shown to be uniquely an adsorption mechanism. On the contrary, it is possible to derive special cases of Equation 3, such as the classical Langmuir equation... 

The Langmuir equation (Eq. 5.1), derived originally to describe the adsorption of gases on solids, assumes that the adsorbed entity is attached to the surface at specific, homogeneous, localized sites, forming a monolayer. It is also assumed that the heat of adsorption is constant over the entire monolayer, that there is no lateral interaction between adsorbed species, that equilibrium is reached, and that the energy of adsorption is independent of temperature ... 

In combination with the Langmuir equation, one can derive the following relation between Ns and p ... 

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). 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

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 ... 

There are several theoretical derivations of adsorption isotherms. The simple Langmuir equation [Eq. (4.2), named after the American chemist and 1932 Nobel laureate Irving Langmuir], describes the formation of a monolayer on a surface [81]. This corresponds to the type I isotherm shown in Figure 4.15. Here Vis the amount adsorbed, Vm is the amount adsorbed in one monolayer, p is the pressure and b is the adsorption coefficient, which depends exponentially on the heat of adsorption. 

This is the famous Saha-Langmuir equation. In it, g+/g0 is the ratio of the statistical weights of the ionic and atomic states, is the work function of the surface, / is the first ionization potential of the element in question, k is the Boltzmann constant, and T is the absolute temperature. Note that gjg0 is close to 1 for electronically complex elements for simpler elements it can take on a variety of values depending on how many electronic states can be populated in the two species for alkali atoms, for example, it is often Vi. Attainment of thermodynamic equilibrium was assumed in the derivation of this equation, and it is applicable only to well-defined surfaces. 

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