Langmuir derivation

I. Langmuir Derived the concept of monolayer adsorption, formed on energetically homogeneous solid surfaces.Was awarded the Nobel Prize in chemistry in 1932 1918... 

In the Langmuir derivation the adsorbed molecules are allowed to interact with the adsorbent but not with each other The adsorbed layer is assumed to be ideal. This necessarily limits adsorption to a monolayer. Once the surface is covered with adsorbed molecules, it has no further influence on the system. The assumption that adsorption is limited to monolayer formation was explicitly made in writing Equations (72) and (73) for the saturation value of the ordinate. Ii is an experimental fact, however, that adsorption frequently proceeds to an extent that exceeds the monolayer capacity of the surface for any plausible molecular orientation at the surface. That is, if monolayer coverage is postulated, the apparent area per molecule is only a small fraction of any likely projected area of the actual molecules. In this case the assumption that adsorption is limited to the monolayer fails to apply. A model based on multilayer adsorption is indicated in this situation. This is easier to handle in the case of gas adsorption, so we defer until Chapter 9 a discussion of multilayer adsorption. 

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. 

Langmuir derived a different adsorption isotherm by assuming that adsorption is terminated upon completion of a monomolecular adsorbed gas layer [43]. He did this by asserting that any gas molecule that strikes an adsorbed atom must reflect from the surface. All the other assumptions (i.e., homogeneous surface and noninteracting adsorbed species) used to obtain Eq. 3.85 were also maintained. If oq is the surface coverage of a completely covered surface, the concentration of surface sites available for adsorption, after adsorbing a molecules, is ao — a. Of the total flux F incident on the surface, a fraction [(a/incident flux will be available for adsorption. Equation 2.84 should thus be modified as... 

Vaporization in a Foreign Gas In addition to the case of vaporization in vacuum, Langmuir derived two other equations for the rate of vaporization in a foreign gas [23]. In a foreign gas environment, the vaporization rate is limited by the diffusion of molecules from the near-surface layer. The thickness of this layer is approximately equal to the mean free-path length, and the molecules in the layer are in their equilibrium concentration. These assumptions are validated by methods of statistical mechanics. Invoking the Pick s first law for one-dimensional diffusion and the Clapeyron-Mendeleev equation, the molecular flux is... 

Except in the rare cases where covalent bonds are made, adsorption is a reversible process, and equilibrium is established according to the mass-action law. In 1918 Langmuir derived the following equation from this law to permit a... 

Dachs, J. and Bayona, J.M. (1997). Langmuir-derived model for diffusion- and reaction-limited adsorption of organic compounds on fractal aggregates. Environ. Sci. Technol., 31, 2754-2760. 

The American scientist I. Langmuir derived this equation based on certain assumptions. More important of these assumptions are... 

Langmuir derived his famous adsorption theory based on the assumption that the adsorption rate is proportional to the concentration in the solution and the fraction of free non-covered area of the adsorbent. The desorption rate, on the other hand, is considered to be proportional to the fraction of surface covered by adsorbate molecules. At equilibrium, the adsorption and desorption rates are equal. The Langmuir theory is derived based on many simplifying assumptions ... 

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 ... 

There are three forms of the Langmuir-Szyszkowski equation, Eq. III-57, Eq. Ill-107, and a third form that expresses ir as a function of F. (n) Derive Eq. III-57 from Eq. Ill-107 and (b) derive the third form. 

Langmuir also gave needed emphasis to the importance of employing pure substances rather than the various natural oils previously used. He thus found that the limiting area (at the Pockels point) was the same for palmitic, stearic, and cerotic acids, namely, 21 per molecule. (For convenience to the reader, the common names associated with the various hydrocarbon derivatives most frequently mentioned in this chapter are given in Table IV-1.)... 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Derive Eq. XI-IS, assuming a Langmuir adsorption process described in Eq. XI-2, where ka and kd are the adsorption and desorption rate constants. Treat the solution... 

The derivation that follows is essentially that given by Langmuir [9] in 1918, in which one writes separately the rates of evaporation and of condensation. The surface is assumed to consist of a certain number of sites S of which S are occupied and Sq = S - S arc free. The rate of evaporation is taken to be proportional to 5, or equal tokiSi, and the rate of condensation proportional to the bare surface So and to the gas pressure, or equal to k PSo. At equilibrium. 

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... 

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. 

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... 

Derive the probable rate law for the reaction CO + j02 = CO2 as catalyzed by a metal surface assuming (a) an Eley-Rideal mechanism and (b) a Langmuir-Hinshelwood mechanism. 

Jones R, Tredgold R H, Hoorfar A, Allen R A and Hodge P 1985 Crystal-formation and growth in Langmuir-Blodgett multilayers of azobenzene derivatives—optical and structural studies Thin Solid Films 134 57-66... 

Tredgold R H, Allen R A and Hodge P 1987 X-ray-diffraction and optical studies of Langmuir-Blodgett films formed from azobenzene derivatives Thin Solid Films 155 343-52... 

Ogawa K, Yonehara FI, Sho]i T, Kinoshita S, Maekawa E, Nakahara FI and Fukuda K 1989 Inplane anisotropy in Langmuir-Blodgett films of a oopper phthalooyanine derivative Thin Solid Films 178 439-43... 

Langmuir referred to the possibility that the evaporation-condensation mechanism could also apply to second and higher molecular layers, but the equation he derived for the isotherm was complex and has been little used. By adopting the Langmuir mechanism but introducing a number of simplifying assumptions Brunauer, Emmett and Teller in 1938 were able to arrive at their well known equation for multilayer adsorption, which has enjoyed widespread use ever since. 

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