Solid surfaces 132 Langmuir isotherm

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 ... 

Figure 5.19 shows an idealized form of the adsorption isotherm for physisorption on a nonporous or macroporous solid. At low pressures the surface is only partially occupied by the gas, until at higher pressures (point B on the curve) the monolayer is filled and the isotherm reaches a plateau. This part of the isotherm, from zero pressures to the point B, is equivalent to the Langmuir isotherm. At higher pressures a second layer starts to form, followed by unrestricted multilayer formation, which is in fact equivalent to condensation, i.e. formation of a liquid layer. In the jargon of physisorption (approved by lUPAC) this is a Type II adsorption isotherm. If a system contains predominantly micropores, i.e. a zeolite or an ultrahigh surface area carbon (>1000 m g ), multilayer formation is limited by the size of the pores. 

The Langmuir Isotherm. The most simple assumption in adsorption is that the adsorption sites, S, on the surface of a solid (adsorbent) become occupied by an adsorbate from the solution, A. Implying a 1 1 stoichiometry... 

The adsorption of a number of organic pollutants on various solid surfaces was found to fit the Langmuir-model isotherm [139,143-145]. 

A typical N2 adsorption measurement versus relative pressure over a solid that has both micropores and mesopores first involves essentially a mono-layer coverage of the surface up to point B shown in isotherm IV (lUPAC classification) in Figure 13.1. Up to and near point B the isotherm is similar to a Langmuir isotherm for which equilibrium is established between molecules adsorbing from the gas phase onto the bare surface and molecules desorbing from the adsorbed layer. The volume of adsorbed N2 that covers a monolayer volume, hence the surface area of N2 can then be determined from the slope of the linearized Langmuir plot when P/V is plotted against P ... 

In a simple solid, a Langmuir adsorption isotherm (Moore 1950) can be used to obtain the surface area assuming monolayer coverage (where the adsorption of gas molecules is measured at a temperature above its condensation into a liquid... 

If 0 is in the middle range, 0.2 < 0 < 0.8, say, the appropriate isotherm could still be that of Langmuir. This would be likely to be the case for the academic systems of liquid electrodes (mercury and occasionally, liquid gallium). On solid surfaces, the effects of heterogeneity will apply and the appropriate isotherm will be that of Temkin. 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

The Langmuir isotherm is used to describe single-layer adsorption based on the concept that a solid surface possesses a finite number of sorption sites. When these active sites are filled, the site will no longer sorb solute from the solution ... 

Solid surfaces are usually heterogeneous therefore, since adsorption at the more active sites is favoured, heats of both monolayer physical adsorption and chemisorption might, in this respect, be expected to become significantly less exothermic as the surface coverage increases, as, for example, shown at low pressures in Figures 5.12a and 5.12b. This, in turn would cause the initial slope of an adsorption isotherm to be steeper than that predicted according to the Langmuir equation or the BET equation. 

There is an interesting parallel between substrate binding and adsorption. Since each enzyme molecule has one active site, and since these active sites all have the same structure, we can think of enzyme molecules in solution as a surface with many equivalent adive sites. In this case, k2 in the Michaelis-Menten kinetics (Eq. 5.1 see Chapter 2 for a detailed discussion) represents the rate of adsorption, k x the rate of desorption, and k2 the rate of the surface readion followed by fast product desorption. Moreover, this system fits the assumptions of the Langmuir isotherm (all sites identical, one molecule per site, no lateral interadions) even better than the adive sites on some real solid catalysts ... 

In general, the adsorption of ionic surfactants follows the Langmuir isotherm, as discussed in Section 4.1. The adsorption of the surfactants onto the solid surfaces is dependent on the orientation and the packing efficiency of the solid surfaces. The onset of the adsorption plateau may occur at the critical micelle concentration (c.m.c.) of the surfactant in water, as shown in Figure 4.28. If the adsorption isotherm... 

The reversible Type I isotherm (Type I isotherms are sometimes referred to as Langmuir isotherms, but this nomenclature is not recommended) is concave to the p/pa axis and na approaches a limiting value as p/p° — 1. Type I isotherms are given by microporous solids having relatively small external surfaces (e.g. activated carbons, molecular sieve zeolites and certain porous oxides), the limiting uptake being governed by the accessible microporc volume rather than by the internal surface area. 

When molecules M are physisorbed onto some solid surface A (even if irregular), at equilibrium the adsorption and desorption rates are equal, and the Langmuir adsorption isotherm of 1916 is 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) ... 

In some cases, adsorption of analyte can be followed by a chemical reaction. The Langmuir-Hinshelwood (LH) and power-law models have been used successfully in describing the kinetics of a broad range of gas-solid reaction systems [105,106]. The LH model, developed to describe interactions between dissimilar adsorbates in the context of heterogeneous catalysis [107], assumes that gas adsorption follows a Langmuir isotherm and that the adsorbates are sufficiently mobile so that they equilibrate with one another on the surface on a time scale that is rapid compared to desorpticm. The power-law model assumes a Fre-undlich adsorption isotherm. Bodi models assume that the surface reaction is first-order with respect to the reactant gas, and that surface coverage asymptotically approaches a mmiolayer widi increasing gas concentration. 

A Ithough the adsorption of polymers onto solid surfaces has been thor-oughly studied (I), relatively few studies can be found in the literature on the adsorption of proteins onto polymer surfaces. In 1905, Landsteiner and Uhliz (2) discussed the interaction of serum proteins with synthetic surfaces. Blitz and Steiner (3) showed that albumin adsorption onto solid surfaces increased with increasing albumin concentration and that adsorption was nearly irreversible. Hitchcock reported (4) that adsorption of egg albumin onto collodion membranes followed a Langmuir isotherm with maximum adsorption occurring near the isoelectric point. Later, Kemp and Rideal (5) reported that protein adsorption onto solids conforms with Langmuir adsorption. 

Figure 13.29. Schematic sorption isotherms of a metal ion (Me) on an oxide (XO ) at constant pH (a) adsorption only (H) adsorption and surface precipitation via ideal solid solution (c) adsorption and heterogeneous nucleation in the absence of a free energy nucleation barrier (AG 0) adsorption and heterogeneous nucleation of a metastable precursor (e) same as in (3) but with transformation of the precursor into the stable phase. The arrows show the isotherm evolution for continual addition of dissolved Me. The initial isotherm with the slope of 1 (in the double logarithmic plot) corresponds to a Langmuir isotherm (surface complex formation equilibrium). [Me]s , = solubility concentration of Me for the stable metal oxide [Me]p = solubility concentration of Me for a metastable precursor (e.g., a hydrated Me oxide phase). (From Van Cappellen, 1991.)...

For dilute additive solutions, consumption can easily become limited by diffusion of the adsorbate to the interface. A particularly tractable situation occurs at the dilute limit of the Langmuir isotherm where C(- is proportional to the surface coverage, and in the limit of diffusion controlled adsorption CsoHd is directly proportional to the additive flux and thus the bulk electrolyte concentration. Such transport limited incorporation was reported in some radiotracer studies of thiourea incorporation in nickel and copper plating in the 1950-1960s [1-4, 17, 130, 131, 141-146], Consistently, a common observation was that the additive concentration in the solid was proportional to the additive concentration at the interface and inversely proportional to metal deposition rate, i, [1-4, 130, 131, 141, 142] such that ... 

Originally derived for the study of gas-solid equilibria, the T6th isotherm [64] accounts for adsorption on a heterogeneous surface, with no adsorbate-adsorbate interactions. It has three parameters. The heterogeneous surface has a unimodal adsorption energy distribution with a width related to the value of the parameter t. Like the Langmuir isotherm, it can be extended to the case of liquid-solid equilibrium. Its equation is... 

In the theories of gas adsorption the applicability of Freundlich equation was long ago associated with the energetic heterogeneity of the adsorption sites on the actual solid surfaces. It was also known, that Freundlich equation is a simplified form of a more general isotherm equation which is now commonly called Langmuir-Freundlich isotherm. Thus, in 1980 Sposito [13] suggested that it was high time it were used also in the case of ion adsorption at water/oxide interfaces. 

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