Langmuir equation two-surface

G. Sposito, On the use of the Langmuir equation in the interpretation of adsorption phenomena. II The two-surface Langmuir equation. Soil Sci. Soc. Am. J. 46 1147 (1982). See also I. M. Klotz, Numbers of receptor sites from Scatchard graphs Facts and fantasies. Science 217 1247 (1982). (In the literature of macromolecular chemistry, Fig. 4.2 is known as a Scatchard plot.)... 

Figure 6.4. Fit of tlie two surface Langmuir adsorption isotherm equation to zinc adsorption by two soil horizons. (From Shuman, 1975.)...

Gas A, by itself, adsorbs to a of 0.02 at P = 200 mm Hg, and gas B, by itself, adsorbs tod = 0.02 at P = 20 mm Hg Tisll K in both cases, (a) Calculate the difference between (2a and (2b> the two heats of adsorption. Explain briefly any assumptions or approximations made, ib) Calculate the value for 6 when the solid, at 77 K, is equilibrated with a mixture of A and B such that the final pressures are 200 mm Hg each, (c) Explain whether the answer in b would be raised, lowered, or affected in an unpredictable way if all of the preceding data were the same but the surface was known to be heterogeneous. The local isotherm function can still be assumed to be the Langmuir equation. 

Some early observations on the catalytic oxidation of SO2 to SO3 on platinized asbestos catalysts led to the following observations (1) the rate was proportional to the SO2 pressure and was inversely proportional to the SO3 pressure (2) the apparent activation energy was 30 kcal/mol (3) the heats of adsorption for SO2, SO3, and O2 were 20, 25, and 30 kcal/mol, respectively. By using appropriate Langmuir equations, show that a possible explanation of the rate data is that there are two kinds of surfaces present, 5 and S2, and that the rate-determining step is... 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

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. 

In the Langmuir-Hinshelwood (L-H) mechanism for surface-catalyzed reactions, the reaction takes place between two surface-adsorbed species [4,5], As a substitute for concentration, we use surface coverage, and the rate is expressed in this term. We consider that the elementary reaction in the L-H mechanism is the bimolecular surface reaction expressed by the following equations ... 

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. 

Let us now show that eq 10 leads to the same result as the Langmuir equation for the force between two identical planar surfaces and arbitrary surface conditions. Using eq 10, one obtains the following expression for the force per unit area, between two identical planar surfaces... 

K C or occurs, where the Langmuir constant has different dimensions in these two cases. As to the surface concentration, conversion of the surface concentration (in mol m ) into the dimensionless fractional coverage = / r tmax) requires knowledge of the plateau adsorption or, for that matter, of the molecular cross-section in a compact monolayer,. This quantity is not a priori known, but can be obtained by linearization (sec. II. 1.4a). In fact, linearization is a useful exercise anyway, because it tells us whether the Langmuir equation applies at all. Below we shall mainly use the dimensionless sets 0 lx ] and as being the most general. As only one adsorbing component will be... 

For common adsorbates the equilibrium constants of reactions involving only solution species are available from literature for less common adsorbates they can be determined in separate experiments that do not involve the adsorbent. The equilibrium constants of (hypothetical) surface reactions are the adjustable parameters of the model, and they are determined from the adsorption data by means of appropriate fitting procedure. With simple models (e.g. the model leading to Langmuir equation which has two adjustable parameters) the analytical equations exist for least-square best-fit model parameters as the function of directly measured quantities, but more complicated models require numerical methods to calculate their parameters. 

It is instructive to compare the Langmuir equation, Eq. (8-1), with Eq. (9-4). If the latter is correct, then k in Eq. (8-1) is a function of surface coverage. However, the reason a and E are functions of 9 is that the first two postulates of the Langmuir treatment (see Sec. 8-4) are not satisfied experimentally that is, in real surfaces all sites do not have the same activity, and interactions do exist. 

The catalysts were characterized by using various techniques. X-ray diffraction (XRD) patterns were recorded on a Siemens D 500 diffractometer using CuKa radiation. The specific surface areas of the solids were determined by using the BET method on a Micromeritics ASAP 2000 analyser. Acid and basic sites were quantified from the retention isotherms for two different titrants (cyclohexylamine and phenol, of p/Ta 10.6 and 9.9, and L ,ax 226 and 271.6 nm, respectively) dissolved in cyclohexane. By using the Langmuir equation, the amount of titrant adsorbed in monolayer form, Xm, was obtained as a measure of the concentration of acid and basic sites [11]. Also, acid properties were assessed by temperature-programmed desorption of two probe molecules, that is, pyridine (pKa= 5.25) and cyclohexylamine. The composition of the catalysts was determined by energy dispersive X-ray analysis (EDAX) on a Jeol JSM-5400 instrument equipped with a Link ISI analyser and a Pentafet detector (Oxford). 

The thermodynamics and dynamics of interfacial layers have gained large interest in interfacial research. An accurate description of the thermodynamics of adsorption layers at liquid interfaces is the vital prerequisite for a quantitative understandings of the equilibrium or any non-equilibrium processes going on at the surface of liquids or at the interface between two liquids. The thermodynamic analysis of adsorption layers at liquid/fluid interfaces can provide the equation of state which expresses the surface pressure as the function of surface layer composition, and the adsorption isotherm, which determines the dependence of the adsorption of each dissolved component on their bulk concentrations. From these equations, the surface tension (pressure) isotherm can also be calculated and compared with experimental data. The description of experimental data by the Langmuir adsorption isotherm or the corresponding von Szyszkowski surface tension equation often shows significant deviations. These equations can be derived for a surface layer model where the molecules of the surfactant and the solvent from which the molecules adsorb obey two conditions ... 

For ideal (with respect to the enthalpy) surface layers of a surfactant capable of adsorbing in two states (1 and 2) with different partial molar areas cOj (coi > 2) and different adsorption equilibrium constants, Eqs. (2.26) and (2.27) can be transformed into a generalised von Szyszkowski-Langmuir equation of state [25]... 

As aggregation of the insoluble component occurs only when its surface concentration is sufficiently high, the description of the two components based on Volmer s equation seems to be more appropriate than that based on the Szyszkowski-Langmuir equation. If a first-order phase transition does not occur in the monolayer, i.e. no aggregates are formed, then the simultaneous solution of Volmer s equation (2.159) for the components 1 and 2, and Pethica s equation (2.152) yields the adsorption isotherm for the soluble component 2 (see [156])... 

The prediction of multicomponent equilibria based on the information derived from the analysis of single component adsorption data is an important issue particularly in the domain of liquid chromatography. To solve the general adsorption isotherm, Equation (27.2), Quinones et al. [156] have proposed an extension of the Jovanovic-Freundlich isotherm for each component of the mixture as local adsorption isotherms. They tested the model with experimental data on the system 2-phenylethanol and 3-phenylpropanol mixtures adsorbed on silica. The experimental data was published elsewhere [157]. The local isotherm employed to solve Equation (27.2) includes lateral interactions, which means a step forward with respect to, that is, Langmuir equation. The results obtained account better for competitive data. One drawback of the model concerns the computational time needed to invert Equation (27.2) nevertheless the authors proposed a method to minimize it. The success of this model compared to other resides in that it takes into account the two main sources of nonideal behavior surface heterogeneity and adsorbate-adsorbate interactions. The authors pointed out that there is some degree of thermodynamic inconsistency in this and other models based on similar -assumptions. These inconsistencies could arise from the simplihcations included in their derivation and the main one is related to the monolayer capacity of each component [156]. 

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... 

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