Langmuir adsorption isotherm derivation

We can compare the coverage of mobile surface species predicted by Eq. 11.99 with the coverage calculated for immobile species (Langmuir adsorption isotherm) derived in Section 11.5.3. To do this comparison, we take the low-pressure limit of Eq. 11.79, in which the denominator becomes unity. The ratio of the coverages in these two limiting cases is... 

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 Langmuir adsorption isotherm is easy to derive. Again we assume that the catalyst contains equivalent adsorption sites, and that the adsorbed molecules do not interact. If the adsorbed molecules are in equilibrium with the gas phase, we may write the reaction equation as... 

Derive the Langmuir adsorption isotherm for the molecular adsorption of CO on a metal with equivalent adsorption sites. Do the same for the dissociative adsorption of H2, and, finally, for the case when CO and H2 adsorb together on the same surface. 

Derive Langmuir adsorption isotherm for the following cases ... 

Note the similarity of this expression to that for 9a, derived by the Langmuir adsorption isotherm. (ES)/(E0) plays a role analogous to 04, while S0 plays a role akin to the gas pressure. Although the expression is formally similar, we do not mean to imply that the two types of catalytic reactions proceed by similar molecular steps. 

Electrochemical impedance spectroscopy was used to determine the effect of isomers of 2,5-bis( -pyridyl)-l,3,4-thiadiazole 36 (n 2 or 3) on the corrosion of mild steel in perchloric acid solution <2002MI197>. The inhibition efficiency was structure dependent and the 3-pyridyl gave better inhibition than the 2-pyridyl. X-ray photoelectron spectroscopy helped establish the 3-pyridyl thiadiazoles mode of action toward corrosion. Adsorption of the 3-pyridyl on the mild steel surface in 1M HCIO4 follows the Langmuir adsorption isotherm model and the surface analysis showed corrosion inhibition by the 3-pyridyl derivative is due to the formation of chemisorbed film on the steel surface. 

For the simulation of SMB-separations efficient software packages,based on the Triangle-Theory, are commercially available. The number of columns, the column dimensions, the theoretical number of plates in the columns, the feed concentration, the bi-Langmuir adsorption isotherm parameters and the number of cycles need to be defined by the user. Then the separation is simulated and values for the flow rate ratios, the flow rates, the switching time and the quality of the separation, purity and yield, are calculated. Based on these values an actual separation can be performed. However, some optimization/further development is usually necessary, since the simulations are based on an ideal model and the derived parameters and results therefore can only be taken as indications for the test runs. 

If a molecule dissociates on being adsorbed, the process is referred to as dissociative adsorption. The Langmuir adsorption isotherm for dissociative adsorption, derived ... 

The binding energy in field adsorption can be derived from consideration of the kinetics of field adsorption. Specifically, it can be determined from a temperature dependence of the probability of field adsorption on an adsorption site, or the degree of coverage of field adsorption on a plane. As will be shown, a consideration of the probability of field adsorption based on adsorption time and desorption time leads to an equation equivalent to the Langmuir adsorption isotherm, but specific to the problem of field adsorption.112115 Let us focus on one surface atom. The average time it takes to have an image gas atom field adsorbed on the surface atom, ra, is... 

An expression that is equivalent to the Langmuir adsorption isotherm is readily derived within mass-action kinetics form that we have adopted. Write the adsorption process as... 

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

The Langmuir adsorption isotherm can be derived [134,417] using the statistical thermodynamics techniques discussed in Chapters 8 and 9. The assumptions necessary are basically the same as were used in deriving the Langmuir adsorption isotherm in Section 11.4.1. That is, adsorption is assumed to occur on a fixed array of surface sites there is assumed to be no interaction between adsorbed species the particular sites that are filled are assumed to be random and adsorbed species are immobile, corresponding to a chemisorbed species. 

Theory of Vapor-Particle Partitioning of PAH Compounds in the Atmosphere. Yamasaki et al. (64) treated semivolatile PAHs as inherently vapor-phase materials, existing in the particle phase only because of their tendency to sorb on nonvolatile materials. Under this assumption, the proportion in each phase should vary with the available surface area for sorption and with the ambient temperature. They assumed that sorption followed a Langmuir adsorption isotherm, requiring that there be only a low fractional coverage of the particulate matter with semivolatile material. In this case, competition for sorption sites can be ignored. At equilibrium, the rate of sorption equals the rate of evaporation. They derived this relationship ... 

The Adsorption of Hydrolyzed Al(III). O Melia and Stumm (12) have shown that specific adsorption of hydrolyzed Fe(III) species accounts for the observed coagulation and restabilization of silica dispersions. A model was formulated on the basis of the Langmuir adsorption isotherm and was shown to explain the observations adequately. The authors derive a relationship between the surface area concentration of the dispersed phase S (in meter2/liter and the applied coagulant ion concentration Cte (in M) necessary to reach a certain fraction of surface coverage. The extent of destabilization or of restabilization can be concluded from the amount of surface coverage on the colloidal particle ... 

To illustrate the analogy more clearly, it is necessary to consider the derivation of the Langmuir adsorption isotherm. We can incorporate the above assumptions into an equilibrium expression which equates the rate of adsorption racis to that of desorption rdes of gas molecules of type J. The desorption rate is directly proportional to the fraction of monolayer sites occupied /, and is expressed as... 

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 ... 

In Langmuir model, the maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [378], The basic idea in the BET theory was to assume a Langmuir adsorption for each of the layers (Fig. 9.8). 

The Langmuir adsorption isotherm is based on the characteristic assumptions that (a) only monomolecular adsorption takes place, (b) adsorption is localised and (c) the heat of adsorption is independent of surface coverage. A kinetic derivation follows in which the velocities of adsorption and desorption are equated with each other to give an expression representing adsorption equilibrium. 

A frequently used adsorption model that allows for adsorption in multilayers has been introduced by Brunauer, Emmett and Teller [10] and is known as the BET equation. With the exception of the assumption that the adsorption process terminates at monolayer coverage, these authors have retained all the other assumptions made in deriving the Langmuir adsorption isotherm. Hence all objections to the application of the Langmuir equation apply here, too. 

Derive the values needed to plot the Freundlich and Langmuir adsorption isotherms using the batch adsorption test data... 

Kariapper and Foster derived a simple model considering the effect of several process parameters. The particle deposition rate is again defined as a Langmuir adsorption isotherm, where the measure of the particle cathode interaction k depends on ... 

Fowler, R. M. 1935. A statistical derivation of the Langmuir adsorption isotherm. Proc. Camb. Phil. Soc. 31 260-264. 

The site balance and the Langmuir adsorption isotherm can be used to derive the forward rate expression ... 

The Langmuir adsorption isotherm is represented hy P — K C/(Cm — U), where K is a function of T only, and Cm represents the maximum concentration of adsorbed species that can be accommodated as a monolayer on the surface. Derive the corresponding two-dimensional equation of state. Determine the molar enthalpy and entropy of the adsorbate, using the perfect gas approximation for the gaseous phase. 

In contrast to the appendix, the area A is now written explicitly. Equation [3.4.33] needs no further comment, but the Langmuir expression, [3.4.34] does. In [I.A1.2b] it was written as =-fcTln(l-fl), where is the area of one site. Recall that the Langmuir adsorption Isotherm and equation of state were derived... 

Other justifications are (i) the fact that the Langmuir adsorption isotherm equation can also be derived klnetically, without making the restriction that there are localized sites (see sec. 11.1.5a), (ii) that thermodynamic derivations without this restriction can also be given (see above), and (iii) the experience that lattice theories can often represent properties of liquids satisfactorily. 

As noted in Chapter 1, the most simple and theoretically sound model for drug-receptor interaction is the Langmuir adsorption isotherm. Other models, based on receptor behavior (see Chapter 3), are available. One feature of all these models (with the exception of some instances of the operational model) is that they predict symmetrical curves. A symmetrical curve is one where the half-maximal abscis-sal point (EC50, concentration of x that yields 50% of the maximal value of y) and the inflection point of the curve (where the slope is zero) are the same see Figure 12.17A. However, many experimentally derived dose-response curves are not symmetrical because of biological factors in the system. Thus, there can be curves where the EC50 does... 

Langmuir Adsorption Isotherm Equation The Langmuir adsorption isotherm equation was developed originally to describe gas adsorption onto clean surfaces and can be derived theoretically based on rates of evaporation and condensation (Adamson, 1976). The Langmuir adsorption isotherm equation is... 

The form of the competitive Langmuir adsorption isotherm will not be derived here, but it has been derived by Adamson (1976) for two competing gases on a surface. 

Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tension and surface chemical potential, can be derived from an expression for the surface free energy, F, which corresponds to a given physical model. This derivation helps us obtain (or identify) the self-consistent system of equations, referring to a given model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say, Langmuir adsorption isotherm with Frumkin surface tension isotherm) is incorrect and must be avoided. 

The Langmuir adsorption isotherm was developed by Irving Langmuir in 1916 from kinetic considerations to describe the dependence of the surface fractional coverage of an adsorbed gas on the pressure of the same gas above the adsorbent surface at a constant temperature. The Langmuir isotherm expression was re-derived thermodynamically by Volmer and statistically mechanically by Fowler. In his original treatment, Langmuir made several assumptions for his model ... 

The comparison of the empirical Szyszkowski equation (II. 18) with the Gibbs equation (II.5) indicates that Langmuir adsorption isotherm (11.22) is well suited also for the description of adsorption at the air - surfactant solution interface. It is interesting to point out that at the gas - solid interface, for which eq. (11.22) was originally derived various deviations from Langmuirian behavior are often observed. 

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