Adsorption isotherm: Frumkin Langmuir

The basic assumption of the Langmuir adsorption isotherm is that the adsorbed molecules do not interact. This condition is not always fulfilled for adsorption, particularly on electrodes. The Frumkin adsorption isotherm includes interaction between molecules in the adsorption film,... 

Figure 4.6 Dependence of the surface coverage on the bulk concentration of the quinone (where and A denote the areas under the anodic and cathodic peaks, respectively) and hydroquinone (where denotes both anodic and cathodic data) forms of 1,2,4-AQASH. The supporting electrolyte is 1.0 M HCIO4. The dashed lines represent the best fits to the Frumkin adsorption isotherm where error bars are not shown, the errors determined from at least three independently formed monolayers are comparable to the sizes of the symbols. Reprinted with permission from R.J. Foster, T.E. Keyes, M. Farrell and D. O Hanlon, Langmuir, 16, 9871 (2000). Copyright (2000) American Chemical Society...

In the special case of Langmuir isotherm we have P = 0, and then =1.) The Bntler eqnation is nsed by many authors as a starting point for development of thermodynamic adsorption models. It shonld be kept in mind that the specific form of the expressions for n, and which are to be snbstituted in Equation 5.16, is not arbitrary, but must correspond to the same thermodynamic model (to the same expression for F,— in our case Equation 5.11). At last, snbstitnting Equation 5.16 into Equation 5.9 we derive the Frumkin adsorption isotherm in Table 5.2, where K is defined by Equation 5.3. 

This is known as the Frumkin adsorption isotherm. Obviously, for / = 0 it reduces to the Langmuir form. If y is negative, va > vd, and the cost of adsorption increases faster than desorption and relative to the case of constant Et, the system is repulsive. For y > 0, on the other hand, the system is cohesive. ... 

Adsorption layers of the same kind as at fluid interfaces are also formed at low-energy solid -water surfaces, as it was established on PE, polystyrene, paraffin, carbon black, and other related materials. The classical Langmuir or Frumkin adsorption isotherm is often applicable to describe this behaviour. Studies on surfactant adsorption at various solid surfaces have been summarised in a great number of reviews [2, 7, 8, 54, 98, 101, 111, 121, 126, 141, 144, 145, 177, 186, 190, 194-198]. The adsorption at the solid/liquid interfaces is governed by a number of factors ... 

For a modelling of adsorption processes the well-known integro-differential equation (4.1) derived by Ward and Tordai [3] is used. It is the most general relationship between the dynamic adsorption r(t) and the subsurface concentration e(0,t) for fresh non-deformed surfaces and is valid for kinetic-controlled, pure diffusion-controlled and mixed adsorption mechanisms. For a diffusion-controlled adsorption mechanism Eq. (4.1) predicts different F dependencies on t for different types of isotherms. For example, the Frumkin adsorption isotherm predicts a slower initial rate of surface tension decrease than the Langmuir isotherm does. In section 4.2.2. it was shown that reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. In this paragraph we will give experimental evidence, that changes in the molar area of adsorbed molecules can cause sueh effectively faster adsorption processes. 

The broadening and narrowing of the surface redox waves are linked to repulsive and attractive interactions. The numbers indicated for each curve are related to the interaction parameter of the Frumkin adsorption isotherm (g) g = 0 for the absence of interaction (Langmuir isotherm), g < 0 and g > 0 for the repulsive and attractive interactions, respectively. 

Surfactant keeps emulsion droplets and latex particles colloidally stable against coalescence/aggregation. The surfactant plays another important role in emulsion polymerisation besides stabilisation. Surfactant is critically involved in the nucleation mechanism (i.e., how the particles are formed) of the polymer latex particles (418,419). The amount of surfactant used is critical in controlling the latex particle size distribution. As surfactant is added to an emulsion, some remains dissolved in the aqueous phase, and some adsorbs onto the surface of the emulsion droplets according to an adsorption isotherm (e.g., Langmuir, Freundhch, or Frumkin adsorption isotherms) (173). 

The preceding equations were written assuming the Langmuir adsorption isotherm for both species. In the case of the Frumkin adsorption isotherm with negative interaction terms the situation is more complex and multi-steady-state curves can be obtained. Such a situation was discussed by Berthier et al. [222]. 

This yields in the current simple case a uniform profile in the bulk, < (x > 0) = and recovers the Frumkin adsorption isotherm (or the Langmuir one, if )8 = 0) [15] at the interface. 

The copper will start depositing at around = 0.4V,but since the bisulfate is in its disordered phase, there should be no cooperative effects due to the copper-copper attraction on the surface, and therefore, to a first approximation we may assume that the electrodeposition of copper follows a Langmuir-like ( or Frumkin-Langmuir [115]) adsorption isotherm... 

Sinee replaeement of an adsorbed solvent moleeule by an adsorbate molecule generally results in a ehange of the eomposition of the first layer of the eleetrolyte solution being in contact with the electrode, the different values of the dieleetrie properties of the solvent and adsorbate moleeules as well as their different size result in a change of the value of Cdl- Evaluation of the changes of Cdl as a function of the concentration of the adsorbate in the solution phase results in adsorption isotherms. Various isotherms (Frumkin, Langmuir etc.) have been used in the further evaluation [75Damj. Beeause of interferenees of the kineties of the adsorption proeess in particular by slow transport and slow adsorption tensammetry... 

FIG. 3 Adsorption isotherm of ONS at the polarized water-DCE interface (square) experimental, (dotted) Langmuir isotherm, and (solid) Frumkin isotherm. (From Ref. 55, cop5right American Chemical Society.)... 

CI2 evolution reaction, 38 56 electrochemical desorption, 38 53-54 electrode kinetics, 38 55-56 factors that determine, 38 55 ketone reduction, 38 56-57 Langmuir adsorption isotherm, 38 52 recombination desorption, 38 53 surface reaction-order factor, 38 52 Temkin and Frumkin isotherm, 38 53 real-area factor, 38 57-58 regular heterogeneous catalysis, 38 10-16 anodic oxidation of ammonia, 38 13 binding energy quantification, 38 15-16 Haber-Bosch atrunonia synthesis, 38 12-13... 

With this in mind, some important adsorption isotherms were introduced, and we found that each of them describes important characteristics of the adsorption process (Table 6.10). Thus, the Langmuir isotherm considers the basic step in the adsorption process the Frumkin isotherm was one of the first isotherms involving lateral interactions the Temkin is a surface heterogeneity isotherm and the Flory-Huggins-type isotherms include the substitution step of replacing adsorbed water molecules by the adsorbed entities (Fig. 6.98). 

This equation is sometimes called the Frumkin-Fowler-Guggenheim (FFG) isotherm [374— 376], For j3 = nEP/RT < 4 lateral interactions cause a steeper increase of the adsorption isotherm in the intermediate pressure range. Characteristic of all Langmuir isotherms is a saturation at high partial pressures P/Po —> 1. 

The amount of adsorbed species is given in terms of the coverage or fraction of the electrode surface covered by the adsorbate, 0. The relationship between the coverage and the concentration in solution of a species under equilibrium conditions is called an adsorption isotherm. The ones most employed in electrochemistry are the Langmuir and Frumkin isotherms, deduced from statistical consideration and assuming absence or presence of interactions between the adsorbed molecules in the former and the latter, respectively. 

In the limit of a - 0, the ideal Langmuir adsorption isotherm is obtained. See - Frumkin isotherm, and for the role of surface heterogeneity - Temkin isotherm. Refs. [i] Horanyi G (2002) Specific adsorption. State of art Present knowledge and understanding. In Bard A], Stratmann M, Gileadi M, Urbakh M (eds) Thermodynamics and electrified interfaces. Encyclopedia of electrochemistry, vol. I. Wiley-VCH Verlag, Weinheim, pp 349-382 [ii] Calvo EJ (1986) Fundamentals. The basics of electrode reactions. In Bamford CH, Compton RG (eds) Comprehensive chemical kinetics, vol. 26. Elsevier, Amsterdam, pp 1-78... 

When reactants or intermediates are adsorbed, the rate of the reaction may no longer be related to the concentration by a simple law. This situation is best understood where a reactant is nonspecifically adsorbed in the outer -> Helmholtz plane. The effect of such adsorption on the electrode kinetics is usually termed the -> Frumkin effect. Physical and chemical adsorption on the electrode surface is usually described by means of an -> adsorption isotherm and kinetic equations compatible with various isotherms such as the - Langmuir, -> Temkin, -> Frumkin isotherms are known. 

The adsorption of PNP to a wide variety of interfaces has been studied by SHG. The adsorption isotherm at the dodecane/water interface is shown in Figure 1.4. The adsorption isotherms typically used to fit the surface tension and SHG data at the fiquid/liquid interfaces are in terms of the interfacial coverage 0 and the bulk phase concentration c, the Langmuir (13) and the Frumkin isotherms (14)... 

The adsorption of either ions or neutral molecules on the electrode surface depends on qn, i.e., on the apphed electric potential. Correspondingly, the electric field at the electrochemical interface is an additional free-energy contribution that either favors or restricts the adsorption of species on the electrode from the ionic conducting phase. A variety of adsorption isotherms has been proposed to account for the behavior of different electrochemical systems. Among them are the Langmuir, Frumkin, and Temkin isotherms [2]. Frumkin and Temkin isotherms, at variance with the Langmuir one, include effects such as adsorbate—adsorbate or adsorbate—surface interactions. 

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 parameters F, and K are the same as in Tables 5.2 through 5.4. Setting Q = 0 (assuming equilibrium surface-subsurface), from each expression in Table 5.5 we deduce the respective equilibrium adsorption isotherm in Table 5.2. In addition, for P = 0 the expressions for Q related to the Frumkin and van der Waals model reduce, respectively, to the expressions for Q in the Langmuir and Volmer models. For Fj F both the Frumkin and Langmuir expressions in Table 5.5 reduce to the Henry expression. 

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