Adsorption isotherm Frumkin equation

The above relationships were derived for low electrode coverages by the adsorbed substance, where a linear adsorption isotherm could be used. Higher electrode coverages are connected with a marked change in the surface charge. The two-parallel capacitor model proposed by Frumkin and described by the equation... 

The adsorption isotherm was modeled in order to deduce a numerical value for A. Following the model of Frumkin and Fowler reported in [21], is given by the following set of equations ... 

In Eq. 16, hi is another adsorption constant (independent of surface coverage) and is equal to the product of hi in Eq. 11 and the base of natural logarithm (= 2.718). For systems containing only one surfactant. Pi = Pu = 0, and Eqs. 15 and 16 reduce to the well-known Frumkin equation of state and adsorption isotherm described as... 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

The maximum surface concentration of benzoic acid obtained by extrapolation of the experimental data is rmax = 5.1 X 1014 molecules cm-2. Determine the parameters P and A in the Frumkin equation of adsorption. Calculate the Gibbs energy of adsorption. Compare the results with the Langmuir isotherm. (Sobkowski)... 

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. 

This empirical equation of the adsorption isotherm, giving the relationship between 6 and the pressure, excellently represents many characteristics of chemisorption. Equation (72) is introduced by Frumkin and Slygin (366), who derived it from their electrochemical investigations on hydrogen electrodes. The equation has played an extensive role in the successful theory of ammonia catalysis of Temkin (367) and it has in literature been termed the Temkin equation (368), although Temkin himself and other Russian investigators call it the logarithmic adsorption isotherm. 

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. 

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. 

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. 

The Frumkin equation of state and adsorption isotherm (2.37) - (2.38) involve one extra parameter a. Thus, the Frumkin model can better fit experimental data. The effect of the parameter a for fixed co values is illustrated by Fig. 2.2. 

The equation of state and adsorption isotherm for the Frumkin model (which becomes the Langmuir model for a = 0, cf Chapter 2) are... 

It was shown in Chapter 2 that the theoretical models defined by Eqs. (3.1)-(3.10) can be used also to describe the behaviour of the solutions of ionic surfactant RX in absence and presence of inorganic electrolyte XY. In this case, the Frumkin constant, in addition to the Van der Waals interaction, involves also the inter-ion interaction in the surface layer. Now instead of the concentration c the corresponding adsorption isotherms should be a function of the mean ionic products c = f (Crx xy rx > where f is the average activity coefficient of ions in the solution bulk. An equation accurately representing measured values of f. is the Debye-Hiickel euqation corrected for short-range interactions... 

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 Frumkin equation of state and adsorption isotherm involve an additional interaction parameter a (for a = 0 we obtain the Langmuir model). 

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]. 

Whatever is the mechanism of OHad and Oad formation, the principal point of this adsorption theory is that Pt surface atoms are stable in their positions in the lattice and the integrity of the Pt lattice is retained after an adsorption-desorption cycle in the voltammetry studies. The formation of adsorbed layer is described quantitatively by adsorption isotherms at equilibrium, e.g., Temkin or Frumkin adsorption isotherms and corresponding kinetic equations such as the Elovich equation. Intrinsic to the adsorption theory is the concept of the maximum surface concentration corresponding to monolayer coverage by the ad-particles on all available adsorption sites. " ... 

Conway et using their concept of various structural configurations of adsorbed OH and O, were able to simulate with a computer the linear sweep voltammograms on Pt in 0.5 M H2SO4. They used kinetic equations for the electrochemical adsorption compatible with the Frumkin adsorption isotherms for various forms of adsorbed oxygen. This simulation involved 12 adjustable parameters, a fact which reduces the value of this simulation as evidence for the proposed mechanism. 

This uncertainty makes the kinetic analysis in terms of Frumkin and Temkin adsorption isotherms also uncertain. As long as there is no clear evidence to the contrary, it may be safer to rely on a simple form of a kinetic equation consistent with the Langmuir adsorption isotherm. Therefore, it is usually accepted that in not too concentrated acids the OER occurs on Pt according to... 

In the extraction systems the surfactant is distributed between the phases. Boguslavsky et al. systematically studied the adsorption of tetra-alkylammonium salts in such systems at the water-nitrobenzene interface [11, 76, 90, 91]. The values of the Volta potential at this interface are consistent with the thermodynamic distribution theory. The adsorption isotherms are formally described by the Frumkin equation with the increasing size of tetra-alkylammonium cation, the repulsion between the adsorbate particles increases. It testifies to the fact that the cations of alkylammonium... 

The adsorption isotherms of tetra-alkylammonium salts (beginning with Hept4N ) from benzene at the water interface formally follow the Frumkin equation ... 

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