Frumkin model

These two equations present the extension of the Frumkin model to the adsorption of one-surfactant system with two orientational states at the interface. The model equations now contain four free parameters, including cou co2, and b. The equations are highly nonUnear, and regression used in the analysis of surface tension data involves special combinations of Eqs. 23 and 24, which produces a special model fimction used in the least-square minimization with measured surface tension data. Since the model function also contains surface... 

It is noted that the investigation of a mixed adsorption layer of CioEs and TPeAB (tetrapentyl ammoniiun bromide) [35] shows evidence for attractive forces / > 0), which suggests that the presence of the ionic surfactant can prevent aggregation in the extended S-L adsorption layer. Therefore, the main question of interest concerns how the Frumkin model and the aggregation model are related. One can find from Eq. 29 that the size of the elementary adsorption cell increases with the aggregation munber resulting in a reduction in the munber of cells. Negative has the same effect of de-... 

Finally, Gugala etal. [132] have analyzed the effect of the concentration of the base electrolyte on adsorption of 1,1,3,3-tetramethyl-2-thiourea at mercury electrode. The Flory-Huggins and the Frumkin models were utilized for the quantitative analysis of the adsorption data. 

Recently, Japaridze etal. [170] have investigated adsorption of some aromatic compounds, including naphthalene, naphthonitrile, naphthylamine, anthracene, and phenathrene at the mercury electrode I ethylene glycol solution interface. The analysis of the differential capacity data obtained at the HMDE has revealed that adsorption of the above-mentioned compounds obeys the Frumkin model, with attractive interactions of the particles in the adsorption layer. The results for ethylene glycol were compared with those for other nonaqueous solvents and their role in determining the adsorption mode was discussed. 

Few outer-sphere electrode reactions have precursor-state concentrations that are measurable [21] so that it is usual to estimate wp and ws from double-layer models. The simplest, and by far the most commonly used, treatment is the Frumkin model embodied in eqns. (8) and (8a) whereby, as noted in Sect. 2.2, the sole contributor to wp and ws is presumed to be electrostatic work associated with transporting the reactant from the bulk solution to the o.H.p. at an average potential Gouy-Chapman (GC) theory [58],... 

All these findings may point to limitations of the classical Frumkin model for correction of the double-layer influence on electrode kinetics in nonaqueous solvents, although it works well in aqueous solution. In the present author s opinion these rather surprising results may follow from some kind of compensation effects. For instance, ion-pair formation in these solutions by decreasing the effective charge of the reactant could reduce the double-layer effect. 

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... 

The temperature of the solutions in all the experiments presented was almost the same, 20 - 25°C. The correspondence between the data reported by different authors is remarkably well. For the processing of experimental data the Frumkin model was employed and the resulting model parameters are presented in Table 3.1. Here the value e characterises the deviation between the theoretical isotherm and the experimental values. Figures 3.2-3.4... 

Therefore, as the intermolecular Van der Waals interaction in the monolayer of normal alcohols becomes larger with increasing number of methylene groups, different theoretical models should be applied to describe the adsorption the Langmuir model for n < 3, the Frumkin model for the intermediate n values, and aggregation and cluster models for n - > 10. 

The data in Table 3.3 obtained from the Frumkin model parameters for diols show, that the molar area of diols is roughly two times larger than that of the normal alcohols. This means that the two polar OH groups are localised in the surface layer. 

It is seen from Fig. 3.9 that for lower acids (Cj-Cg) the corresponding theoretical curves are indistinguishable, i.e. hoth theoretical models provide good description of the adsorption behaviour. However, for decanoic and lauric acids the aggregation model leads to essentially smaller deviation from the experimental data (by a factor of 2 for lauric acid) then the Frumkin model. Similarly to the normal alcohol series, the increase of the Frumkin constant a with n -takes place, and for even homologues the value of a is essentially higher, cf. Fig. 3.9. 

The surface tension isotherms for alkyl dimethyl phosphine oxides (C DMPO) in the interval from Cj to Ci6 at 25 °C are shown in Fig. 3.11. It should be noted that the experimental data reported in [36] for Cj, Ciq, Cp and C[4 are in a good agreement with the data presented in [37, 38], and therefore the results for these experimental data are also shown in Fig. 3.11. The parameters of the Frumkin and reorientation models are summarised in Tables 3.6 and 3.7. Both isotherms agree well with the experimental data. Small differences between the calculated isotherms exist only for nc > 13, while for lower nc the curves for the two models perfectly coincide. It follows then that neither of the two models can be preferred if one takes into account only the agreement between the experimental and theoretical data. However, the negative values of the Frumkin constant a for lower homologues, and the unusual shape of this dependence on nc (cf Fig. 3.12) indicate that for the Frumkin model the coincidence with the experiment is only formal. 

Fig. 3.19. Dependence of AG° (O) and standard free energy of adsorption AG° ( ) and AG jIA) on nc for calculated from the Frumkin model.

It is seen that for the Frumkin model, the two methods give somewhat different values of the standard free energy of adsorption, while for the reorientation model the values obtained by these two methods are equal. This fact can be easily explained equating the AG values given by Eqs. (3.11) and (3.16) to each other, one obtains ... 

A satisfactory agreement with the experiment can also be formally achieved in the framework of the Frumkin model. This, however, results in physically unrealistic values of intermolecular interaction constant for CnEOj a = -(4 6) for the water/air interface, and a = -10.8 for the water/hexane interface as shown in [57]. A negative value of the constant a corresponds to a repulsion between adsorbed molecules, which is characteristic to solutions of ionic surfactants, where the parameter a compensates the Coulomb interaction. In these cases, however, the value of this parameter is usually much lower than that estimated for CnEOg. We therefore conclude that for the non-ionic surfactant, the Frumkin constant a should be regarded to as a pure fitting parameter, which has no physical meaning. 

We compare now the adsorption behaviour of oxyethylated alcohols (with C,oEOg as an example) at the water/air and water/hexane interfaces, with reference to the data reported in [57]. The experimental and theoretic isotherms at the water/hexane interface are shown in Fig. 3.33. It was mentioned above that the experimental data agree satisfactorily with the Frumkin model for a physically unrealistic value of a = -10.8. Comparing the reorientation model parameters for CiqEOj at the two interfaces (cf. Fig. 3.33 and Table 3.13), one can see that the molar areas are almost the same, while the value of a for the water/hexane interface is 2.5 times higher than that for the water/air interface. Thus the adsorption activity of the oxyethylene groups at the water/hexane interface is significantly higher than that at the water/air interface. 

Fig. 3.41. Dependence of the equilibrium surface tension on c in the solutions of sodium salts of fatty acids, data reported in ( ), [65] and (<>), [33] at 25 C and sodium ions concentration 0.1 M, theoretical curves calculated from the Frumkin model using the model parameters listed in Table 3.17.

Fig. 3.46. Dependence of surface tension for C S04Na solutions on the mean ionic activity c, numbers correspond to the carbon atoms number, without salt ( ) - [68] (O) - [67] ( ) - [69] ( ) - [75] and with additions of inorganic salt (O) - 0.03 M NaCl [70] (A) - 0.1 M NaCI and (A) - 0.5 M NaCl [68], theoretical curves are calculated from the Frumkin model with the parameters listed in Table 3.18.

Fig. 3.52. Dependence of surface tension for C TAB solutions as a function of the mean ionic activity c ( .0,A, 0) at 20°C [80] ( ) at 20 C [81] (A) at 25 >C [82] ( ) at 30°C [82], numbers denote the number of carbon atoms, the theoretical curves were calculated from the Frumkin model using the parameters listed in Table 3.21.

It is seen that the calculated deviation for the reorientation model is two times lower than for the Frumkin model. The dependencies of the isotherm parameters of C TAB on n are similar to those obtained for other surfactants the Frumkin constant a increases with n (cf Table 3.21), the minimum area of the surface active ion C0 2 is almost independent of n, and the molar area in the unfolded state [Pg.246]

C12ACI (A without and A with NaCl to give total Cci=20 mmol/I) Cncr5 mmol/1, at 20 °C [33], the theoretical curves were calculated from the Frumkin model using the parameters listed in Table 3.23. 

Table 3.23. The parameters ofthe Frumkin model, Eqs. (3.1), (3.2), calculated forCioACI.

It should be noted first that the Frumkin model is the most general one with respect to its application to surfactants of different nature. In spite of the fact that, e.g., for oxyethylated nonionic or ionic surfactants this model is essentially biased, in the majority of practical cases it can be recommended irrespectively of the nature of the surfactant. In the Frumkin model, three parameters are necessary to describe the adsorption and surface tension isotherm. Leaving aside the molar area co which can be estimated from the molecular geometry [16, 84], we concentrate on the results which follow from our development for the parameters a and b for surfactant molecules with linear hydrocarbon chain. Figure 3.59 illustrates the dependence of the Frumkin constant a on the molar area co of various surfactants at n<- = 10. Note that for ionic surfactants the co values are equal to the doubled values of co, from corresponding tables. 

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