Frumkin constant

The presence of the second polar group leads to a Frumkin constant a which is much lower than that for alcohols. However, the most significant decrease is obtained for the adsorption equilibrium constant b, as demonstrated in Fig. 3.7. 

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. 

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. 

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... 

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]

Figure 3.57 presents experimental data for decyl ammonium chloride (CioACl) and dodecyl ammonium chloride (C12ACI) in presence of 0.005 M HCl [33]. The molar area to, is somewhat lower than that for C TAB. At the same time, the Frumkin constant a is somewhat higher which underlines the interrelation of these parameters. The dependencies of b on n, for CnACl and CnTAB are compared with the data for fatty acids in Fig. 3.58. 

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. 

As one can see the Frumkin model does not reflect all the details of the adsorption process. In some cases, the reorientation and aggregation models lead to better results. The scope of the data available is still insufficient to formulate a criterion for the best choice of the adsorption model. However, it follows from the results summarised in this chapter, that the reorientation model can be successfully applied to oxyethylated surfactants, and also for surfactants with relatively large molar area (o > 2.5T0 m /mol). At the same time, the surfactant molecules with relatively high values of the Frumkin constant and low molar area (m < 2.5-10 m /mol) are more capable for aggregation in the surface layer. 

In some cases, a better agreement with the experimental surface tension isotherms and other data (dynamic surface tension, optical methods) is provided by the reorientation or aggregation model, respectively. It follows from the presented results that the reorientation model is more appropriate for oxyethylated surfactants and for surfactants which possess relatively high molar area, ro > 2.5-lO m /mol. At the same time, the aggregation and cluster models describe better the behaviour of surfactants with a relatively large Frumkin constant and low molar area, (o< 2.5-10 m /mol. 

In the treatment of the kinetics of the electron transfer illustrated in Section 4.1, it has been assumed that the propulsive force for the electron transfer was the electrochemical potential E i.e. a quantity directly related to 4>M — < >s). However, since the solvated ions cannot enter the inner layer of the double layer (IHP), the true propulsive force should be < )M — standard rate constant, k°, and the exchange current, i0, should become respectively ... 

In the case of a Frumkin isotherm, the equilibrium constant y " is given by... 

In many works in which the adsorption process is described using the Frumkin isotherm, the solvent activity is considered as constant for every value of ag in view of the fact that the ratio of the molar fractions Xg/Xg is small. Hence,... 

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 double-layer influence on the electrode reaction of Zn(II)/Zn(Hg) on DME in NaNOs solutions was studied in the concentration range from 0.01 to 1 M, using dc and ac polarography [30]. The apparent rate constants of the Zn(II)/Zn(Hg) system increase with dilution of the NaN03 supporting electrolyte. However, after the Frumkin correction, the rate constant was virtually independent of the supporting electrolyte concentration. 

Consider, now the dependence of 0 upon potential under the condition that AGe varies with 0. It will he less dramatic (i.e., d dV will be much smaller) than in the situation represented by the original, simpler, corresponding, Langmuir equation [Eq. (7.153)]. If the latter isotherm is applicable to a variation of 0 with potential at constant concentration, the surface is effectively either empty of intermediate (0 1) or near to 0 = 1. With Frumkin-Temkin in control, 0 varies linearly and more slowly with V than it does with the Langmuir equation. Thus, from (7.156), at constant cp... 

If the relative values of the rate constants among the consecutive or parallel steps in reactions such as that of hydrogen evolution have the most decisive influence here, Frumkin-Temkin should be used. If they lead to a situation in which the intermediate radical coverage tends toward zero or one, the matter is decided. As remarked above, for 0 — 0 or 0 — 1, the Frumkin and Temkin isotherms coincide in effect with that of Langmuir. 

The addition of the inert electrolyte affords other advantages. The most important point is that the conductivity of the solution increases (and thus the ohmic drop decreases through a decrease of the resistance of the cell, Rccw see Sect. 1.9). Moreover, the diffuse double layer narrows, being formed mainly by the ions of the inert electrolyte (with a sharp potential drop over a very short distance from the electrode surface). This makes the capacitance more reproducible and the Frumkin effects less obtrusive. Activity coefficients of the electroactive species are also less variable (and, therefore, quantities like formal potentials and rate constants), since... 

There seems to be a close relation between this factor a and the index j8, in Bronsted s relation1 k = cK between the dissociation constant K of an acid and its catalytic activity k this had been suggested by Frumkin 2 the reader must consult the originals for further elaboration. 

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