Frumkin equation

An adsorption equation known as the Frumkin isotherm has the form... 

Even with the simpler Frumkin or Fowler-Guggenheim approach (Eqs. 6.50 and 6.52), treating the coadsorption and surface reaction of different adsorbates leads immediately to mathematically intractable expressions and to the introduction of new parameters, whereas equation... 

Reactant concentrations Cyj in the bulk solution, as well as the Galvani potential between the electrode and the bulk solution (which is a constituent term in electrode potential E), appear in kinetic equations such as (6.8). However, the reacting particles are not those in the bulk solution but those close to the electrode surface, near the outer Helmholtz plane when there is no specific adsorption, and near the inner Helmholtz plane when there is specific adsorption. Both the particle concentrations and the potential differ between these regions and the bulk solution. It was first pointed out by Afexander N. Frumkin in 1933 that for this reason, the kinetics of electrochemical reactions should strongly depend on EDL structure at the electrode surface. 

The same system has been studied previously by Boguslavsky et al. [29], who also used the drop weight method. While qualitatively the same behavior was observed over the broad concentration range up to the solubility limit, the data were fitted to a Frumkin isotherm, i.e., the ions were supposed to be specifically adsorbed as the interfacial ion pair [29]. The equation of the Frumkin-type isotherm was derived by Krylov et al. [31], on assuming that the electrolyte concentration in each phase is high, so that the potential difference across the diffuse double layer can be neglected. 

The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

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... 

Since C < Co, this work is positive, and the coverage decreases away from the pzc. Equations (4.13) and (4.17) can be combined with the Frumkin isotherm, resulting in ... 

The Frumkin Equation (also referred to as the Frumkin-Fowler-Guggenheim, FFG, equation) has been specifically developped to take lateral interactions at the surface into account. In the FFG equation, the term 6 / (1-6) in (4.10b) is multiplied by the factor exp(-2 a0) which reflects the extent of lateral interactions... 

Example 4.1 Surface Complex Formation, Langmuir Equation and Frumkin Equation... 

Fig. 4.8 compares data on the adsorption of lauric acid (C12) and caprylic acid (Cs) at a hydrophobic surface (mercury) as a function of the total bulk concentration for different pH-values. As is to be expected the molecular species becomes adsorbed at much lower concentrations than the carboxylate anions. The latter cannot penetrate into the adsorption layer without being accompanied by positively charged counterions (Na+). As was shown in Fig. 4.4, the adsorption data of pH = 4 can be plotted in the form of a Frumkin (FFG) equation. Fig. 4.9 compares the adsorption of fatty acids on a hydrophobic model surface (Hg) with that of the adsorption on Y-AI2O3. 

It is necessary to remind ourselves, that the adsorption of humic acids or fulvic acids correspond to the adsorption of a mixture of adsorbates. Adsorption equations derived for the adsorption of a single adsorbate (Langmuir, Frumkin or Gibbs Equation) cannot be used for mechanistic interpretation of the data even if these data can be fitted to such equations (Tomaic and Zutic, 1988). 

In electrochemical conditions, the electrons are transferred from the metal to the solution rather than to a vacuum. Moreover, the metal/solution interface is charged and the potential difference between the metal and the solution should be taken into account. The situation is simplified when the work function and uncharged interface are considered. The relationship between the work function and potential of zero charge was propos nearly 30 years ago by Bockris and Argade and by Frumkin (see e.g., Ref. 66) and later intensively discussed by Trasatti (e.g., Refs. 5, 21, 67). The relationship is given by the equation... 

On the other hand, the data for some organic compounds are often fitted into the Frumkin isotherm equation in a modified form... 

The intrinsic parameter, characterizing the type of interactions, is the Frumkin interaction parameter a, which is positive for attractive forces and negative for repulsive forces. In addition, 9 = is the fraction of the electrode covered with deposited material, and f ax is the maximal surface coverage. Combining (2.93) and (2.94) with (2.102), the following integral equation is obtained as a general solution ... 

Consider a system in which a potential difference AV, in general different from the equilibrium potential between the two phases A 0, is applied from an external source to the phase boundary between two immiscible electrolyte solutions. Then an electric current is passed, which in the simplest case corresponds to the transfer of a single kind of ion across the phase boundary. Assume that the Butler-Volmer equation for the rate of an electrode reaction (see p. 255 of [18]) can also be used for charge transfer across the phase boundary between two electrolytes (cf. [16, 19]). It is mostly assumed (in the framework of the Frumkin correction) that only the potential difference in the compact part of the double layer affects the actual charge transfer, so that it follows for the current density in our system that... 

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 ... 

The first attempt to account for surface contamination in creeping flow of bubbles and drops was made by Frumkin and Levich (FI, L3) who assumed that the contaminant was soluble in the continuous phase and distributed over the interface. The form of the concentration distribution was controlled by one of three rate limiting steps (a) adsorption-desorption kinetics, (b) diffusion in the continuous phase, (c) surface diffusion in the interface. In all cases the terminal velocity was given by an equation identical to Eq. (3-20) where C, now called the retardation coefficient , is different for the three cases. The analysis has been extended by others (D6, D7, N2). 

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... 

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... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

To resolve the problem of negative /3 values obtained with the Frumkin theory, the improved Szyszkowski-Langmuir models which consider surfactant orientational states and aggregation at the interface have been considered [17]. For one-surfactant system with two orientational states at the interface, we have two balances, i.e., Ft = Fi + F2 and Ftco = Ficoi + F2C02, which can be used in conjunction with Eq. 24 to derive two important equations for determining the total surface excess and averaged molecular area required in the calculation of surface tension, i.e.,... 

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. 

A. N. Frumkin, Surface Tension Curves of the Higher Fatty Acids and the Equations of Conditions of the Surface Layer, Z Physik. Chem. 116 466 (1925). 

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)... 

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