Frumkin-type isotherm

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. 

However, a number of treatments specifically oriented in this direction have been developed (Conway and Gileadi, 1962 Srinivasan and Gileadi, 1967 Conway and Stoneharl, 1977). In these, a Frumkin-type isotherm (6.8.9) is used, i.e.,... 

Let us assume now that the perm-selectivity is broken down that is, an excess of MA electrolyte is present in the film, so that the change in the concentration of A induced by doping, is small compared to the excessive electrolyte concentration. This leads to an invariance of the chemical potential of instead of the condition of perm-selectivity (Equation 11.9), so that a Frumkin-type isotherm (Equation 11.14) appears instead ofthe non-Nernstian isotherm (Equation 11.13). 

Lucassen-Reynders (1976, 1981) derived more general thermodynamic relationships which take into consideration different molecular self-areas of solvent co, and solute molecules (B2. Her relationship reduces to a Frumkin-type isotherm by setting (0 = 2 = 1... 

When lateral interactions exist between O and R in the film, the shape of the i-E curve depends upon the energies of the interactions of O with O, R with R, and O with R. The exact shape of the curve depends upon how these interactions are taken into account. For example, if a Frumkin-type isotherm (35, 36) is assumed, the relevant expression is ... 

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). 

Underpotential deposition usually follows Frumkin-type adsorption isotherms due to strong lateral interactions and the interaction parameter, g, varies stepwise with coverage which is a function of the electrode potential. This is due to important structural and electric changes operative in the upd layers as coverage increases. 

The von Szyszkowski isotherm establishes the connection between the change in surface tension y and the surfactant bulk concentration. Stauff (1957) has evaluated the parameters of this semi-empirical adsorption isotherm and has shown that it is in agreement with interfacial thermodynamics. Frumkin s isotherm has often recently been used to describe the adsorption of different types of surfactants, for example by Lunkenheimer (1983), Miller (1986), Wiisteneck et al. (1993), and others. One of the main aims of this book is to show that in the many... 

Based on a Frumkin type adsorption isotherm (cf. Chapter 2), which after rearrangement into a c(r)-form reads... 

Thus, deviations from the ideal Langmuir isotherm can be caused both by intermolecular interactions, which result in an enthalpy of mixing, and by area differences between molecules, which produce a non-ideal entropy of mixing [18]. For a simple case where the interactions are of the Frumkin type and the partial molar areas of solvent and surfactant are constant the entropic effect of area differences results in typical features of macromolecular adsorption, e.g., a steep initial increase of adsorption ( high affinity adsorption) and a very slow rise once the surface is approximately half filled [18]. 

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. 

It is instructive to consider what will happen as we go through a potential region where coverage has intermediate values, i.e., 0.1 < t ai < 0.9. In this case, it is possible that we will go through a condition where AG ads is coverage-dependent, i.e., some generalized isotherm of Frumkin type ... 

Adsorption isotherm To a first approximation it can be assumed that the coverage of the electrode 0 is proportional to the adsorbed amount r, 0 = T/T = TS, where is the maximum adsorbed amount per 1 cm and S the area covered by 1 mole of adsorbed substance [3], [1]. The dependence of the surface coverage on the volume concentration c at constant temperature is called the adsorption isotherm. There may be different types of adsorption isotherms according to the properties of the systems studied and the theoretical approach (see [1-5]). Only the proper model for the given experimental conditions should be used [73-82]. For adsorption of organic molecules the Frumkin adsorption isotherm is generaly used [1], [3] ... 

For the incommensurate case, or for the liquid mercury electrode, Frumkin s isotherm describes a gas-liquid transition, and not the solid-liquid transition [135]. This opens interesting possibilities about different types of transitions that are possible, and that we hope to discuss in the future. In the voltammogram the area under the spike is proportional to the charge transferred, not to the coverage of the surface, because the charge per adatom on the surface is not necessarily an entire number equal to the stoichiometric electrovalence [118]. 

Irreversible reaction, 1251, 1419 Isoconic, definition, 933, 978, 982 Isotherm, 932, 964, 1197 applicability, 941 and charge transfer, 954, 955 Conway and Angersein-Kozlowska, 943 definition, 933 in electrode kinetics, 1197 Flory—Huggins type, 941,942, 944, 965 Frumkin, 938, 942, 965 Frumkin-Temkin, 1197, 1198 Habib-Bockris, 943... 

The AGf, AGAa AGAs. and AGss values, and, correspondingly, In fi and a values depend on the electric state of the surface, i.e., on the electrode potential or charge. This isotherm was deduced by Frumkin [i] (and named after him soon) as a general case of the -> Langmuir isotherm, which corresponds to a = 0. A statistical derivation of the Frumkin isotherm is available [ii] various model considerations and relations to other types of isotherms are discussed in [iii]. Another typical form of the Frumkin isotherm is... 

Of course, other types of adsorption isotherms commonly represent the adsorption processes in heterogeneous catalysis, for example, the Freundlich isotherm and, especially in electrode processes, the Temkin and Frumkin isotherms (99). In the latter case... 

The maxima of fo.ads at certain A (Fig. 3.41) more or less coincide with the inflection points of the corresponding I E) isotherms (Fig. 3.36), indicating thatf ct turns maximal for an occupation of half of the adsorption sites of a given pattern. Such behavior is expected if the adsorption and desorption rates are steeply increasing functions of -6 and 9, respectively, near the point of half-coverage. This is true even in a primitive Frumkin or Langmuir-type adsorbate, because in that case... 

As the theoretical models are comparatively complex, only numerical methods allow to interpret experimental data. A software package is available that allows to make model calculations for any type of the above discussed diffusion-controlled mechanism [223]. In addition to the theory for a Langmuir isotherm, where the collocation solution by Ziller and Miller can serve as analytical solution, the programme gives access also to calculations based on the Frumkin, the reorientation and aggregation isotherms. 

Figure 21 shows model predictions based on Eq. (53) when X ranges from 0 to 5. It is seen that in general the heterogeneity effect does not seriously disturb the adsorption properties for values of X smaller than 2 - 3. It is also interesting to observe that the model predicts Frumkin s type adsorption isotherms even at high X values (Figure 21B). This may explain the validity ofthe Frumkin isotherm in a number of studies using solid polycrystalline electrodes of high hydrogen overpotential. ...

Sorption isotherms at constant pH (most often used types are Langmuir, Freimdlich, and Frumkin) are a step toward a more realistic description of surface phenomena, expressing the Kj as a function of sorbent concentration. Their to the best semi-empirical foimdations allow them to be rather simple in terms of defining equations and amoimt of model parameters. Of course this often does not reflect reality with the necessary accuracy, so their usefulness is restricted to only a few cases. 

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