The Frumkin Isotherm

We proceed here to derive the expressions for the adsorption pseudocapacitance when the Frumkin isotherm applies, which we shall denote as Cp. 

the adsorption pseudocapadtance under Frumkin conditions is given by  

---

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

The Frumkin isotherm can be regarded as a general isotherm from which both the Temkin and Langmuir isotherms can be obtained as special cases. Szklarska-Smialowski and Wieczorek found that the adsorption of various aliphatic compounds (acids, alcohols and amines) on steel in H2SO4 conformed with the Frumkin isotherm. 

FIG. 24 ct-c Isotherms at 22°C for purified alkanesulfonates. The curve is fitted by the Frumkin isotherm. , Sodium n-decanesulfonate x, sodium /i-dodecanesulfonate. 

The parameters of the Frumkin isotherms shown in Fig. 24 are listed in Table 16. 

Kakiuchi et al. [75] used the capacitance measurements to study the adsorption of dilauroylphosphatidylcholine at the ideally polarized water-nitrobenzene interface, as an alternative approach to the surface tension measurements for the same system [51]. In the potential range, where the aqueous phase had a negative potential with respect to the nitrobenzene phase, the interfacial capacity was found to decrease with the increasing phospholipid concentration in the organic solvent phase (Fig. 11). The saturated mono-layer in the liquid-expanded state was formed at the phospholipid concentration exceeding 20 /amol dm, with an area of 0.73 nm occupied by a single molecule. The adsorption was described by the Frumkin isotherm. 

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 ... 

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,... 

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

The study of thiourea adsorption on an Hg electrode from ethanolic solutions shows that different supporting electrolytes can make a comparison of the adsorption parameters more difficult. The data obtained for various electrolytes (KF, KPFe, LiCl, NH4NO3, and KCNS) suggest that coadsorption, size, and polarizability of ions strongly influence the interfacial behavior of TU, at high concentration of the ions in particular. The effect of the electrolyte on AG° and parameter A in the Frumkin isotherm is illustrated in Table 6. 

The description of a mixed adsorption layer of ionic and nonionic surfactants requires the appropriate adsorption isotherms. For example, the Frumkin isotherm gives... 

The adsorption of reactant in the Cd(II)-I system [44], and the influence of electrolyte concentration (1-6 M NaCl04) on the parameters of the Frumkin isotherm system were investigated [45]. The obtained data indicated that the maximum surface concentration of Cdl2 (/kaax = 1 x... 

Sottomayor etal. [273] have investigated adsorption of n-hexanol on Au(lll) electrode from 0.2 M NaCl04 aqueous solution, applying chronocoulometry to measure dynamic capacitance and capacitive charge. They have found that in this case, adsorption may be described by the Frumkin isotherm and have determined the corresponding parameters. The obtained results were discussed and... 

Gomez etal. [160] have studied specific adsorption of potassium penicillin G (salt K) on Hg electrode from electrocapillary and capacity measurements. The Frumkin isotherm has been found applicable to the quantitative description of the data repulsive interaction parameter equaled —3.5 and standard free Gibbs energy of adsorption at the zero charge was —38.6 kJ mol h depended linearly... 

The Frumkin isotherm is one of the earliest isotherms (1925) that deals with lateral interactions among adsorbed species (Fig 6.98). The isotherm can be written as59... 

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). 

What isotherm does the Frumkin isotherm become when (a) A =0 and (b) 0 = 1/2 (Gamboa-Aldeco)... 

More elaborate treatments consider the lateral repulsion between reactant and adsorbate [120—123]. Parsons used a generalized version of the Frumkin isotherm, obtaining for the rate coefficient in the presence of adsorbed substance [124]... 

If there is interaction between the particles (attractive or repulsive), the Frumkin isotherm may apply ... 

Adsorptive accumulation — Organic substances which exhibit -> surface activity and electroactivity can be electrochemically analyzed by adsorptive accumulation on the surface of a an electrode, e.g., mercury electrode, followed by the reduction, or oxidation of the adsorbate using -> voltammetry [i,ii]. Also, the adsorption of highly stable and inert -> complexes of metal ions with surface-active organic ligands is utilized for the determination of trace metals [iii]. In all these methods the maximum voltammetric response is linearly proportional to the surface concentration of the adsorbed analyte at the end of the accumulation period [iv]. In the majority of cases, the adsorption on mercury can be described by the -> Frumkin isotherm /icx=o = 0exp(ad)/(1- 9), where f is the adsorption constant, cx=o is the concentration of the dissolved compound at the electrode surface, 6 = T/rmax is the surface coverage, T is the surface concentration of the adsorbed compound, rmax is the maximum surface concentration and a is the Frumkin... 

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... 

The adsorption of PNP to a wide variety of interfaces has been studied by SHG. The adsorption isotherm at the dodecane/water interface is shown in Figure 1.4. The adsorption isotherms typically used to fit the surface tension and SHG data at the fiquid/liquid interfaces are in terms of the interfacial coverage 0 and the bulk phase concentration c, the Langmuir (13) and the Frumkin isotherms (14)... 

The formation of dimers in solution at higher concentrations indicates an interaction between the dye molecules and so the b term in the Frumkin isotherm is not a surprise. The adsorption behaviour suggests that the dye is highly surface-active and formation of a monolayer is almost complete by the transition at ca. 6 pM. The standard free energy of adsorption derived from the isotherm is consistent with a monolayer coverage forming even at the micromolar aqueous concentrations. Once the first layer is substantially complete, subsequent adsorption takes place on this layer to form a second-ordered layer. 

According to the thermodynamics of the interface, the adsorption induces the variation in the interfacial tension, y, a decrease (increase) in the case of the positive (negative) adsorption. In the case of the Frumkin isotherm, the decrease in y is given by [19]... 

An exaggerated emphasis on heats of chemisorption has probably been harmful in the proper understanding of the role of chemisorption in surface phenomena. Thus the marked nonuniformity of all surfaces with respect to heats of chemisorption has led to rather elaborate treatments where models of surface heterogeneity (statistical distribution of energy sites) or, less successfully, specific forces of interaction between adsorbed species have been invoked to explain the non-Langmuirian adsorption isotherms. For instance the Frumkin isotherm can be obtained with a linear variation of heats of adsorption with coverage, and the Freundlich isotherm is attributed to an exponential variation of heats of adsorption. 

---