Logarithmic isotherm

For different acceptor particle adsorption isotherms expressions (1.85) - (1.89) provide various dependencies of equilibrium values of <7s for a partial pressure P (ranging from power indexes up to exponential). Thus, in case when the logarithmic isotherm Nt InP is valid the expression (1.85 ) leads to dependence <75 P" often observed in experiments [20, 83, 155]. In case of the Freundlich isotherm we arrive to the same type of dependence of - P" observed in the limit case described by expression (1.87). 

An isotherm test can determine whether or not a particular contaminant can be adsorbed effectively by activated carbon. In very dilute solutions, such as contaminated groundwater, a logarithmic isotherm plot usually yields a straight line represented by the Freundlich equation62 63 ... 

Lithium as anode during electrodepostion. 1343 Logarithmic isotherm, 941... 

This equation is logically called the logarithmic isotherm. It shows that under the conditions chosen (constant V and 0.2 < 0 < 0.8), 0 is linear with In cr... 

Langmuir equation for a uniform surface, but by the Zel dovich and Rogin-skii equation or by the Bangham equation. Adsorption equilibrium is described not by the hyperbolic Langmuir isotherm, but by the Freundlich isotherm or the logarithmic isotherm (40). 

Distribution functions (93) and (94) are substantiated by the fact that they lead to the most frequently observed adsorption isotherms for the region of medium surface coverages, viz., the Freundlich isotherm corresponds to distribution (93) with > 0 and T < as it has been demonstrated by Zel dovich (43) and the logarithmic isotherm corresponds to distribution (94) (40). Besides that, if we accept Assumption 1, then distribution (94) will give the equation of adsorption kinetics by Zel dovich and Roginskil and distribution (93) will result in the equation of adsorption kinetics by Bangham. Finally, if Assumption 1 is correct, then distribution (93), including distribution (94) as its particular case, follows from the kinetics of fractional order reactions (44). 

In the original derivation of (305), it was supposed (40) that the nitrogen adsorption equilibrium on the catalyst follows the logarithmic isotherms (i.e., that the surface is evenly nonuniform). In this case y — 0 and, according to (143) and (164), m — a, n — / . Experiments with iron catalyst promoted with A1203 and K20 gave m = 0.5. This was interpreted as a = 0.5 (93). 

Experimental results of this work (124) were found to agree with (352) at n = 0.5. Since according to the data on adsorption-chemical equilibrium (344), the logarithmic isotherm is valid i.e., y = 0, (158) gives n = / . Therefore, as usual, a = / = 1/2. 

Significantly later, foreign scientists reached a similar conclusion regarding the Freundlich isotherm. In the USSR, a theory of adsorption on an inhomogeneous surface was developed independently by M. I. Temkin of the Karpov Physico-Chemical Institute in connection with electrochemical research by Academician A. N. Frumkin. M. I. Temkin s work on a logarithmic isotherm was cited in [74] and published in [75]. The theory of adsorption and catalysis on an inhomogeneous surface was especially extensively developed by S. Z. Roginskii. 

Disregarding for a moment the electrochemical aspect of this isotherm, we note that 0 is proportional to logC, (as opposed to the Langmuir isotherm, where it is proportional to a linear function of the concentration.) A simitar "logarithmic isotherm" was developed by Temkin. His derivation is much more complex, but in the final analysis it is based on the same physical assumptions. It has, therefore, become common to refer to Eq. 141 as the Temkin isotherm, although Temkin has never used it in this form. It is this approximate form of the Frumkin isotherm which is applied to electrode kinetics, as we shall see below. 

Equation (3.86) relates the rate of adsorption to the fugacity of adsorbed layer. Expressing fugacity via surface coverage, using for instance logarithmic isotherm at medium coverage 0 = (1 / /) In(avp) we have... 

For intermediate coverages 0.2 < 0 < 0.8, variations of 0/(1 — 6) can be neglected as compared with the exponential term in 6 and the well-known Temkin-type logarithmic isotherm (48) is obtained ... 

Strictly speaking, this conclusion is not restricted to a specific form of the logarithmic isotherm. Indeed, in the case of an equilibrium, chemical (and electrochemical, in the case of an electrode reaction) potentials (or, respectively, /I) throughout the volume and on the surface are equal. In the region of intermediate coverages, a variation in the adsorbed substance s chemical potential, due to the variable term RT In 6/(1 - 6), is small therefore, the essential role in the equalization of [jl is played by a variation in A with 0. This means that at a given value of /x, A for various metals will be practically the same although 6 will be different. This result is obtained with any form... 

By integration of the hydrogen region of I, curves [15], obtained in sulfuric acid and zinc sulfate and cadmium sulfate solutions at 28, 40, and 60 C, hydrogen adsorption isotherms were plotted with 6 against log p (p = pressure), from which it was seen that they obey a logarithmic isotherm equation ... 

Thomas[116] suggested that a slope of about 60 mV may be caused by various mechanisms, in particular, by a slow recombination under the condition that the hydrogen adsorption isotherm is logarithmic. However, for low surface coverages, which are always observed in the case of mercury[117], the logarithmic isotherm is not applicable and hence this explanation for a 60 mV slope cannot be correct. 

Several authors have obtained a linear dependence between (p and log i for a platinum anode. Stout[370] obtained a value of b equal to 57-60 mV, while according to Thomas[371], b lies between 43 and 56 mV. Similar results were obtained for Pd, Rh, Ir, Fe, and Ni[370-372]. The discharge process is preceded by a charging which is probably associated with the adsorption of N3 radicals on the electrode surface. This adsorption is satisfactorily described by a logarithmic isotherm[371]. 

---