Kinetics Elovich equation

In practice the kinetics are usually more complex than might be expected on this basis, siace the activation energy generally varies with surface coverage as a result of energetic heterogeneity and/or sorbate-sorbate iateraction. As a result, the adsorption rate is commonly given by the Elovich equation (15) ... 

Kinetics of Adsorption and Desorption and the Elovich Equation C. Aharoni and F. C. Tompkins... 

The two-constant equation and the simple Elovich equation efifect-tively described the change of Co in the main source, the ERO fraction in both soils (Fig. 6.30). This suggests that the diffusion processes controlled the rate of Co transformation. It is noted above that the kinetics of Mn transformation in the three main fractions (EXC, CARB, and ERO), in the... 

Reaction kinetics. The time-development of sorption processes often has been studied in connection with models of adsorption despite the well-known injunction that kinetics data, like thermodynamic data, cannot be used to infer molecular mechanisms (19). Experience with both cationic and anionic adsorptives has shown that sorption reactions typically are rapid initially, operating on time scales of minutes or hours, then diminish in rate gradually, on time scales of days or weeks (16,20-25). This decline in rate usually is not interpreted to be homogeneous The rapid stage of sorption kinetics is described by one rate law (e.g., the Elovich equation), whereas the slow stage is described by another (e.g., an expression of first order in the adsorptive concentration). There is, however, no profound significance to be attached to this observation, since a consensus does not exist as to which rate laws should be used to model either fast or slow sorption processes (16,21,22,24). If a sorption process is initiated from a state of supersaturation with respect to one or more possible solid phases involving an adsorptive, or if the... 

The Elovich equation was developed to determine the kinetics of heterogeneous chemisorption of gases on sohd surfaces. This equation assumes a heterogeneous distribution of adsorption energies, where the energy of activation E) increases linearly with surface coverage (Rao et al. 1989). A simplified Elovich equation used to study the rate of soil chemical processes is given by... 

Cells R, Cox L, Hermosin MC, Cornejo J (1996) Retention of metamitron by model and natural particulate matter. Intern J Environ Anal Chem 65 245-260 Chaney RL (1989) Toxic element accumulation in soils and crops protecting soil fertility and agricultural food chains. In Bar Yosef B, Barrow NJ, Goldschmid J (eds) Inorganic contaminants in the vadose zone Springer, Heidelberg, pp 140-159 Charlatchka R, Cambier P (2000) Influence of reducing conditions on solubility of trace metals in contaminated soils. Water, Air Soil PoUut 118 143-167 Chien SH, Clyton WR (1980) Application of Elovich equation to the kinetics of phosphate release and sorption in soils. Soil Sci Am J 44 265-268... 

Polysopoulos NA, Keramidas VZ, Pavlatou A (1986) On the limitation of the simplified Elovich equation in describing the kinetics of phosphate sorption and release from soils. J Soil Sci... 

In the cases of Cr203, NiO, and Fe2Ch it was possible (24) to follow the kinetics of the adsorption of 02 at constant pressures around 6 cm. It was found that the uptake q followed the Elovich equation ... 

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

The Elovich equation has been used to describe the kinetics of P04 sorption and desorption on soils and soil minerals (Atkinson et al., 1970 Chien and Clayton, 1980 Chien et al., 1980 Sharpley, 1983), potassium reactions in soils (Sparks et al., 1980b Martin and Sparks, 1983 Sparks and Jardine, 1984 Havlin and Westfall, 1985), borate dissolution from... 

Atkinson, R. J., Hingston, F. J., Posner, A. M., and Quirk, J. P. (1970). Elovich equation for the kinetics of isotope exchange reactions at solid-liquid interfaces. Nature (London) 226, 148-149. 

A number of additional equations are often used to describe reaction kinetics in soil-water systems. These include the Elovich equation, the parabolic diffusion equation, and the fractional power equation. The Elovich equation was originally developed to describe the kinetics of gases on solid surfaces (Sparks, 1989, 1995 and references therein). More recently, the Elovich equation has been used to describe the kinetics of sorption and desorption of various inorganic materials in soils. According to Chien and Clayton (1980), the Elovich equation is given by... 

Rudzinski W. and Panczyk T. Sur ce Heterogeneity Effects on Adsorption Equilibria and Kinetics Rationalisation of Elovich Equation. In SCHWARZ J. and CONTESCU C. (eds.) Surfaces of Nanoparticles and Porous Materials (Marcel Dekker, 1999), pp.355-390. 

Kinetics of several organosilicon compound chemisorption from the gas phase on the non-porous silica (Cab-O-Sil M5) surface is described by the well-known Elovich equation [27] ... 

Rather than trying to describe the sparse and occasional results hitherto reported, this work is devoted to the foundation of a general and comprehensive theory of adsorption-desorption kinetics. The emphasis, however, will not be concentrated on the formal mathematical aspects of the theory, but rather on its physical bases in particular, the validity of the methods developed will be checked by verifying how they are able to explain the observed experimental behaviours. Among them one plays a special role — the time-logarithm law known as Elovich equation. 

The kinetics of adsorption and desorption and the Elovich equation have been the matter of a comprehensive review by Aharoni and Tompkins in 1970 [14]. At that time, however, concepts now pervasive in physical chemistry of surfaces like fractality were not known, the mathematical theory of adsorption equilibrium on heterogeneous surfaces was at its beginning, and the notion of equilibrium surfaces had not demonstrated yet its usefulness in the understanding of adsorption phenomena on real surfaces. In view of these facts there is a space for another work, which however does not intend to be as comprehensive as that of Aharoni and Tompkins, but rather aims to study the Elovich behaviour met in new situations, to elucidate the theoretical origin of Eq. (3), and to relate the macroscopic empiric parameters te, and t and to microscopic quantities. 

Let us first consider second order kinetics. The Elovich equation (3), expressed in terms of S, reads... 

Figure 2 shows the distribution functions calculated for second-order kinetics in the frame of the condensation approximation for Elovich equations with different values of l — it is immediately seen that the larger is the ratio (i.e., the more accurate is the... 

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