Adsorption coefficient equation used

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

The adsorption coefficients (K) were determined using the equation for the Freundlich adsorption isotherm ... 

TABLE IV. Adsorption Coefficients for Butylate, Alachlor, and Metolachlor in Keeton Soil at Various Temperatures Obtained Using the Freundlich Equation. ... 

In this equation, B stands for adsorption coefficients and C for concentrations. The thermodynamic control imposes the use of very high pressures, low space velocities, and very active catalysts. For the specific case of aromatic saturation and in the presence of H2S or any other sulfur compound, NiW is the recommended catalyst [66], However, in those cases where a precious metal catalyst may be used then, it becomes the preferred choice [67],... 

Isotherm Subtraction. A second method (7) of determining the net proton coefficient from adsorption data is an adaptation of the thermodynamics of linked functions as applied to the binding of gases to hemoglobin (19). The net proton coefficient determined by this method is designated, Xp- The computational procedure makes a clear distinction between the influence of adsorption density and pH on the magnitude of the net proton coefficient. The fundamental equation used in the calculation of Xp is... 

A similar system, (CH3)2C=CH X, was studied by Endrysova and Kraus (55) in the gas phase in order to eliminate the possible leveling influence of a solvent. The rate data were separated in the contribution of the rate constant and of the adsorption coefficient, but both parameters showed no influence of the X substituents (series 61). A definitive answer to the problem has been published by Kieboom and van Bekum (59), who measured the hydrogenation rate of substituted 2-phenyl-3-methyl-2-butenes and substituted 3,4-dihydro-1,2-dimethylnaphtalenes on palladium in basic, neutral, and acidic media (series 62 and 63). These compounds enabled them to correlate the rate data by means of the Hammett equation and thus eliminate the troublesome steric effects. Using a series of substituents with large differences in polarity, they found relatively small electronic effects on both the rate constant and adsorption coefficient. 

The decrease in hydrogenation rate with increasing size of the alkyl group (Table VI) (93-96) can be correlated by the Taft equation. However, the correlation of the data by Smith and Pennekamp (93) (series 70) using the polar parameter Taft equation is applied that also includes steric constants. Similarly, Kieboom (34) has discussed Yoshida s correlation of series 73 based on main conclusion has been that the data do not allow a clear distinction between steric and polar effects. It seems that both operate in the same direction. Series 72 and 73, in which the rate data have been separated into the rate constants and adsorption coefficients, show opposite trends with the latter parameter. A similar problem has been encountered by Volter, Hermann, and Heise (100) and by Najemnik and Zdrazil (103) in the series of methylbenzenes (Table VII) and is discussed in this connection. 

A quantitative correlation of structural effects of four esters and four alcohols in the vapour phase transesterification on a macroreticular ion exchanger at 120°C was made using the Taft equation [441]. The authors found that rate coefficients [from eqn. (27)] yielded better correlation with steric (Es) than with polar (a ) parameters, while there was no significant difference between the correlations of the adsorption coefficients of alcohols, Kb, with both parameters. The correlations with Es yielded the slopes 1.4 and 0.6 for the reactivity of the esters and the alcohols, respectively, and —0.4 for the adsorptivity of the alcohols. The observed... 

Illustrative calculations appear below for estimating the particulate fraction (c()) of p,p -DDT in urban air at 20°C using the Junge-Pankow adsorption model (Equation 3), the Mackay adsorption model (Equation 15), and the octanol-air partition coefficient model (Equation 25). Table 10.3 lists values of P3 and Koafor p,p -DDT and other POPs of different chemical classes. All model calculations are for an urban air TSP = 80 pg/m3 (Shah et al., 1986). 

In the above equations the symbols A, B, C, D designate phenol, hydrogen, cyclohexanone and cyclohexanol. Table 5.7 presents the model parameters at 423 K and 1 atm. The model takes into account the effect of the products on the reaction rate in the region of higher conversion. This feature is particularly useful for describing the product distribution in consecutive catalytic-type reactions. Note that the adsorption coefficients are different in the two reactions. Following the authors, this assumption, physically unlikely, was considered only to increase the accuracy of modeling. 

The effect of using the sodium adsorption isotherm (equation 6) to determine the apparent diffusion coefficient can be seen by... 

Another kinetic method of determining relative adsorption constants does show that toluene is adsorbed more strongly than benzene on the platinum metals. The method determines the effect of the partial pressure of toluene on the rate of hydrogenation of benzene. With the assumption that adsorption is reversible, the ratio of adsorption coefficients, br/bB, is evaluated by the use of equation (37) where U°B and Ub(T) represent the rate of hydrogenation of benzene in the absence of toluene and in its presence at stated partial pressures of toluene (Ft) and benzene (Pb). This method avoids the determination of the relative individual rate constants that is required by the method of Wauquier and Jungers. Xylene inhibits the hydrogenation of alkenes on Pd more effectively than does benzene, which is consistent with the effect of alkyl groups on the basicity of benzene. ... 

Measurements have been made of tracer or intrinsic diffusion coefficients of water in a number of zeolites. Most refer to crystals nearly saturated with zeolitic water, while smoothed rather than adsorption areas were used in calculating the coefficients. From the temperature dependence of Da or Da, the activation energy, E, may be found using the Arrhenius equations Da = Do exp — E/RT or Da = Do exp — E/RT. Some results for tracer diffusion are summarized in Table VI. (iS). These shed considerable light upon certain aspects of intracrystalline diffusion for small polar molecules ... 

The Dubinin-Radushkevitch equation (TOZM theory) was applied to calculate the micropores volume. Affinity coefficient for nitrogen was taken as 0.33, and adsorption phase densi at adsorption temp atures was taken as 0.808 g/cm [11], Micropores size was estimated from adsorption energy defined from the Dubinin-Radushkevitch equation using correlation X=10/E (run), where X is a semi-width of a slit micro-pore [12],... 

The measured data also were used (700) in a quantitative representation of the effect of structure on the reactivity and adsorptivity of substrates by means of the Taft-Pavelich equation (22). The adsorption data suffered from a larger scatter than the rate data. No substrate or substituent could be detected that would fail to satisfy completely the correlation equations. In the correlation of the initial reaction rates and relative adsorption coefficients the parameter p was negative, while the parameter S was positive. In correlations of the reaction rates obtained by the hydrogenation of a similar series of substrates on the same catalyst in a number of solvents, the parameters p and had the same sign as in the hydrogenation in solvent-free systems, while in the correlation of the adsorption coefficients the signs of the parameters p and in systems with solvents were opposite to those in solvent-free systems. This clearly indicates that solvents considerably affect the influence of the structure of substrates on their reactivity. 

Distribution Coefficient The simplest and most widely used adsorption isotherm equation is a linear function. This adsorption isotherm equation is... 

Due to the effects of molecular size and shape and pore structure on the kinetics, the model cannot be used for general predictive purposes. In practice, in order to predict PAC adsorption, a series of experiments must first be carried out using the compound of interest, the activated carbon to be applied, and the water in which it is to be used. Equilibrium parameters, determined from the Freundlich adsorption isotherm equation, are used as input into a computer-based HSDM, which uses the method of least squares to minimize the difference between the experimental kinetic data points and the HSDM fit of the data [10]. When the best fit is achieved, the resultant kinetic parameters (liquid film mass transfer coefficient, k(, and the surface diffusion coefficient, DJ can then be used for the prediction of adsorption behavior under different conditions. 

The activity coefficients/A and/B can be evaluated by equation 2.24. The standard free energies of adsorption calculated by use of equation 2.33b are independent of the ionic strength of the solution. 

Currently no adequate quantitative theory of the discrete-ion potentials for adsorbed counterions at ionized monolayers exists although work on this problem is in progress. These potentials are more difficult to determine than those for the mercury/electrolyte interface because the non-aqueous phase is a dielectric medium and the distribution of counterions in the monolayer region is more complicated. However the physical nature of discrete-ion potentials for the adsorbed counterions can be described qualitatively. This paper investigates the experimental evidence for the discrete-ion effect at ionized monolayers by testing our model on the results of Mingins and Pethica (9, 10) for SODS. The simultaneous use of the Esin-Markov coefficient (Equation 3) and the surface potential AV as functions of A at the same electrolyte concentration c yields the specific adsorption potentials for both types of adsorbed Na+ ions—bound and mobile. Two parameters which need to be chosen are the density of sites available to the adsorbed mobile Na+ ions and the capacity per unit area of the monolayer region. The present work illustrates the value... 

Adsorption. The solution used to evaluate the pesticide transport equation, Equation 4a, assumes a linear adsorption isotherm that is constant with depth. However, linearity may not be the case for some pesticides and the adsorption coefficient will almost never be constant with depth. The rationale for using a linear model is initially based on the Freundlich isotherm... 

Methods using regression equations with soil adsorption coefficients or bioconcentration factors (Nos. 4 and 5 in Table 1-1) are not recommended because of the relatively large method errors that would be involved. 

If the range of temperatures is adequate, and measurement errors are small enough to establish the temperature coefficients of the equilibrium constant, then this single ramping can be used to calculate all of the equilibrium constant, the desorption, and the adsorption rate constants for the system. In practice the best operating conditions for this type of run are at high values of transit time, i.e. under conditions where Wfc is large but there is a premium on accurate data to avoid the instabilities that result from the form of the equations used. 

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