Empirical distribution coefficient

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

Figure 4 Seawater cadmium reconstructed from foraminifera using a depth-dependent empirical distribution coefficient versus estimated bottom-water cadmium (sources Boyle, 1988, 1992).

Figure 5 Seawater zinc reconstmcted from foramini-fera using an empirical distribution coefficient which depends on the degree of carbonate unsaturation versus estimatedbottom-waterzinc(afterMarchittoera/.,2000).

The quantities in brackets represent activies of ions in solution and of components in the solid phase. Application of the defining equation directly would require knowing activity coefficients for the ions in solution and also the Henry s law coefficient for the trace carbonate in solid solution. A practical approach is to rewrite equation (13) in terms of an effective or empirical distribution coefficient... 

The thermodynamic distribution coefficient requires that equilibrium be maintained between the crystallizing solid and the parent liquid. However, diffusion rates in the solid are so slow that there is negligible interchange between trace elements in the crystal and trace elements in solution except at the surface. The measured empirical distribution coefficients describe an instantaneous or surface partitioning. This requires that a careful distinction be made between a static system, such as a closed pocket or pond from which crystals are growing and a flow-through system in which the growing crystals are continuously bathed in fresh solution. 

For the determination of the exchange reaction s direction and equilibrium concentrations of its ions it is necessary to know activities of the adsorbed ions and their corresponding equilibrium constants. However, methods to determine ion activity in exchange capacity are not yet available. That is why instead of equilibrium constants in exchange reactions are used empirical distribution coefficients. 

This equation, although originating from the plate theory, must again be considered as largely empirical when employed for TLC. This is because, in its derivation, the distribution coefficient of the solute between the two phases is considered constant throughout the development process. In practice, due to the nature of the development as already discussed for TLC, the distribution coefficient does not remain constant and, thus, the expression for column efficiency must be considered, at best, only approximate. The same errors would be involved if the equation was used to calculate the efficiency of a GC column when the solute was eluted by temperature programming or in LC where the solute was eluted by gradient elution. If the solute could be eluted by a pure solvent such as n-heptane on a plate that had been presaturated with the solvent vapor, then the distribution coefficient would remain sensibly constant over the development process. Under such circumstances the efficiency value would be more accurate and more likely to represent a true plate efficiency. 

The Langmuir equation has a strong theoretical basis, whereas the Freundlich equation is an almost purely empirical formulation because the coefficient N has embedded in it a number of thermodynamic parameters that cannot easily be measured independently.120 These two nonlinear isotherm equations have most of the same problems discussed earlier in relation to the distribution-coefficient equation. All parameters except adsorbent concentration C must be held constant when measuring Freundlich isotherms, and significant changes in environmental parameters, which would be expected at different times and locations in the deep-well environment, are very likely to result in large changes in the empirical constants. 

The semi-empirical descriptions of adsorbate/solid interactions are based on net changes in system composition and, unlike surface complexation models, do not explicitly identify the details of such interactions. Included in this group are distribution coefficients (Kp) and apparent adsorbate/proton exchange stoichiometries. Distribution coefficients are derived from the simple association reaction... 

The results just obtained for < y) are, however, rarely used in applications because (v ) and T are generally not known. The Gaussian dispersion parameters aj and al are, in a sense, generalizations of (Cj) and particle displacement variances o-y and a-] are not calculated by Eq. (8.8). Rather, they are treated as empirical dispersion coefficients the functional forms of which are determined by matching the Gaussian solution to data. In that way, the empirically determined a-y and deviations from stationary, homogeneous conditions which are inherent in the assumed Gaussian distribution. 

HLB values decrease as the solubility of the surface-active agent decreases in water. Solubility of cetyl alcohol in water (at 25°C) is less than a milligram per liter. It is thus obvious that, in any emulsion, cetyl alcohol will be present mainly in the oil phase, while SDS will be mainly found in the water phase. Empirical HLB values are found to have significant use in emulsion technology applications. It was shown that HLB is related, in general, to the distribution coefficient, KD, of the emulsifier in the oil and water phases ... 

In chromatography one traditionally avoids the use of empirical parameters, such as e, and prefers Kd relative to a certain internal standard usually a low-molecular-weight substance. In this case, the distribution coefficients of macromolecules, Kd, are a function of t — tab, where tad and t are the elution times of the standard in a given mixture and in a mixture corresponding to the critical conditions. Close to the critical conditions, Kb is much more sensitive to a change in the composition of the mobile phase than to the retention times of low-molecular-weight substances, and so precision of Kd determined from t — tab will hardly be higher than that of eab calculated from the semiempirical Equation (3.16). 

Sorption is most commonly quantified using distribution coefficients (Kd), which simplistically model the sorption process as a partitioning of the chemical between homogeneous solid and solution phases. Sorption is also commonly quantified using sorption isotherms, which allow variation in sorption intensity with triazine concentration in solution. Sorption isotherms are generally modeled using the empirical Freundlich equation, S = K CUn, in which S is the sorbed concentration after equilibration, C is the solution concentration after equilibration, and Kt and 1 In are empirical constants. Kd and K are used to compare sorption of different chemicals on one soil or sorbent, or of one chemical on several sorbents. Kd and K are also commonly used in solute leaching models to predict triazine interactions with soils under various environmental conditions. 

Here, Vi is the molar volume of the solute (as a measure of the size of the cavity to accommodate the solute i in the solvent), d is an empirical parameter which takes also account for polarizability n, a and / characterize respectively the acidity or basicity which in general represents the ability to form hydrogen bonds, and the C s are solvent characteristics independent of the solute. Meyer and Maurer [39] used this equation for 30 systems (371 substances, 947 experimental distribution coefficients) to evaluate generalized solvent Cj parameters. 

This equation is quite accurate in comparison with group contributing methods [40] or other predictive LSER methods [41]. For compounds where the solvatochromic parameters are known, the mean absolute error in log Dy is about 0.16. It is usually less than 0.3 if solvatochromic parameters of the solute and solvent must be estimated according to empirical rules [42], In contrast to the prediction of gas-liquid distribution coefficients, which is usually easier, the LSER method allows a robust estimation of liquid-liquid distribution coefficients. However, these equations always involve empirical terms, despite their being physico-chemically founded thermodynamic models. However, this is considered due to the fundamental character of the solvatochromic scales. 

A convenient way to classify solvents of chromatographic interest in terms of their polarity and the specific chemical interactions is the empirical scheme proposed by Snyder [214,215]. This scheme is based on experimental (gas chromatographic) distribution coefficients for three test solutes ( probes ) on a large number of stationary phases, which were published by Rohrschneider [216]. The probe compounds are ethanol (e), 1,4-dioxane (d) and nitromethane (n). The experimental values for the distribution coefficients undergo several empirical modifications ... 

The dependence of the distribution coefficient of the protein on an ion-exchanger with regard to protein and salt concentration [K(Cm, /)] can be empirically described by the following equation assuming a Langmuir-type adsorption behavior for the protein ... 

In this context, another empirical solvent parameter called SI should be mentioned. SI stands for Solvent /nfluence (in Russian, BP for Ejmstsae FacTBopHTena). This parameter was introduced by Shmidt et al. in 1967 and was derived from the study of many different extraction equilibria, i.e. of the distribution of organic and inorganic compounds between two immiscible liquid phases [298-301]. It was found that in the extraction of metal salts using various extraction reagents, the distribution coefficients of the extractable compound depend on the specific electrophilic and/or nucleophilic properties of the solvents used as diluent. From a large number of well-studied extraction systems, Eq. (7-12d) has been derived. 

Yan [38] further simplified the equations for batch extractions by assuming an irreversible, first-order extraction reaction between the solute and the carrier, irreversible first-order stripping reaction between the complex carrier and the internal reagent and constant distribution coefficients. Weiss et al. [39] proposed an empirical model for the extraction of mercury. Recently, Baneijea et al. [40] and Chakraborty et al. [4] presented an unsteady-state mathematical models to explain type 2 facilitation. 

The distribution coefficient, or partition coefficient, Kj, (v/w), parameterises the ratio of adsorbed particulate concentration, P (w/w), to dissolved concentration, C of a constituent, and may be determined empirically... 

Methodology for acquiring such information is relatively well-established and interpretations are typically noncontroversial. Results, however, are limited to the system studied. This may be adequate for the task at hand, but the prospect of individually describing every sorbate/sorbent combination usually encourages attempts toward predictive modeling whereby applicability is broadened to systems not actually studied. Prediction, in turn, requires information not directly available from empirical studies sorption mechanisms must be deduced and system parameters such as rate constants and distribution coefficients must be defined. In many cases, thermodynamic properties of the system are also useful for modeling input. 

The smoothed bootstrap has been proposed to deal with the discreteness of the empirical distribution function (F) when there are small sample sizes (A < 15). For this approach one must smooth the empirical distribution function and then bootstrap samples are drawn from the smoothed empirical distribution function, for example, from a kernel density estimate. However, it is evident that the proper selection of the smoothing parameter (h) is important so that oversmoothing or undersmoothing does not occur. It is difficult to know the most appropriate value for h and once the value for h is assigned it influences the variability and thus makes characterizing the variability terms of the model impossible. There are few studies where the smoothed bootstrap has been applied (21,27,28). In one such study the improvement in the correlation coefficient when compared to the standard non-parametric bootstrap was modest (21). Therefore, the value and behavior of the smoothed bootstrap are not clear. 

A commonly used system measures — directly or otherwise — partition between water and octan-l-ol to derive the distribution coefficient (Pow), and then applies an empirical formula to translate these values into bioconcentration factors (BCF) using a range of benchmark compounds. As would be expected, the numerical relationships depend on the organism used so that different equations result. Some equations that have been used for different organisms are the following (Mackay 1982 Hawker and Connell 1986) ... 

In preliminary design work, it is convenient to correlate distribution coefficients on a mass-fraction basis. An empirical correlation technique that is simple to use and often highly effective is... 

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