Thermodynamics distribution coefficient

The thermodynamic distribution coefficient is introduced when one of the components can be considered as a solute in each phase, and when we choose the reference states of that component to be the infinitely dilute solution in each phase. For discussion, we designate the first and second components as those that form the solvents and the third component as the solute. Equations (10.245), (10.246), (10,248), and (10.249) are still applicable when we choose the pure liquid phase as the standard state for each of the two components. When we introduce expressions for the chemical potential of the third component into Equation (10.247), Equation (10.250) becomes... 

With the definition of the thermodynamic distribution coefficient, Equation (10.252) can be written as... 

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. 

The corresponding thermodynamic distribution coefficient K is defined as the concentration of component (X) in the stationary phase divided by its concentration in the mobile phase. 

Using the thermodynamic distribution coefficient for the choice of the best LLE parameters for quantitative extraction of pesticides from water is usually not done. The time to determine the distribution coefficient of a pesticide under different water quality conditions and with different solvents has seemed to justify picking these parameters arbitrarily. Then the parameters chosen are tested by fortifying, usually in distilled water. As indicated above, the arbitrary approach does not work well. A systematic procedure based upon the distribution coefficient is strongly suggested to replace the arbitrary approach. In time a Standard Method for choosing the parameters of LLE for quantitative pesticide analysis may evolve from careful comparative study of parameters. 

In order to achieve separation, one must first have retention. Earlier, the thermodynamic distribution coefficient was defined as a measure of the degree of retention for compound X. The capacity factor k is a more practical quantity that can be determined directly from the chromatogram. It is given by... 

In Figure 8.2-9 the phase diagram for an eutectic solidifying binary nrixture is shown. According to the given phase diagram (binary system without formation of mixed crystals) this soUd should contain less A than the feed. In this case the thermodynamic distribution coefficient for an impurity component i is defined by... 

Kinetics of mass transfer prevents a growing crystal from achieving the thermodynamically possible value of the distribution coefficient. Figure 8.2-12 shows that the impurities enrich close to a growing crystal surface (phase boundary Ph). Due to the single-sided mass transfer, impurity i enriches near the growing phase boundary to y,. The definition of the thermodynamic distribution coefficient... 

The thermodynamic distribution coefficient is only a constant in the region close to the target component, that is, at ideal dilution with respect to the impurity (solidus and liquidus lines are linear). For values of keq equal to unity, no purification is achieved for keq less than unity, a purification can be obtained with highest purity for keq close to zero. 

The thermodynamic distribution coefficient fe,.q characterizes the maximum purity achievable by crystallization it is a substance-spedfic parameter. Under real process conditions, the amount of impurities incorporated in the crystallized material usually exceeds the amount predicted from thermodynamics. This arises from different factors related to nonequilibrium, that is, kinetically controlled growth processes that commonly occur in technical crystallization processes. 

Burton et al. [4, 5] introduced an expression that quantitatively relates the effective distribution coefficient keff to the thermodynamic distribution coefficient keq (Equation 7.4) ... 

Retention in i-LC is thus directly related to the thermodynamic distribution coefficient. K can be expressed in terms of fundamental thermodynamic properties, that is, the partial molar free energy associated with the transfer of one mole of analyte from the mobile phase to the stationary phase (Ag), the corresponding partial molar enthalpy, and the corresponding entropy effect (As), according to Eq. (2), where R is the gas constant and T the absolute temperature. 

By measuring the retention volume of a solute, the distribution coefficient can be obtained. The distribution coefficient, determined over a range of temperatures, is often used to determine the thermodynamic properties of the system this will be discussed later. From a chromatography point of view, thermodynamic studies are also employed as a diagnostic tool to examine the actual nature of the distribution. The use of thermodynamics for this purpose will be a subject of discussion in the next chapter. It follows that the accurate measurement of (VV) can be extremely... 

The distribution coefficient is an equilibrium constant and, therefore, is subject to the usual thermodynamic treatment of equilibrium systems. By expressing the distribution coefficient in terms of the standard free energy of solute exchange between the phases, the nature of the distribution can be understood and the influence of temperature on the coefficient revealed. However, the distribution of a solute between two phases can also be considered at the molecular level. It is clear that if a solute is distributed more extensively in one phase than the other, then the interactive forces that occur between the solute molecules and the molecules of that phase will be greater than the complementary forces between the solute molecules and those of the other phase. Thus, distribution can be considered to be as a result of differential molecular forces and the magnitude and nature of those intermolecular forces will determine the magnitude of the respective distribution coefficients. Both these explanations of solute distribution will be considered in this chapter, but the classical thermodynamic explanation of distribution will be treated first. 

Classical thermodynamics gives an expression that relates the equilibrium constant (the distribution coefficient (K)) to the change in free energy of a solute when transferring from one phase to the other. The derivation of this relationship is fairly straightforward, but will not be given here, as it is well explained in virtually all books on classical physical chemistry [1,2]. 

In contrast molecular interaction kinetic studies can explain and predict changes that are brought about by modifying the composition of either or both phases and, thus, could be used to optimize separations from basic retention data. Interaction kinetics can also take into account molecular association, either between components or with themselves, and contained in one or both the phases. Nevertheless, to use volume fraction data to predict retention, values for the distribution coefficients of each solute between the pure phases themselves are required. At this time, the interaction kinetic theory is as useless as thermodynamics for predicting specific distribution coefficients and absolute values for retention. Nevertheless, it does provide a rational basis on which to explain the effect of mixed solvents on solute retention. 

The basic principle of chromatography separations can be described by thermodynamics using the distribution coefficient K (12) ... 

Diphenylcarbazide as adsorption indicator, 358 as colorimetric reagent, 687 Diphenylthiocarbazone see Dithizone Direct reading emission spectrometer 775 Dispensers (liquid) 84 Displacement titrations 278 borate ion with a strong acid, 278 carbonate ion with a strong acid, 278 choice of indicators for, 279, 280 Dissociation (ionisation) constant 23, 31 calculations involving, 34 D. of for a complex ion, (v) 602 for an indicator, (s) 718 of polyprotic acids, 33 values for acids and bases in water, (T) 832 true or thermodynamic, 23 Distribution coefficient 162, 195 and per cent extraction, 165 Distribution ratio 162 Dithiol 693, 695, 697 Dithizone 171, 178... 

The classic thermodynamic expression for the distribution coefficient (K) of a solute between two phases is given by... 

Liquid-Fluid Equilibria Nearly all binary liquid-fluid phase diagrams can be conveniently placed in one of six classes (Prausnitz, Licntenthaler, and de Azevedo, Molecular Thermodynamics of Fluid Phase Blquilibria, 3d ed., Prentice-Hall, Upper Saddle River, N.J., 1998). Two-phase regions are represented by an area and three-phase regions by a line. In class I, the two components are completely miscible, and a single critical mixture curve connects their criticsu points. Other classes may include intersections between three phase lines and critical curves. For a ternary wstem, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature due to the compressibility of the solvent. 

To a first approximation the three terms in equation (1.46) and (1.47) can be treated as independent variables. For a fixed value of n Figure 1.8 Indicates the influence of the separation factor and capacity factor on the observed resolution, when the separation factor equals 1.0 there is no possibility of any separation. The separation factor is a function of the distribution coefficients of the solutes, that is the thermodynamic properties of the system, and without some... 

Distribution coefficient (Kd)—Describes the distribution of a chemical between the solid and aqueous phase at thermodynamic equilibrium, is given as follows ... 

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. 

Thermodynamic calculations based on the compositional dependence of the equilibrium constant are applied to solubility data in the KCl-KBr-H20 system at 25°C. The experimental distribution coefficient and activity ratio of Br /Cl in solution is within a factor of two of the calculated equilibrium values for compositions containing 19 to 73 mole percent KBr, but based on an assessment of uncertainties in the data, the solid solution system is clearly not at equilibrium after 3-4 weeks of recrystallization. Solid solutions containing less than 19 and more than 73 mole percent KBr are significantly farther from equilibrium. As the highly soluble salts are expected to reach equilibrium most easily, considerable caution should be exercised before reaching the conclusion that equilibrium is established in other low-temperature solid solution-aqueous solution systems. 

Finally, it is not appropriate to derive thermodynamic properties of solid solutions from experimental distribution coefficients unless it can be shown independently that equilibrium has been established. One possible exception applies to trace substitution where the assumptions of stoichiometric saturation and unit activity for the predominant component allow close approximation of equilibrium behavior for the trace components (9). The method of Thorstenson and Plummer (10) based on the compositional dependence of the equilibrium constant, as used in this study, is well suited to testing equilibrium for all solid solution compositions. However, because equilibrium has not been found, the thermodynamic properties of the KCl-KBr solid solutions remain provisional until the observed compositional dependence of the equilibrium constant can be verified. One means of verification is the demonstration that recrystallization in the KCl-KBr-H20 system occurs at stoichiometric saturation. 

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