Distribution coefficient equation

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. 

Table VI summarizes values of the activity coefficient ratio Ygr-/YC].- in the saturated solution for each average solid composition (as calculated from the model of Table II), the calculated provisional equilibrium distribution coefficient (Equation 12) and the provisional equilibrium aqueous solution activity ratio of Br to Cl- (Equation 13) based on the data of Table V.

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

A simple review [6] of the definition of a distribution coefficient (equation 5) enables the calculation of the column capacity for a cation from equilibrium batch tests, provided that the concentration of the ion being scavenged by the zeolite is low in the solution being treated. This is particularly useful when removal of radioisotopes from aqueous nuclear waste is the intended use [30]. 

Boundary conditions with distribution coefficient. Equations for boundary conditions in Eqs. (7.7-7) and (7.7-10) were derived for the distribution coefficient K given in Eq. 

The equilibrium constant for this reaction hears a strong resemblance to the Henderson and Kracek compound distribution coefficient (equation 5) The equilibrium constant in (l6) should be less dependent on liquid composition than (9)> since the aQ2-in the liquid phase, which is difficTilt to estimate, cancels. Likewise, the compound distribution coefficients should show less... 

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

The distribution coefficient, represents the fractional volume of a specific stationary phase explored by a given solute, represented by equation 3 ... 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

Nonideal Separations. In numerous iastances, the ideal equations 2 and 4 have been verified experimentally. However, ia other experiments different results were obtained, reflecting failure of one ore more of the assumptions made ia deriving equations 2 and 4. Likewise, much theoretical work is concerned with modified assumptions, iucluding varyiag distribution coefficient k (19), eutectic-forming phase behavior (4,20—21), varyiag mass of 2one (22), and soHd-state diffusion (23). 

Equation (1) can be viewed in an over-simplistic manner and it might be assumed that it would be relatively easy to calculate the retention volume of a solute from the distribution coefficient, which, in turn, could be calculated from a knowledge of the standard enthalpy and standard entropy of distribution. Unfortunately, these properties of a distribution system are bulk properties. They represent, in a single measurement, the net effect of a large number of different types of molecular interactions which, individually, are almost impossible to separately identify and assess quantitatively. 

In the experiments of Katz et al., that validated the relationship given in equation (10), the distribution coefficients (K) were referred to the solvent phase (mobile... 

It should be recalled that the distribution coefficients are referenced to the solvent mixture and not the stationary phase and are thus the inverse of the distribution coefficient employed in the chromatography elution equation. 

Bearing in mind that the distribution coefficients given in the above equation are the reciprocal of the distribution coefficients given by Katz et al, then... 

Testing the applicability of equation (10) to liquids where the solvent components associate with themselves is experimentally difficult. Katz et al. attempted to do this by measuring the distribution coefficients of some solutes between hydrocarbon and... 

It is seen from the above equation that the band velocity is inversely proportional to the distribution coefficient with respect to the stationary phase. It follows that any changes in the distribution coefficient (K) will result directly in changes in the band velocity (Z). Consequently, if the isotherm is linear, then all concentrations will travel at the same velocity and the peak will be symmetrical. 

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 rate at which two constituents separate in the column is determined by the ratio of the two corresponding distribution coefficients, where the distribution coefficient is given by the equation... 

The distribution coefficient can be determined by batch experiments in which a small known quantity of resin is shaken with a solution containing a known concentration of the solute, followed by analysis of the two phases after equilibrium has been attained. The separation factor, a, is used as a measure of the chromatographic separation possible and is given by the equation,... 

An important relationship exists between the weight distribution coefficient and the volume of eluant (Vmax) required to reach the maximum concentration of an eluted ion in the effluent. This is given by the equation ... 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Eqs. (1-76) show that these values of the coefficients produce the Navier-Stokes approximations to pzz and qz [see Eq. (1-63)] the other components of p and q may be found from coefficient equations similar to Eq. (1-86) and (1-87) (or, by a rotation of coordinate axes). The first approximation to the distribution function (for this case of Maxwell molecules) is ... 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

This equation has been used for estimating migration velocities of radionuclides (e.g. 66). Here Pr is the density of the rock (kg/m3), p the density of water, e the fissure porosity, af the specific surface of fissures in the bedrock (m2/m3) and ap the specific surface of particles used in the Kd determinations (m2/m3). The distribution coefficient Kd represents ar. equilibrium value for the particular rock under the pertinent conditions. 

Coefficients, distribution—See Distribution coefficients Coefficients of free energy of formation equation, Pu oxide vapor. 127/... 

The problem is now solved. To find the F-value for the electron distribution of equation (20) with given nv n2, and l, it is necessary to expand the bracketed expression in equation (32), to collect the imaginary terms, and multiply the coefficient of un l x vn l 1 by the factors given in equations (32) and (21). 

In fundamental SEC studies retention is often described in terms of a distribution coefficient. The theoretical distribution coefficient Kg is defined as the ratio of solute concentration inside and outside of the packing pores under size exclusion conditions. The experimental distribution coefficient as defined in Equation 1, is a measurable quantity that can be used to check the theory. 

For a series of n types of interaetions, the expression of the distribution coefficient is given by Equation 4.15 ... 

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