Distribution coefficients ideal solid solutions

One of the most successful applications of crystal field theory to transition metal chemistry, and the one that heralded the re-discovery of the theory by Orgel in 1952, has been the rationalization of observed thermodynamic properties of transition metal ions. Examples include explanations of trends in heats of hydration and lattice energies of transition metal compounds. These and other thermodynamic properties which are influenced by crystal field stabilization energies, including ideal solid-solution behaviour and distribution coefficients of transition metals between coexisting phases, are described in this chapter. 

There have been many modifications of this idealized model to account for variables such as the freezing rate and the degree of mix-ingin the liquid phase. For example, Burton et al. [J. Chem. Phy.s., 21, 1987 (1953)] reasoned that the solid rejects solute faster than it can diffuse into the bulk liquid. They proposed that the effect of the freezing rate and stirring could be explained hy the diffusion of solute through a stagnant film next to the solid interface. Their theoiy resulted in an expression for an effective distribution coefficient k f which could be used in Eq. (22-2) instead of k. 

The initial predictive method by Wilcox et al. (1941) was based on distribution coefficients (sometimes called Kvsi values) for hydrates on a water-free basis. With a substantial degree of intuition, Katz determined that hydrates were solid solutions that might be treated similar to an ideal liquid solution. Establishment of the Kvsj value (defined as the component mole fraction ratio in the gas to the hydrate phase) for each of a number of components enabled the user to determine the pressure and temperature of hydrate formation from mixtures. These Kysi value charts were generated in advance of the determination of hydrate crystal structure. The method is discussed in detail in Section 4.2.2. 

It is observed frommFigure 7 that, first,Pthe value of r icu is not unity and, second, that there exists a partial cancellation of the composition dependence in the liquid phase activity coefficient product by that found in the solid solution. The second observation suggests that the liquid and solid solution model selection process should be insensitive with respect to liquidus and solidus data alone. Indeed, the assumption of ideal solution behavior in both phases closely predicts the correct distribution coefficient, yet experimental measurements of the solution thermochemical properties clearly indicate moderate negative deviations from ideal behavior. 

The noncavitating pressure distribution for the Venturi is shown in Fig. 3. The data are plotted in terms of a pressure coefficient Cp as a function of the axial distance from the minimum pressure point. Cp is conventionally defined as the difference between the local wall and free-stream static-pressure head ijix — ho) divided by the velocity head F /2g. Free-stream conditions are measured in the approach section about 1 in. upstream from the quarter roimd. The solid line (Fig. 3) represents a computed ideal flow solution. The dashed line represents experimental data obtained with nitrogen and water in the cavitation tunnel and from a scaled-up aerodynamic model studied in a large wind tunnel. The experimental results shown are all for a Reynolds number of about 600,000. The data for the various fluids are in good agreement, especially in the critical minimum-pressure region. The experimental pressure distribution shown here is assumed to apply at incipient cavitation, or more exactly, to the single-phase liquid condition just prior to the first visible cavitation. 

It describes the thermodyrranric equiUbrium between the concentrations of impurities X in the solid and y, in the solution. The thermodyrranric eqirilibriirm is orrly achieved at low crystal growth rates v 0. In case of systems without formation of mixed crystals this distribution coefficient should ideally be close to zero. In reahty, however, the crystallized solid will not possess a distribution coefficient of... 

Note 5.2 (Distribution coefficient Kd and the classical diffusion equation). If the mixture of a fluid (i.e., solution) is dilute, which is what is assumed for an ideal solution, the adsorption can be treated as being the distribution in the solid phase, where a distribution coefficient Kd is introduced as follows ... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

---