Supersaturation thermodynamic analysis

Apart from the purely thermodynamic analysis, the description of the -> electro crystallization phenomena requires special consideration of the kinetics of nucleus formation [i-v]. Accounting for the discrete character of the clusters size alteration at small dimensions the atomistic nucleation theory shows that the super saturation dependence of the stationary nucleation rate /0 is a broken straight line (Figure 2) representing the intervals of Ap within which different clusters play the role of critical nuclei. Thus, [Ap, Apn is the supersaturation interval within which the nc -atomic cluster is the critical nucleus formed with a maximal thermodynamic work AG (nc). 

The wet synthesis of CdS nanoparticles used in this work is based on the reaction between a dissolved cadmium salt (CdCl2) and a S-containing compound (thiourea (NH2)2CS) in an aqueous solution. Chemical deposition of CdS nanoparticles in the CdCl2 - NH3 - NaOH - (NH2)2CS - H2O bath was described elsewhere [3]. In the present work all the baths had the same composition and were prepared from solutions of cadmium chloride CdCl2 (0.005 mold-1), ammonia NH3-H2O (1.5 moll"1), sodium hydroxide NaOH (0.074 mold-1) and thiourea (NH2)2CS (0.025 mol-F1) using distilled water. The synthesis temperature was varied from 294 to 325 K. The primary concentrations of the precursors have been chosen according to the thermodynamic analysis [4]. A supersaturation of the solution with Cd(OH)2 takes place in the baths. It means that the mechanism of the cadmium sulfide formation could involve the stage of Cd(OH)2 formation. When the deposition process of CdS particles in the solution completed, the residue was filtered at an ambient pressure and dried at room temperature. 

Veverka, V., Sohnel, O., Bennema, P. and Garside, J. (1991) Concentration gradients in supersaturated solutions a thermodynamic analysis. AIChEJ, 37, 490-498. 

In the context of the refined , free-of-drawbacks approach, only one way can be proposed concentration zones from da to da(a ) and from to (/S ), formed in the process of meta-quasi-equilibrium diffusion, are metastable and must eventually decay a into a+, /S into /S-i-1. At that, the problem of intermediate phases rise at interdiffusion is reduced to the problem of supersaturated solid solutions decomposition. A thermodynamic analysis of this situation was made in [39]. The kinetic difficulties here are presented by concentration supersaturations c -Ca and which may turn out to be quite small when compared to the differences required for new phase formation. 

To circumvent such seemingly insurmountable difficulties, approximations and simplifications can often be made. For example, even though the formation of a solid from a saturated solution can involve several successive reactions, one may judiciously select for analysis or prediction the one that presents the slowest rate. The so-called Oswald s phase rule states that a supersaturated solution that undeigoes a sudden alteration that takes it out of such a state will produce a metastable solid (instead of the expected thermodynamically stable solid). It is a very useful rule, although it is not always obeyed. Three limiting cases are shown in Figure 5.8 for different values of the overall crystallization rate (nucleation -1- crystal growth), v. 

From this analysis it is clear that the trade-off between kinetics and thermodynamics is not at all obvious. Consider a monotropic, dimorphic system (for simplicity) whose solubility diagram is shown schematically in Fig. 2.10. It is quite clear that for the occurrence domain given by solution compositions and temperatures that lie between the form II and I solubility curves only polymorph I can crystallize. However, the outcome of an isothermal crystallization that follows the crystallization pathway indicated by the vector in Fig. 2.10 is not so obvious since the initial solution is now supersaturated with respect to both polymorphic structures, with thermodynamics favouring form I and kinetics (i.e. supersaturation) form II. 

Having in mind Equation 13.3, such hysteresis cannot be explained by thermodynamics alone but has to involve a description of the kinetics of first-order phase transformation as well. In order to develop a theoretical description, we appHed the previous thermodynamic approach to the study of the kinetic decoding of back and forth transitions during temperature cycling of nanopowders and presented a numerical analysis within the framework of the standard kinetic equation approach [93-95]. As a continuation of our approximation in Section 13.4, we arbitrarily assumed that the new thermodynamically advantageous phase has strict stoichiometry Ci = 0.5. The composition in each initially supersaturated solid particle is equal to Co (so that the new phase nucleus has a composition different from that of the parent phase, Ci Co). 

An interesting kinetic study deals with the solution-mediated phase transformation of COT and COD into the thermodynamically stable COM [50]. The experimental conditions were adjusted so that either COT or mixtures of COD and COM crystallized initially as confirmed by X-ray diffraction powder patterns. The systems were then aged in contact with the mother liquid, and the transformation of COT or COD into COM was followed by monitoring the total crystal volume as a function of time (by Coulter counter) and determining (by thermo-gravimetric analysis) the relative proportion of the crystal hydrates at fixed time intervals. In addition, supersaturation profiles (i.e., activity products) were determined by solution calcium analysis. In all cases the transformation was completed within approximately 80-100 h. 

Mo is removed from sulfidic waters by sorption of thiomolybdate ions on sinking particles. Rather, they postulated that Mo is incorporated into an Fe-Mo sulfide mineral phase that precipitates in euxinic waters below the depth where FeS becomes supersaturated and that solubility of this phase controls the Mo concentration in the sulfidic water column. This may be the same Fe-Mo phase that was experimentally precipitated by Flelz et al. [35] and was shown by EXAFS analysis to contain Mo tetrahedrally coordinated by S, as appears to be the case in some samples of black shale. To understand better the conditions in which Mo might precipitate as an Fe-Mo sulfide, Helz et al. [17] used thermodynamic methods to calculate the solubility of the putative sulfide phase as a function of pH, [Mo], and [H2S]. Their calculations predicted that only a narrow range of pH-[Mo]-[H2S] conditions will result in complete insolubility of Mo. Their model succeeded in predicting the Mo concentrations in deep waters in several other modern anoxic and euxinic areas. Helz et al. [17] concluded that the complete removal of Mo from the water column in the Black Sea is not representative of the general behavior of Mo in euxinic basins over time, but is instead the exception to the rule. 

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