Step rule

When a compound that can form several modifications crystallizes, first a modification may form that is thermodynamically unstable under the given conditions afterwards it converts to the more stable form (Ostwald step rule). Selenium is an example when elemental selenium forms by a chemical reaction in solution, it precipitates in a red modification that consists of Se8 molecules this then converts slowly into the stable, gray form that consists of polymeric chain molecules. Potassium nitrate is another example at room temperature J3-KN03 is stable, but above 128 °C a-KNOs is stable. From an aqueous solution at room temperature a-KN03 crystallizes first, then, after a short while or when triggered by the slightest mechanical stress, it transforms to )3-KN03. 

We might take a purist s approach and attempt to use kinetic theory to describe the dissolution and precipitation of each mineral that might appear in the calculation. Such an approach, although appealing and conceptually correct, is seldom practical. The database required to support the calculation would have to include rate laws for every possible reaction mechanism for each of perhaps hundreds of minerals. Even unstable minerals that can be neglected in equilibrium models would have to be included in the database, since they might well form in a kinetic model (see Section 26.4, Ostwald s Step Rule). If we are to allow new minerals to form, furthermore, it will be necessary to describe how quickly each mineral can nucleate on each possible substrate. 

Ostwald s step rule holds that a thermodynamically unstable mineral reacts over time to form a sequence of progressively more stable minerals (e.g., Morse and Casey, 1988 Steefel and Van Cappellen, 1990 Nordeng and Sibley, 1994). The step rule is observed to operate, especially at low temperature, in a number of min-eralogic systems, including the carbonates, silica polymorphs, iron and manganese oxides, iron sulfides, phosphates, clay minerals, and zeolites. 

Various theories, ranging from qualitative interpretations to those rooted in irreversible thermodynamics and geochemical kinetics, have been put forward to explain the step rule. A kinetic interpretation of the phenomenon, as proposed by Morse and Casey (1988), may provide the most insight. According to this interpretation, Ostwald s sequence results from the interplay of the differing reactivities of the various phases in the sequence, as represented by Ts and k+ in Equation 26.1, and the thermodynamic drive for their dissolution and precipitation of each phase, represented by the (1 — Q/K) term. 

To investigate the kinetic explanation for the step rule, we model the reaction... 

Nordeng, S. H. and D. E Sibley, 1994, Dolomite stoichiometry and Ostwald s step rule. Geochimica et Cosmochimica Acta 58, 191-196. 

The Ostwald Step Rule, or the rule of stages postulates that the precipitate with the highest solubility, i.e., the least stable solid phase will form first in a consecutive precipitation reaction. This rule is very well documented mineral formation via precursors and intermediates can be explained by the kinetics of the nucleation process. The precipitation sequence results because the nucleation of a more soluble... 

The "classical" theory of nucleation concentrates primarily on calculating the nucleation free energy barrier, AG. Chemical interactions are included under the form of thermodynamic quantities, such as the surface tension. A link with chemistry is made by relating the surface tension to the solubility which provides a kinetic explanation of the Ostwald Step Rule and the often observed disequilibrium conditions in natural systems. Can the chemical model be complemented and expanded by considering specific chemical interactions (surface complex formation) of the components of the cluster with the surface ... 

Ostwald proposed that when two or more new phases may form from existing phase or phases, that is, when new phases are more stable than the existing phase(s), the least stable new phase would form first and then transform into more stable phases. This is called the Ostwald rule, the Ostwald step rule, or the law of successive reactions. An alternative statement of the Ostwald rule is as follows ... 

Secondary phases predicted by thermochemical models may not form in weathered ash materials due to kinetic constraints or non-equilibrium conditions. It is therefore incorrect to assume that equilibrium concentrations of elements predicted by geochemical models always represent maximum leachate concentrations that will be generated from the wastes, as stated by Rai et al. (1987a, b 1988) and often repeated by other authors. In weathering systems, kinetic constraints commonly prevent the precipitation of the most stable solid phase for many elements, leading to increasing concentrations of these elements in natural solutions and precipitation of metastable amorphous phases. Over time, the metastable phases convert to thermodynamically stable phases by a process explained by the Guy-Lussac-Ostwald (GLO) step rule, also known as Ostwald ripening (Steefel Van Cappellen 1990). The importance of time (i.e., kinetics) is often overlooked due to a lack of kinetic data for mineral dissolution/... 

But as the extraction of sulfur by zinc from the disulfide solid solution series is much faster than the disulfides can equilibrate, the step rule is surpassed (Ostwaldsche Stufenregel). These phenomena were described in detail by Moh4. ... 

The application of the GLO Step Rule and, for that matter, the interpretation of activity-ratio diagrams in general are influenced by the existence of varying degrees of crystallinity or of particle size in soil minerals, with a corresponding variation in their solubility.15 For example, in the case of Fig. 3.5, very poorly crystallized forms of gibbsite and kaolinite, alluded to above, would require... 

B. S. Hemingway, Gibbs free energies of formation for bayerite, nordstrandite, Al(OH)2+, and Al(OH)2 aluminum mobility, and formation of bauxites and laterites, Adv. Phys. Geochem. 2 285(1982). The mechanistic basis of the GLO Step Rule is discussed in Chap. 6 of W. Stumm, Chemistry of the Solid-Water Interface, Wiley, New York, 1992. See also Chap. 5 in W. Stumm and J. J. Morgan, op. cit.12... 

In 1897, Friedrich Wilhelm Ostwald (1853-1932) published his now famous study of crystallization processes, which led to the Ostwald rule of stages or Ostwald step rule (Ostwald, 1897). Ostwald noticed that the course of transformation of unstable (or metastable) states into stable states normally occurs in stages,... 

Use a high-resolution step as soon as possible. This rule is in accord with the preceding use of the minimum number of separation steps rule, which anticipates that product yields decline exponentially. 

The preceding information can serve as an introduction to the methods of linear programming including the step-by-step rule approach used for a simplex algorithm. The reader is referred to any of the many standard texts on linear programming for proof of the theorems and rules used in this treatment and further extensions of the methods of linear programming. ... 

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