The Principle of Reaction Invariants

Thus far, the attainable region has been shown for the analysis of systems with two key compositions to be tracked. In the following, the principle of reaction invariants is used to reduce the composition space in systems of larger dimension. 

Let the reacting system consist of n, moles of each species i, each containing Ojj atoms of element j. The molar changes in each of the species due to reaction are combined in the vector An, and the coefficients Oy form the atom matrix A, noting that since the number of gram- 

Assuming that A is square and nonsingular, an expression for the changes in the number of moles of each dependent species is obtained by algebraic manipulation  

The dimension of i is equal to the number of species minus the number of elements (atoms) in the species. When this dimension is two or less, the principle of reaction invariants permits the application of the attainable region to complex reaction systems. This is illustrated in the following example, introduced by Omtveit et al. (1994). 

RUMPLE 6.5 Attainable Region for Steam Reforming of Methane 

---

Glasser et al. (1987) and Hildebrandt et al. (1990) demonstrated this two-dimensional approach on a number of small reactor network problems, with better results than previously reported. Moreover, Omtveit and Lien (1993) were able to consider higher-dimensional problems as well through projections in concentration space that allow a complete two-dimensional represention. These projections were accomplished through the principle of reaction invariants (Fjeld et al., 1974) and the imposition of system specific constraints. 

By evoking the principle of reaction invariants, the number of species that need to be tracked for this system is reduced to two so that the attainable region can be shown in two dimensions. Accordingly,... 

In contrast to the well-known Wigner-Witmer rule [24] that multiplicity is retained during a chemical reaction, the origin of the spin exclusion principle is both the retention and the distortion of symmetry. The fact is that from the viewpoint of rigorous invariance the initially singlet system cannot spontaneously pass into the triplet state. Spin-orbital and spin-lattice interactions probably play the main role in this transition. 

However, it is these very procedures which exploit the unique capability of DSC. There has been something of a drive towards obtaining kinetic constants from a single dynamic experiment. Although the results obtained in this way may fulfil a useful function further tests are invariably needed to explore the possibility of limitations to their applicability. Advantages have been claimed for sample controlled kinetic experiments in which the experimental conditions are varied in order to maintain the rate of reaction constant. This has proved a popular method of temperature control in thermogravimetry although in principle it can be applied to DSC. 

Growth. Thin films are invariably grown on surfaces. Catalysis. A catalyst is either the surface itself or it is supported on a surface, with usually one or both of these components being a ceramic. Reactions occur at reactive sites, but the principle of catalysis is that the material facilitates, but is not chemically changed by, the process. 

The only theory of the precipitation reaction which, following the program begun by Arrhenius, has been developed by straightforward application of the principles of chemical equilibrium is that of Pauling, Pressman, Campbell and Ikeda (19). This theory applies only to relatively simple systems, namely, those composed of bivalent antigen and bivalent antibody, univalent hapten, certain soluble complexes, and precipitate with invariant composition AB. 

The goal of discrete lumping is to group species based on their reactivity so that each lump follows the principle of invariant response the rate of reaction of the lump should depend only on the sum of the species and not on the individual species it contains. For a system of unimolecular, reversible reactions, equation (1) becomes... 

For Sn2 reactions in solution, there are four main principles that govern the effect of the nucleophile on the rate, though the nucleophilicity order is not invariant but depends on substrate, solvent, leaving group, and so on. 

The concept of minimum AE and maximum Emw is illustrated with the generalized sequence shown in Scheme 4.7 under stoichiometric conditions with complete recovery of reaction solvents, catalysts, and post-reaction materials. Markush structures are used to show both variable R groups and necessarily invariant atoms. This analysis is useful in studying combinatorial hbraries where a constant scaffold structure is selected and then is decorated with, in principle, an unlimited number of possible R groups. 

The writers have found in their laboratory that invariably after a certain burnoff (depending upon the reactor, temperature, and sample), a subsequent extended period of constant reaction rate, expressed in grams of carbon reacting per unit time, is attained. In this bumoff region, there obviously is equilibrium between the rate of formation of the surface-oxygen complex and its removal with a carbon atom. It is felt that this is the reaction rate most characteristic of a given temperature and should be used in kinetic calculations. In principle, Wicke (31) concurs with this reasoning and reports reactivity data only after the sample has attained a total surface area which is virtually constant. 

A distinction between solid/fluid and solid/solid boundaries is irrelevant from the point of view of transport theory. Solid/fluid boundaries in reacting systems are, for example, (A,B)/A, B, X (aq) or (A,B)/X2(g). More important is the distinction according to the number of components. In isothermal binary systems, the boundary is invariant if local equilibrium prevails. In higher than binary systems, the state of the a/fi interface is, in principle, variable and will be determined by the reaction kinetics, including the diffusion in the adjacent bulk phases. 

Eqns. (32) give (s - 1) conditions that must be satisfied and though they are generally non-linear in concentration they can in principle be solved to give as many as (s - 1) of the concentrations ca, a = s + 1,..., s, in terms of the others. In addition there are the (s - 5) invariants of the mechanism which again may yield as many as (s - r) of the ca in terms of the cs, but it is impossible to say in general how many of such relations are independent. Between them the invariants and Eqns. (31) might determine as many as (s -r + s - 1) of the (s - s) ca s and if the actual number so determined is not sufficient existential hypotheses must be introduced to provide further equations. The kinetics of the reaction 01 can then be made entire by substitution for the c° in Eqn. (33) which then becomes a function of the cs. 

---