Stoicheiometry

Thiophosphoryl chloride and phenylphosphonothioic dichloride give adducts with a,a -bipyridyl and ethylenediamine, the stoicheiometry... 

With cobalt(ii) chloride, complexes of the monophosphazene, PhgP=NH (= L), of stoicheiometry CoCl2L,THF and C0CI2L2 have been isolated. Their i.r. and u.v. spectra were compared with analogous complexes of triphenylphosphine oxide. 

An approximate indication of likely instability in a compound may be gained from inspection of the empirical molecular formula to establish stoicheiometry. 

Diamantopoulou, E. et al., Polyhedron, 1994, 13(10), 1593 A study of the 1-methylbenzotriazole complexes, of various stoicheiometry and hydration, showed several to be explosive. 

The above evidence strongly suggests that the pseudocationic reactions involve the ester 1-phenylethyl perchlorate and its oligomeric homologues as catalyst. It also shows that the ester is only stable when an excess of styrene is present in the reaction mixture. Spectroscopic and conductimetric studies on the present system confirmed this interpretation and indicated that at least four molecules of styrene are required for the stabilisation of one molecule of ester. Details of the experiments carried out to investigate the stoicheiometry of ester stabilisation will be given in a later paper. The mode of this stabilisation is not clear at present and we do not known the location of the four styrene molecules with respect to the ester. 

The kinetics and stoicheiometry of the oxidation of oxovanadium(iv) by peroxide have been studied and continuous-flow e.s.r. techniques used to investigate the intermediate which appears to be a vanadium(v) complex with a paramagnetic ligand, formulated as either [OVOO,aq] or [V02(H02). aq]. Six- and seven-co-ordinate vanadium(v) has been identified by. Y-ray crystallography in the peroxo-complexes (NH4)[V0(02)2-Nh ]36o (NH4)4[0 V0(02)2 2] respcctively. and seven-co-ordination... 

The heats and entropies of formation of the sulphato-complexes [Cr(en)3-(S0.3) ] " (n = 1—3) have been reported and the complex with n = 1 has been shown to be the most stable. The reaction of cis- and tra 5-[Cr(en)2 Bt2] with DMSO affords mainly c/s-[Cr(en)2Br(DMSO)] and c/s-[Cr(en)2-(DMS0)2], which have been isolated, as have the thermodynamically less stable corresponding traus-complexes. The stoicheiometry and rate of aquation of rrn s-[Cr(en)2F2] have been determined and the sole pro-... 

The complexes [Mn(L2)(02PH2)2] (L2 = o-phen or bipy) have been re-ported and phenylphosphonic acid forms complexes with Mn in which ligand-bridging gives rise to cross-linked polymers of stoicheiometry MnL... 

M. L. Sinnot and P. J. Smith, Affinity labelling with deaminatively generated carbonium ion. Kinetics and stoicheiometry of die alkylation of methionine-500 of die lacZ p-galactosidase of Escherichia coli by p-D-galactopyranosylmethyl-p-nitrophenyltriazene, Biochem. J., 175 (1978) 525—538. 

Examples of such oxidations are listed in Table 2.3 with furanoses listed first in a roughly alphabetical arrangement mainly by substituent groups of products. Some large-scale oxidations (> 1 g) are listed in the first column. The effective oxidant in most cases is RuO, with a few by [RuO ] or [RuO ] . For oxidations by RuO the overall stoicheiometry of the reaction was shown [312] to be... 

No exact stoicheiometry can be given as the boron-containing product(s) have not been identified. 

By stoicheiometry we understand the calculus of changes in composition that take place by reaction it corresponds to kinematics in the analogy with continuum mechanics for its provides the framework within which chemical motions must take place, irrespective of the forces that bring them about. By kinetics we understand the relations that govern the speed of the composition changes and this bears some analogy to the dynamics of continua. Just as the latter can only be built on a proper understanding of the kinematics, so the analysis of stoicheiometry must precede that of kinetics. 

In the s dimensional space of concentrations the state of the system can be represented by the point c = (c1,. .., cs). Evidently this point can only move in an r dimensional subspace if the composition changes are due to only r independent reactions. The orientation of this subspace is given by the matrix of stoicheiometric coefficients, but its position is fixed by the initial composition. Indeed the stoicheiometry of the possible reactions can be determined from the atomic matrix by the following theorem this does not of course imply that all, or even any, of the possible reactions do take place. 

We wish to show next that the stoicheiometry may be deduced from a given set of kinetics to within the usual arbitrary rotation. Whilst the deduction of the stoicheiometry from the atomic matrix can only give the maximum number of reactions, the deduction from a given kinetics may show that not all the possible reactions are taking place. 

Theorem 14. Ifcs = csyss is any linear transformation of concentration space, the kinetics cs = gs describe the same set of reaction rates f as do the kinetics cs = gs = fasr, provided the stoicheiometry is given by asr = asryss. 

As noticed in a review of P [2]t, an important early reference had been overlooked, namely the two notes of Jouguet [3], Observations sur les principes et les theoremes generaux de la statique chimique (pp. 61-180) and Sur les lois de la dynamique chimique relatives aux sens des reactions irreversibles (pp. 181-194). In these papers (of which I was culpably unaware when writing P) Jouguet carefully establishes the notion of independence of reactions and the invariants of a system of reactions and considers the ramifications of the phase rule. In particular he is quite clear as to the way in which stoicheiometry lays a foundation for thermodynamics and in going on to discuss the nature of equilibrium he shows that its uniqueness is a consequence of its stability. Thus much of the content of 2-4 of P is to be found in Jouguet. His second note, however, is concerned with the direction... 

As well as the abstract account of stoicheiometry there is the metrical aspect. The mean composition in any region may be represented by a point... 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

Stoicheiometry is linear algebra, and the independence of reactions is an important concept. I wrote a paper with Richard Mah [33] that showed how the equations for the concentrations of all species could be reduced in number to that of the independent reactions. A note [38] of about this time suggested that, if a convention were needed, reactions might be orthonormalized. This is a notion, risible to the chemist but obvious to the mathematician, illustrates the difference between formal and practical chemistry.18... 

Happel s use of the stoicheiometric number (Isotopic assessment of catalysts. Academic Press. 1986) should also be mentioned. Arpad Petho has related stoicheiometry to dimensional analysis in Three Lectures on Linear Algebraic Methods in Chemical Engineering [see also Acta. Math. Acad. Sci. Hung. 18 19, (1967)]. 

A study of the 1-methylbenzotriazole complexes, of various stoicheiometry and hydration, showed several to be explosive. 

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