Cross relationship

By 1945, Stacey speculated about the possibility of a structural relationship between Pneumococcus capsular polysaccharides and those produced by other organisms. With Miss Schliichterer, he had examined the capsular polysaccharide of Rhizobium radicicolum. This polysaccharide gave a precipitin reaction in high dilution, not only with Type III Pneumococcus antiserum, but also mixed with antisera from other Pneumococcus types. The chemical evidence indicated that the polysaccharide resembled the specific polysaccharides of Types I and II Pneumococcus. A decade later, the acidic capsular polysaccharide from Azoto-bacter chroococcum, a soil organism, was studied. It, too, produced serological cross-reactions with certain pneumococcal specific antisera. Although the molecular structure of the polysaccharide was not established, adequate evidence was accumulated to show a structural relationship to Type III Pneumococcus-specific polysaccharide. This was sufficiently close to account for the Type III serological cross-relationship. 

Crossing relationships among North American and eastern Asian populations of... 

If self-exchange rate constants for the Cu(II/I) couple are calculated by applying the Marcus cross relationship to the observed second-order... 

The requisite value for k x was only approximately defined by the fitting procedures, and because of uncertainty in the standard potential for the Br2/Br redox couple it was likewise deemed unsuitable to use the value of kx and the principle of detailed balancing to derive the value of k x. Further reason to be doubtful of the derived value of k x was a major disagreement between it and the value predicted by the cross relationship of Marcus theory. 

A literature value for E° for the SCH2COO / SCH2COO redox couple (0.74 V) was then used in conjunction with the cross relationship of Marcus theory to derive a self-exchange rate constant of 1.5 x 105 M-1 s-1 for the SCH2COO / SCH2COO redox couple. 

From the discussion it also follows that there Is a certain freedom of choice regarding the units Into which measured quantities are expressed. Recall, however, the restriction that if cross-relationships are to be derived on the basis of Onsager s reciprocal relationships (secs. 1.6.2b and c) the fluxes J. resulting from forces X, must have such dimensions that the product XJ has the dimensions of entropy production per unit time and unit volume (SI units J m K" s ). 

We have noticed a correlation between the logarithm of the selfexchange rate constant for a series of [CoNe] couples and the degree of substitution at the coordinated amines as judged by the number of amine protons in the complex (Fig. 12). It should be noted that more than half the points that fall on the line have been estimated by application of the cross relationship to the rate of a single cross reaction with (102). Not all the points in the plot are homoleptic complexes some mixed ligand CoNe complexes were included because of the need to try to verify the unexpected observation. The line in Fig. 12 was... 

Cross-Relationships Evaluation of Rate Constants from Isotopic Rate Constants... 

The operative interest of such a cross-relationship is extreme, since when associated with Eqs. (78) through (80) it allows an a priori estimation of the rate constant of any outer sphere electron transfer reaction provided the corresponding standard reduction potentials and isotopic rate constants are known. Yet a caveat in this approach is the necessity that /c = 1 for all reactions, that is, that there is always sufficient overlap between the orbitals [62]. 

A number of ostensibly outer-sphere reactions involving the O2 /02 redox couple were considered to have anomalous rates that were inconsistent with the Marcus model, but many of these deviations are now recognized to arise from size disparities between the small 02 radical and the large reaction partners these size disparities affect the solvent reorganization energies in ways that are not accounted for in the simple Marcus cross relationship.74... 

Coordinated ligands can also undergo one-electron reduction when C02 is the reducing agent. This occurs for the coordinated bpz ligand in [Ru(bpz)3]2 +, 192 and also for an extensive series of related complexes.193 The rates of these reactions depend on driving force in conformity with the Marcus cross relationship. 

In data sets with a large number of variables, collinear data and missing values, projection models based on latent structures, such as Principal Component Analysis (PC A) (6) (7) (1) and Partial Least Squares (PLS) (8) (9) (10), are valuable tools within EDA. Projection models and the set of tools used in combination simplify the analysis of complex data sets, pointing out to special observations (outliers), clusters of similar observations, groups of related variables, and crossed relationships between specific observations and variables. All this information is of paramount importance to improve data knowledge. 

The acid dependence observed in the reduction of trans-[Co(Me4[14]tetraeneN4)(N3)2] by Fe (aq) is attributed to the greater reactivity of the oxidant upon protonation of an azide ligand. The application of the Marcus cross-relationship to the data for the reduction of [Co(tmen)3] (tmen = tetramethylethylenediamine) by [Ru(H20)6] " yields a self-exchange rate constant of 10 s for the [Co(tmen)3] couple. Studies of the spectroscopic... 

The electron exchange rate constants [Rh(dmpe)3]" / " couples have been determined to be 2 x 10 and 4 x 10 M" s", respectively, from the appliction of the Marcus cross-relationship to the reactions with several ruthenium(II) pentaammine complexes. The relative values are consistent with the differences in the M—P bond distance changes (Ado = 0.068 A for Tc and 0.054 A for Re) determined by EXAFS measurements. 

Electrochemical measurements of the Cu(II/I) potentials with the nS4 ligands (n = 12-16) indicate that the Cu(II) and Cu(I) species each exist in two different conformational states [170]. Conformational rearrangement may either precede or succeed electron transfer. Rorabacher and coworkers interpreted their results in light of a square mechanistic scheme that neatly reconciles the sweep rate dependence of the cyclic voltammograms with the requisite change in coordination geometry at Cu. Kinetic studies on the electron transfer [149, 170, 176-177] support this scheme application of the Marcus cross relationship to reduction of Cu(II) and oxidation of Cu(I) yields widely discrepant values, presumably because of the different conformational states involved. 

One of the most important results to evolve from the theoretical treatment of Marcus is now referred to as the Marcus cross relationship. This important relationship was developed later by Ratner and Levine from a thermodynamic perspective, and this formulation provides a simple basis for understanding some of the concepts and assumptions in the more microscopic molecular theory of Marcus that is described later. 

The cross relationship involves the relationship between the free energies or rate constants for the following reactions, where (6.11) and (6.12) are called self-exchange reactions and (6.13) is called the cross reaction ... 

This is the Marcus cross relationship in terms of Ifree energies. 

The thermodynamic development of the cross relationship depends on the assumptions that ... 

This is the cross relationship in terms of rate constants. There is often reason to believe (or need to assume) that = 1, and then Eq. (6.20) reduces to what is often called the simplified Marcus cross relationship. This is particularly useful because a knowledge of any three of the values, AB AA BB o " AB Hows One to predict the fourth. 

The preceding development illustrates the assumptions that are necessary to develop the cross relationship. The more detailed theory that follows provides further understanding in terms of the molecular properties of the reactants and solvent. 

From the detailed theory, Marcus recognized certain simplifications that led to a cross relationship of the same form as that developed by Ratner and Levine. In Marcus terms, this relationship is given by... 

Table 63. Comparison of Some Obsaved Rate Constants (M s", 25 C) with Those Calculated from the Marcus Cross Relationship...

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