Average valence

A fraction y of the Mn4+ ions are replaced by Mn3+. This fraction determines the average valence of the manganese atoms. For each Mn,+ there is a further OH ion in the lattice, replacing an O2 anion in the coordination sphere of the Mn3+ cation. A schematic drawing of the Rue-tschi model is shown in Fig. 4. 

White tin, to which the valency 2-44 has been assigned (Pauling 1938), does not contain bivalent Sn B alone, but also some Sn A. It seems likely that these two kinds of tin atoms occur in the ratio 3 1, leading to the average valency 2-5, a value in essential agreement with that suggested before. 

Zinc and cadmium have the A 3 structure, but with abnormally large axial ratio (1-856 for zinc) instead of the value 1-633 corresponding to close packing. From the distances 2-660 A (for six bonds) and 2-907 A (for the other six) the bond numbers 0-54 and 0-21 have been deduced. If the axial ratio were normal each of the twelve bonds formed by a zinc atom with v = 4-5 would have bond number f. The assumption of the distorted structure permits a split into two classes with the more stable bond numbers and (or, probably, with the average valency of zinc equal to 4). 

The number of ways in which n bonds are arranged around an atom with average valence v in a crystal involves a binomial coefficient ... 

Number of structures per atom with n bonds, average valence v... 

It is evident that the calculation of the number of resonating structures must be made in a different way from that for hypoelectronic metals, because M+ and M form the same number of bonds and are therefore classed together in the calculation of the number of ways of distributing the bonds. We consider first the valence v of a hyperelectronic metal whose neutral atoms form z bonds and whose ions M+ and M" form z + 1 bonds. For any atom, with average valence v, the number of structures, b, having n bonds, is, by the assumption used previously (4), proportional to the probability given by the binomial distribution ... 

The following simplified treatment is presented to illustrate some roughly quantitative aspects of the theory. The value of 0 is taken to be constant, with AO = 120°. The interaction constant p is taken as 0.36 yaV22ipiARi, in which pt is the fraction of ions i in the crystal and ARt is the change in radius. The quantity v v, the cube of the average valence for the metal or alloy, is an approximate representation of the force constant k of the bonds, which enters linearly in the expression for V. The coefficient z has the value +1 for M+ and —1 for M. The number 0.36 has been introduced to give agreement with the observed... 

The structures of the prototype borides, carbides, and nitrides yield high values for the valence electron densities of these compounds. This accounts for their high elastic stiffnesses, and hardnesses. As a first approximation, they may be considered to be metals with extra valence electrons (from the metalloids) that increase their average valence electron densities. The evidence for this is that their bulk modili fall on the same correlation line (B versus VED) as the simple metals. This correlation line is given in Gilman (2003). 

A general agreement may be noticed this may highlight the relevance, and the limits, of the role played by the average valence-electron number per atom... 

Hall et al. (63) found that the active species in the hydrogenolysis of cyclopropane are Mo(IV) in reduced M0O3-AI2O3 catalysts. Also, Burwell and Bowman found that the hydrogenolysis of cyclopropane at 100 C (64) and also propane at 300°C (65) occurs over Mo(IV), Mo(II), and Mo(0) catalysts, which were prepared from Mo(CO)g on AI2O3. The average valence state... 

To illustrate the use of the qualitative ionization scale given in Figure 17, consider the case of layered Li(Nii/2Mni/2)02. Assuming that the oxidation states of Li and O are +1 and —2, respectively (calculated to be true except in extremely electron-deficient cases), Ni and Mn must hold an average valence of +3. Moving up from the bottom of the scale one notes Mnoct is ranked lower than This... 

Some semiconducting compounds can be of the II-VI type, which also has an average valence of four, but these have much more ionic character than Ill-V compounds. Their band gaps are thus larger, and in some cases they may even be viewed as insulators. For example, ZnS, with a band-gap energy of 3.6 eV, is an insulator, whereas ZnSe has an band gap of 2.8 eV, which is closer to a semiconductor. A wide variety of... 

Matsunaga (15) applied the magnetic techniques of Eischens and Selwood and the chemisorption and chemical techniques of Voltz and Weller to a series of chromia-alumina catalysts. He found that in the limit of low chromia contents, where Eischens and Selwood deduced a two-dimensional distribution of chromium ions, treatment with oxygen at 450°C. resulted in an average valence number of six for all of the chromium ions in the sample. 

Nickel and chromium mixed-phase rutile pigments are industrially important. When nickel and chromium are substituted into the rutile lattice, higher valency metals (e.g., antimony, niobium, or tungsten) must also be substituted to maintain an average valency of four. 

Averaged valence electrons per atom for the alloy tested. 

Module type and average valence state of the lanthanide higher oxides... 

It should be noted in this connection that, for ethylene polymerisation in the presence of a non-supported catalyst such as TiCl4-AlR3 (R = Et, z -Bu), polymerisation proceeds at the maximum rate when the average valence state of titanium is 2 [304], However, while catalysts with divalent titanium compounds as precursors are known to be highly active for the polymerisation of ethylene, their activities for the polymerisation of a-olefins are generally low [51,240],... 

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