Lever parameter

Correlations of Metallonitrosyl Properties with the Lever Parameter. 161... 

The dependence of the Pi ligand parameter on the type of binding metal center is a limitation of the Pickett s model, but a similar difficulty is encountered for the Ei Lever parameter (see in the following) for which the need for corrections has been recognized for ligands such as CO and CNR. Such a type of limitation is inherent to any additive model that tries to separate the effects of ligands and metal centers on the redox potential, and add them as independent components, in contrast with the situation of the real molecule in which those effects are mutually dependent. 

The El Lever parameter was also shown [71] to correlate linearly with other parameters that measure the net electron-donor character of a ligand (L), namely the Tolman s electronic parameter TEP) [81] for phosphines and a computed electronic parameter CEP) [71] based (as TEP) on the infrared A v CO) frequency in complexes [NiL(CO)3], which is determined by the electronic effect of L. 

L. Perrin, E. Clot, O. Eisenstein, J. Loch and R. H. Crabtree. Computed ligand electronic parameters from quantum chemistry and their relation to Tolman parameters. Lever parameters, and Hammett constants. Inorg. Chem. 40, 2001, 5806-5811. 

In the majority of cases, the tests are conducted using a dead-weight lever-arm stress-rupture rig with an electric timer to determine the moment of fracture, but a variety of test rigs similar to those shown in Fig. 8.89g are also used. The evaluation of embrittlement may be based on a delayed-failure diagram in which the applied nominal stress versus time to failure is plotted (Fig. 8.103) or the specimen may be stressed to a predetermined value (say 75% of the ultimate notched tensile strength) and is considered not to be embrittled if it shows no evidence of cracking within a predetermined time (say 500 h). Troiano considers that the nature of delayed fracture failure can be described by four parameters (see Fig. 8.103) ... 

Finally, a further refinement of the Lever equation40 allows one, in applying the treatment of ligand electrochemical parameters EL, to account for the presence of eventual aromatic or aliphatic ligands bearing different substituents, i.e. to take into account the Hammett, or Taft, substituent parameters a (above discussed) for a certain ligand. The expression is ... 

Fio. 43. Retention factor fo leveral amino acida data conform to Eq. (92) as shown by the curvee drawn using parameters given by Kroeff and Pietrzyk. The eluites are L-phenylalanine (L-Phe) oL leucine (ot-Leu) OL>valine (OL-Val), and OL-alanine (OL-Ala). Date were taken with 0.04 M phosidwte buftr at 0.2 M ionic strength with NaCl. Reputed with permission from Kroeff and Pietrzyk (208). Anal. Chem. Copyri t 1978 by the American Chemical Society. 

Values of those parameters have already been proposed for many ligands (Ei) and a considerable number of metal centers (S m and 7m), mainly by Lever [64-67, 72] and occasionally by others [14, 15, 24, 27, 29, 35, 38, 41, 48, 71, 73-77]. Examples concerning the former parameter are listed in Table 9 and an extensive list for many carbyne, carbene, vinylidene, allenylidene and alkynyl ligands has been proposed recently by Pombeiro [38]. Tables 10-12 list the proposed S m and 7m values for the binding metal centers so far studied. 

The Lever s model has also been extended to sandwich and half-sandwich complexes with 7T-cyclopentadienyl or 7r-arene ligands [66, 67, 69]. The parameter for the 7r-ligands has been defined [69] on the basis of the low spin Fe hH redox couple, by Eqs. (21) or (22), for homolep-tic sandwich [Fe( 7r-L)2] or mixed sandwich ]Fe( 7r-Li)( 7r-L2)] complexes, respectively. 

Hence, these models are somehow of complementary characters and efforts should be directed toward their developments and wider applications, including those situations in which the additivity fails. Extensions of the Pickett s and Lever s models have already been proposed to overcome their limitations, as discussed above, and in some cases the dependency of the ligand parameters on the metal centers and generalizations to types (with different electron-counts, structures, compositions, etc.) of complexes, metal centers and ligands not included in the original proposals, have already been successfully treated. 

Lever has successfully predicted Mn"/ potentials of 24 Mn-carbonyl complexes containing halide, pseudohalide, isonitrile, and phosphine co-ligands, with additivity parameters derived from the potentials of Ru "/" couples [39]. An important consideration for heteroleptic complexes is the influence of isomerism on redox thermodynamics. For Mn(CO) (CNR)6- complexes, with n = 2 or 3, the Mn"/ potentials for cis/trans and fac/mer pairs differ by as much as 0.2 V [40]. The effect arises from the different a-donor and 7r-acceptor abilities of carbonyl (CO) and isocyanide and their influence on the energy of the highest energy occupied molecular orbital (HOMO). 

Bertini et al.497 and Lever et al.495 measured the electronic spectra of a series of tetragonal nickel(II) ammine complexes and found a relationship between the Ni—N distance and the ea parameter, e decreasing when the Ni—N distance increases. 

Recall that in the moment approach, each phase a is parameterized by Lagrange multipliers kf for the original moments (the ones appearing in the excess free energy of the system) and the fraction of system volume that it occupies. If extra moments are used, there is one additional Lagrange multiplier A, for each of them these are common to all phases. These parameters have to be chosen such that the pressure (44) and the moment chemical potentials nt given by Eq. (42) are equal in all phases. Furthermore, the (fractional) phase volumes v have to sum to one, and the lever rule has to be satisfied for all moments (both original and extra) ... 

This feature took the form of a parameter representing the degree of revenue-maximizing behaviour. The value of this parameter (between 0 and 1 within the mathematical framework of the model) was set so that, where there was a possibility of companies increasing their profits as a result of the policy levers, this would never be implemented in a manner such that it would attract new entry and therefore be self-defeating, i.e. that the firms exhibit limit pricing (see later discussion on profit maximization). As a result, new entry was only assumed to take place when companies did not have the opportunity of increasing their profits as a result of the policy levers introduced. 

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