Reduced coordinates

Figure 6.43 (a) Ring-puckering potential function for cyclopentene. The reduced coordinate z is... 

TABLE 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (T, = T/Tj P, = P/Pc) ... 

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

Reciprocity relationship, radiation 448 Rectangular duct 87 Redman, J. 550, 566 Reduced coordinates 35... 

Simple examples of A, / and reactions are seen to have the same rate law. There remains another important mechanism that differs. This is the pathway through an intermediate of reduced coordination number that is possible if a transition state like d of Fig. 3 occurs. This path may be called dissociative, D. The mechanism may be represented as... 

Intermediate of increased One-step process Intermediate of reduced coordination number coordination number... 

In the particular case of diatomic molecules, the molecular geometry can be described by the reduced coordinate... 

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p[Pg.94]

Figure 3.1b, d, and f shows graphs similar to Figure 3.1a, c, and e, but the coordinates lie in a reduced square having unit dimensions. The reduced coordinates x and y are... 

In aqueous solution, a metal complex is enclosed by several solvation spheres involving many solvent molecules. As recently shown, for mechanistic studies a reduced coordination sphere is sufficient (157,159), and therefore here inclusion of a third solvent molecule is sufficient. Addition of a third water molecule results in the exothermic formation of [Be(H20)2(L)--H20] and [Be(H20)2(LH)--H20]+, where the third hydrogen-bonded... 

We have delineated viable coordinated ligand reactions and their attendant intermediates for the stoichiometric conversion of CO ligands selectively to the C2 organics ethane, ethylene, methyl (or ethyl) acetate, and acetaldehyde. We now outline results from three lines of research (1) T -Alkoxymethyl iron complexes CpFe(C0)2CH20R (2) are available by reducing coordinated CO on CpFe(C0)3+ (1) [Cp = r -CsHs]. Compounds 2 then form t -alkoxyacetyl complexes via migratory-insertion (i,e. CO... 

Once a particular surface structure has been determined to be stable (i.e., autocompensated), the primary factor determining the nature of the surface reconstruction is the energy that can be gained by rehybridizing the surface dangling bond charge density in response to the reduced coordination at the surface. As a consequence, a charge transfer between the atoms at the surface takes place and this results in the formation of new bonds between surface atoms adsorbed to the surface (also known as adatoms). The formation of new bonds on the surface leads to different chemical and physical properties at the surface.11... 

In this approach, the potential energy V is a function of the reduced coordinates and the -matrix. For the kinetic energy, one would only be interested in the motion of the particle relative to the distorted geometry so that a suitable Lagrange function Lq for the system would read as follows ... 

Schematic representation of the cross section of the surface layer of a metal oxide. , Metal ions O, oxide ions. The metal ions in the surface layer (a) have a reduced coordination number. They thus behave as Lewis acids. In the presence of water the surface metal ions may first tend to coordinate H20 molecules (b). For most of the oxides dissociative chemisorption of water molecules (c) seems energetically favored.

Langford and Gray proposed in 1965 (13) a mechanistic classification for ligand substitution reactions, which is now generally accepted and summarized here for convenience. In their classification they divided ligand substitution reactions into three categories of stoichiometric mechanisms associative (A) where an intermediate of increased coordination number can be detected, dissociative (D) where an intermediate of reduced coordination number can be detected, and interchange (I) where there is no kinetically detectable intermediate [Eqs. (2)-(4)]. In Eqs. (2)-(4), MX -i and... 

Figure 5.19 shows the evolution of the gel volume fraction as a function of reduced time K(t — to). The solid line represents the theoretical curve obtained using Eq. (5.22). For (p values between 20% and 80%, the theoretical curve is roughly linear with a slope of 1 (see dashed line). Equivalently, within the same (p range, the volume fraction should vary linearly with time with a slope equal to K. The experimental data (Fig. 5.17) were recalculated in order to be plotted in reduced coordinates. For each initial volume fraction, K is deduced from the initial slope of the curve (p = f t) (for cp between roughly 20% and 60%). All the data lie within a unique curve that is in reasonable agreement with the theoretical one. 

Some information is available to identify neurological effects in humans from inhalation exposures. The available data indicate that coordination and concentration difficulties, headache, intoxication, and/or anorexia may be induced by inhalation of JP-5 vapor (Porter 1990), headaches may be induced by diesel fuel vapor (Reidenberg et al. 1964), and sensory impairment may be induced by deodorized kerosene vapor (Carpenter et al. 1976). In animals, a few studies were found that document neurological effects from inhalation of fuel oils. Acute inhalation of diesel fuel no. 2 vapor produced behavioral changes, tremors, ataxia, reduced coordination, and increased sensitivity to heat in mice (Kainz and White 1984). 

The results above are consistent with the original proposal of Eorster on the importance of the formation of intermediate neutral, possibly reduced coordination num-... 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

It has always been of some interest to examine the extent to which associative activation dominates the mechanism of substitution of four-coordinate planar cP metal complexes. The coordination unsaturation of these formally 16-electron valence shell complexes predicts that a substitution pathway with increased coordination number (18-electron valence shell) will be favoured over one with a reduced coordination number (14 electrons). This was well understood by workers in the field438 long before Tolman94 published his rules. The first attempt to force a dissociative mechanism was made by Basolo and Baddley513>514 who reasoned that since the steric requirements of associative substitution (rates reduced by steric hindrance from the cis position) were opposite to those of a dissociative mechanism (rates increased or unchanged by increased steric hindrance), sufficient congestion in the substrate should reduce the rate of the associative process to the point where dissociative activation took over. If this did not produce a change in mechanism it could at least indicate a lower limit to the difference of the two modes of activation. 

To relieve the strain of sterically demanding ligands, a metal often remains coordinatively unsaturated. Copper(I) halides and phosphines form cubane-like metal cluster compounds, Lm(CuX) ,12 With the bulky trimesitylphosphine, a monomeric two-coordinate [CuBr(Pmes)3] is formed, Br—Cu—P = 173.7°.252 The d(Cu—P) of 2.193 A is comparable to that in normal tet-rameric complexes, but d(Cu—Br) at 2.225 A is shorter, no doubt due to the reduced coordination number. Heating crowded complexes can also result in a reduction in coordination number (see equation 65). 

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