Cone-angle profile

Analyses of cone-angle radial profiles have been used to correlate multinu-clear NMR data with spatial regions of steric overlap (43). Cone-angle radial profiles have the potential to be used to predict relative stabilities of cis and trans isomers and to predict coordination numbers of bulky ligands. However, cone-angle profiles are very sensitive to the conformation of a ligand. 

STM and AFM profiles distort the shape of a particle because the side of the tip rides up on the particle. This effect can be corrected for. Consider, say, a spherical gold particle on a smooth surface. The sphere may be truncated, that is, the center may be a distance q above the surface, where q < r, the radius of the sphere. Assume the tip to be a cone of cone angle a. The observed profile in the vertical plane containing the center of the sphere will be a rounded hump of base width 2d and height h. Calculate q and r for the case where a - 32° and d and h are 275 nm and 300 nm, respectively. Note Chapter XVI, Ref. 133a. Can you show how to obtain the relevent equation ... 

Fig. 2.8.7 (a) Implementation of cone-and-plate device, (b) Velocity image taken across a vertical slice, (c) Velocity profile taken across the cell with 7° cone angle. Note the highly linear variation of velocity (adapted from Ref. [15]). 

For complex chemical systems, the analysis of ligand-property control leads to similar profiles. For the above mentioned ligand-concentration control of the system 1,3-dimethylallyl-methyl-nickel/P-ligands/CO, the corresponding profile is shown in Fig. 1 of Scheme 3.5-5. The increase of the cone angle and of the acceptor strength (high X values) favours C-C bond formation over C=0 insertion (15 over 16) ... 

The second variable feature of phosphines is the steric profile. An approximate measure of this is given by the Tollman cone angle (0, Figure 2.2). This angle is derived by considering the cone which would be defined by the van der Waals surface generated by a phosphine bound... 

Figure 12 Oliver and Smith s presentation of a ligand profile showing the variation of van der Waals contacts with the rotation () about the M-P bond resulting in cone angle 0) variations. (Reprinted with permission from Ref. 86. 1978 American Chemical Society)...

Problem 1.2 (Worked Example) Compute the shear-rate profile in a cone-and-plate geometry. For a cone angle a of 0.1 radians, what percentage increase occurs in shear rate as one migrates from the plate to the cone Hint Look at the component of the momentum balance equation in spherical coordinates.)... 

Figure 4 Generation of a ligand profile for a PR3 ligand. As the ligand rotates about the M-P bond, (j), the half cone angle varies (shown on left). The plot of half cone angle versus (j) is the ligand profile (shown on right).

Figure 5 Generation of a cone-angle radial profile, (a) A sphere of radius d (a variable) is allowed to grow from the metal toward the ligand, (b) The plot of cone angle (or solid angle see Sec. 3.4) versus radial distance, d, is called the cone-angle radial profile. (From Ref. 48.)...

Figure 6 Three-dimensional cone-angle radial profiles, (a) A circle of radius cARP is plotted at each distance, d(see Fig. 5). (b) At each distance d, a surface of cARP versus (f> (the M-P rotational angle see Fig. 4) is plotted. (From Ref. 29.)...

In order to demonstrate the performance of the Doppler-burst envelope integral value method for the estimation of the instantaneous particle velocity vector and the particle mass flux or concentration, measurements were performed in a liquid spray issuing from a hollow cone pressure atomizer (cone angle 60°) and a swirling flow which exhibits complex particle trajectories (Sommerfeld and Qiu 1993). All the measurements were conducted using the one-component phase-Doppler anemometer. The integration of the mass flux profiles provided the dispersed phase mass flow rate which agreed to 10 % with independent measurements of the mass flow rate (Sommerfeld and Qiu 1995). 

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