Geometrical Similarity

Figure 15 shows a stereoscopic view of the crystalline 1 1 complex where R7 = i-CsHn and R8 = (CH2)2Ph 9). The packing mode of the four molecules in the unit cell of this complex corresponds to the association scheme of tetramer 17 (Fig. 8). Of particular interest is that a pair of groups with similar geometrical shape, NMe2 and CHMe2 [part of C6H4NMe2 and (CH2)2CHMe2], are in close contact.

Calderbank et al. (C1-C4), who worked with systems quite similar geometrically to that of Yoshida and Miura, found that the average bubble diameter for air in water at 15°C ranged from 3 to 5 mm. Westerterp et al. (W2-W4) found the range to be 1-5 mm for air in sodium sulfite solution at 30°C. In addition, they noted that any increase in interfacial area between the bubbles and the liquid was due primarily to the increase in gas holdup, and the average bubble diameter was essentially unaffected by the impeller speed and was approximately 4.5 mm (W3). 

Increase the tube diameter, either to maintain a constant pressure drop or to scale with geometric similarity. Geometric similarity for a tube means keeping the same length-to-diameter ratio L/dt upon scaleup. Scaling with a constant pressure drop will lower the length-to-diameter ratio if the flow is turbulent. 

Dynamic. similarity exists if the systems are similar geometrically and kinematically, and moreover, the ratios of forces between corresponding points are equal in both systems (similar field of forces). 

This effect, called the third body effect by Conway and co-workers [101], is however controversial [102], The main argument against this theory is the fact that there is a specificity of catalytic behavior for each kind of metal adatom. Even adatoms producing similar geometrical blocking effects, present different catalytic properties. So, for instance, tin and lead [97] occupy two Pt atoms, but tin produces... 

Gas phase calculations on the CS state of 22 also predict a large change in the cavity size, from 10 A in the ground state, to only about 6 A in the CS state, and that this change is mainly caused by out-of-plane bending of both DC V and DMAn groups.1381 Similar geometric distortions are predicted for other U-shaped dyads in their CS states. 

The restricted effects of the enzymes may therefore be explained by the assumption that the approach of the molecules that cause the chemical process can occur only in the case ofa similar geometric shape. 

I Systems mainly isomorphous to each other - the systems with almost the same number of heterogeneous atoms and summarily similar geometric shapes of interacting orbitals. 

In this section some geometrical factors are discussed, that are observed to influence the adaption of structure types. By analogy and based on similar geometrical arguments prediction of isostructural componnds has been possible and should be further so in most cases. 

Owing to the very similar geometrical shape, size, and equal charge between Cu(CN)2[N=C-Cu-C=N for X-Cu-Y, with obtuse bond angle of 120.1°] and N(CN)2[N=C-N-C=N, bond angle 116.7°], they were nearly freely replaceable with each other in the anion layer, resulting in comparable lattice parameters among k-(ET)2Cu(CN)3, c-(ET)2Cu(CN)[N(CN)]2, and their alloy, K -salt (Fig. 21a-c). 

Although abundant evidence supports the existence of snch an antoimmnne phenomenon, the causative event that heralds this self-directed immune-mediated attack remains uncertain. A number of mechanisms have been proposed to afford a molecular-level explanation of autoimmunity. One such explanation is molecular mimicry. Molecular mimicry occurs when a protein associated with a foreign substance bears structural similarities to a protein found in the host. For example, if a person experiences an infection from bacteria, there is a possibility that a protein in the bacterium shares certain similar geometrical and conformational features with a protein already existing in the person. Thus, an immune response directed against the bacteria will cross-react with organs in the host organism. 

Two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe them have the same numerical value (n, = identical or idem). 

Dimensional analysis is a method for producing dimensionless numbers that completely characterize the process. The analysis can be applied even when the equations governing the process are not known. According to the theory of models, two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe the process have the same numerical value [2], The scale-up procedure, then, is simple express the process using a complete set of dimensionless numbers, and try to match them at different scales. This dimensionless space in which the measurements are presented or measured will make the process scale invariant. 

The calculations described are organized into sections defined by the general formulae of the species. This is to some extent an arbitrary division but serves to group together those molecules of similar geometrical structure. As mentioned above, discussion will be restricted usually to work carried out during the period 1973—6. 

Similar geometric optimization has been reported for bicyclo[3.2.2]nona-6,8-diene (BND). The double bond situated in the opposite direction to the methylene group was found to be more exo-pyramidalized than the other double bond and the electron density (qi, HOMO) of the former double bond in HOMO of the molecule higher than that of the latter double bond. The exo and endo faces of exo-pyramidalized double bonds proved not to be equal and the electron density was found to be higher on the endo faces. The endo molecular complexes with bromine have been found by the HF/321G method to be more stable than their exo congeners this was attributed to electronic and steric factors. As a result, endo-facial stereoselectivity of bromination ( ) predominates.21 A related theoretical study of facial selectivity and regioselectivity of the electrophilic addition of chlorine to exo-tricyclo[4.2.1.02,5]nona-3,7-diene (exo-TND) has also been reported.22... 

Product inhibition occurs if the catalyst binds to the product with a similar tightness to the substrate. In a similar geometric approach of reactants, however, catalysis was successful (Mackay, 1994). Besides the above-mentioned Diels-Alder reaction, the substrate-free porphyrin trimer also catalyzes the acetylation of 4-hydroxymethylpyridine with N-acetylimidazole. Although both reactants are bound to the porphyrin trimer, their ground states are bound less tightly than the... 

---