Point of symmetry

The first condition requires that the reactant concentration at the external surface of the catalyst be fixed. The second condition implies that there can be no diffusive flux through the center of the pellet, since this is a point of symmetry. 

The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (p, 0, 9) illustrated in Fig. 3-27. The relations are x =... 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

The moments of transition time of a dynamical system driven by noise, described by arbitrary potential cp(x) such that cp( oo) = oo, symmetric relatively to some point x = d, with initial delta-shaped distribution, located at the point xo < d [Fig. A 1(a)], coincides with the corresponding moments of the first passage time for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile [Fig. A 1(b)]. 

Calculating from (9.4) the values of arbitrary constants and putting them into (9.3), one can obtain the following value for the probability current Laplace transform G(z, s) (9.2) at the point of symmetry z — 0. 

Thus, we have proved the principle of conformity for both probability densities and moments of the transition time of symmetrical potential profile and FPT of the absorbing boundary located at the point of symmetry. 

Figure 3,10 Solvus and spinodal decomposition fields in regular (B) and subregular (D) mixtures. Gibbs free energy of mixing curves are plotted at various T conditions in upper part of figure (A and C, respectively). The critical temperature of unmixing (or consolute temperature ) is the highest T at which unmixing takes place and, in a regular mixture (B), is reached at the point of symmetry.

The absence of chirality. Achiral molecules have an internal plane or point of symmetry. A molecular configuration is said to be achiral when it is superimposable on its mirror image. See Chirality... 

Isomers that lack internal planes of symmetry or points of symmetry (hence, are chiral) due to a prevention (or significant hindrance) to rotation about a single bond. 

Because proteins are made up of L-amino acids, they exhibit chirality in their structures, lacking planes or points of symmetry. Proteins also can exhibit chirality in their interactions with other chiral molecules as well as prochiral centers in other molecules. This latter point is beautifully illustrated by fumarase s catalysis of the dehydration of L-malate, a molecule containing two seemingly equivalent hydrogen atoms ... 

A molecular structural property wherein a rotation of 180° leads to the same superimposable structure. Because proteins are made up of L-amino acid residues, neither a single subunit nor a multisubunit protein can possess a plane or point of symmetry. However, proteins do exhibit dyad symmetry. This structural property allows many DNA-binding proteins to make equivalent contacts to the two antiparallel chains of the nucleic acid. 

PRODUCT INHIBITION STUDIES Isomers lacking internal planes/points of symmetry,... 

A center (point) of symmetry is a point in the center of a molecule to which a line can be drawn from any atom such that, when extended an equal distance past the center, the line meets another atom of the same kind. 

A) with respect to the operation of inversion about the origin of the system. The electric dipole operator is antisymmetric (A) with respect to inversion at a point of symmetry. The electric quadrupole operator is inversion symmetric (S). A transition is allowed if the product function in the expression for transition moment is symmetric for electric dipole radiation and antisymmetric for electric quadrupole radiation. 

Turning takes place on a potter s wheel and is only suitable for so-called rotation symmetric shapes. This means that the objects which are made, such as plates, dishes and vases, are symmetric along an axis or point of symmetry. 

As you can see in figure 11.1.1 each point X on a rotation symmetric object has a mirror image X with respect to the axis or point of symmetry. 

Sterols are synthesized in nature from squalene and, therefore, ultimately from isoprene. Mevalonic acid is the immediate precursor of the isoprene unit, and the carboxylic acid group is lost as carbon dioxide when two mevalonic acid molecules combine head to tail. Thus, if the a carbon of mevalonic acid is labeled, then this carbon is always adjacent to the carbon bearing a side-chain methyl group. Examination of the way in which six isoprene units are linked in squalene (Example 6.2) shows that they are not all linked head to tail there is a point of symmetry in the structure of squalene (marked in the structure below). At this point a set of three isoprene units, linked head to tail, is joined head-to-head to a similar set of three isoprene units, to give the labeling pattern shown. 

The < vo-methylene structure of the 6-methylenedibenzo[<7,g][l,3,6]trithiocine 11 was determined by X-ray crystallographic analysis. Any axis, plane, or point of symmetry does not exist in the molecule in the crystalline state, although the two benzene rings and two < vo-methylene hydrogen atoms are equivalent in solution as is evident from the H NMR spectrum <1999T10057>. 

Here the tip of the fin, being a point of symmetry, is more convenient for the origin of x. The boundary conditions are then... 

One common characteristic of these length scales is that the dimensionless spatial coordinate is identically zero at the plane, axis, or point of symmetry, and r] reaches its maximum value of unity on the external surface of the catalyst. In terms of a universal correlation, a typical characteristic length scale for all catalyst shapes is L = V cataiyst/>S extemai, where Vcataiyst is the volume of catalyst and 5extemai IS the total external surface area. This definition of L reduces to ... 

---