S„ axis of symmetry

An S point group contains an S axis of symmetry. The group must contain also... 

A helical chain such as Se o is easy to recognize, but it is not always such a facile task to identify a chiral compound by attempting to convince oneself that it is, or is not, non-superposable on its mirror image. Symmetry considerations come to our aid a chiral molecular species must lack an improper (S ) axis of symmetry. 

A chiral molecule lacks an improper (S ) axis of symmetry. 

Performing a symmetry operation on a molecule gives a nuclear configuration that is physically indistinguishable from the original one. Flence the center of mass must have the same position in space before and after a symmetry operation. For the operation C , the only points that do not move are those on the C axis. Therefore, a C symmetry axis must pass through the molecular center of mass. Similarly, a center of symmetry must coincide with the center of mass a plane of symmetry and an S axis of symmetry must pass through the center of mass. The center of mass is the common intersection of all the molecular symmetry elements. 

An improper rotation S is actually a combination of two operations a rotation about a Cn axis and then a reflection through a plane which is horizontal with respect to the axis. This operation is defined as the two procedures together. The molecule has an S axis of symmetry if the combined rotation-reflection gives a result indistinguishable from the start point. After just the rotation the structure may be completely different from the start point neither the C axis nor the mirror plane need be symmetry elements themselves. 

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

A cr orbital can be formed either from two s atomic orbitals, or from one s and one p atomic orbital, or from two p atomic orbitals having a collinear axis of symmetry. The bond formed in this way is called a a bond. A n orbital is formed from two p atomic orbitals overlapping laterally. The resulting bond is called a n bond. For example in ethylene (CH2=CH2), the two carbon atoms are linked by one a and one n bond. Absorption of a photon of appropriate energy can promote one of the n electrons to an antibonding orbital denoted by n. The transition is then called Ti —> 7i. The promotion of a a electron requires a much higher energy (absorption in the far UV) and will not be considered here. 

In Equation 12.6 p, is the permanent dipole moment, h is Planck s constant, I the moment of inertia, j the angular momentum quantum number, and M and K the projection of the angular momentum on the electric field vector or axis of symmetry of the molecule, respectively. Obviously if the electric field strength is known, and the j state is reliably identified (this can be done using the Stark shift itself) it is possible to determine the dipole moment precisely. The high sensitivity of the method enables one to measure differences in dipole moments between isotopes and/or between ground and excited vibrational states (and in favorable cases dipole differences between rotational states). Dipole measurements precise to 0.001 D, or better, for moments in the range 0.5-2D are typical (Table 12.1). 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

Each 5ec-butyl group has a chiral C that can be R or S. Since all four groups are equivalent, the order of writing the designation is immaterial, RRRS is the same as RRSR. The possibilities are RRRR, RRRS, RRSS, SSRR, SSSR, and SSSS. RRRR and SSSS, and RRRS and SSSR, are enantiomeric pairs. These are the four optically active isomers. The mirror image of RRSS is SSRR. These are identical and therefore the isomer is meso. This isomer is a rare example of a compound which is achiral because it has only an improper axis of symmetry—it has no plane or center of symmetry. 

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

Disks fall into the second category of electrode, at which nonlinear diffusion occurs. The lines of flux to a disk electrode (Figure 12.2B) do not coincide with the simple geometries for which we derived Fick s second law, and the diffusion problem must therefore be expressed in two dimensions. Note that a line passing through the center of the disk and normal to the plane of the disk is a cylindrical axis of symmetry, so it is sensible to choose the radial distance from this axis as one of the coordinates for the problem. Diffusion along this radial coordinate, r, is described by Equation 12.7. Also, diffusion along the coordinate, x, normal to the plane of the electrode is described by Equation 12.4. Thus, the form of Fick s second law that must be solved for the disk is ... 

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