Symmetry, centre planes

In a thin flat platelet, the mass transfer process is symmetrical about the centre-plane, and it is necessary to consider only one half of the particle. Furthermore, again from considerations of symmetry, the concentration gradient, and mass transfer rate, at the centre-plane will be zero. The governing equation for the steady-state process involving a first-order reaction is obtained by substituting De for D in equation 10.172 ... 

A question which may sometimes be asked is this If an enantio-morphous crystal- -that is, one possessing neither planes, nor inversion axes, nor a centre of symmetry—is dissolved in a solvent, does the solution necessarily rotate the plane of polarization of light The answer to this question is, Not necessarily . If the molecules or ions of which the crystal is composed are themselves enantiomorphous, then the solution will be optically active. But it must be remembered that enantiomorphous crystals may be built from non-centrosymmetric molecules which in isolation possess planes of symmetry—these planes of symmetry being ignored in the crystal structure such molecules in solution would not rotate the plane of polarization of light. (A molecule of this type, in isolation, may rotate the plane of polarization of light (see p. 91), but the mass of randomly oriented molecules in a solution would show no net rotation.) An example is sodium chlorate NaC103 the crystals are enantiomorphous and optically active, but the solution of the salt is inactive because the pyramidal chlorate ions (see Fig. 131) possess planes of symmetry. 

This may not be obvious in the normal drawing (which has a centre of symmetry), but rotation around the central C-C bond clearly shows the plane of symmetry. Neither plane nor centre of symmetry may be present in a chiral molecule, but a C2 axis is allowed (Chapter 16). 

In the normal alkanes (Table 31) one finds the alternation of the increase of melting point also well known in the fatty acids. This is a consequence of the difference in crystal structure of even and odd molecules resulting from a difference in symmetry (centre or plane through the middle). 

Chiral Used to describe a molecule that does not possess a centre of symmetry, a plane of symmetry, or an alternating axis of symmetry. Such a molecule will rotate plane polarised light. Chloral hydrate The water addition product of chloral, i.e. Cl3CCH(OH)2. 

The simplest symmetry elements are the centre, plane, and axes of symmetry. A cube, for example, is symmetrical about its body-centre, that is, every point (xyz) on its surface is matched by a point (xyz). It is said to possess a centre of symmetry or to be centrosymmetrical a tetrahedron does not possess this type of symmetry. Reflection of one-half of an object across a plane of symmetry (regarded as a mirror, hence the alternative name mirror plane) reproduces the other half. It can easily be checked that a cube has no fewer than nine planes of symmetry. The presence of an -fold axis of symmetry implies that the appearance of an object is the same after rotation through 3607 l a cube has six 2-fold, four 3-fold, and three 4-fold axes of symmetry. We postpone further discussion of the symmetry of finite solid bodies because we shall adopt a more general approach to the symmetry of repeating patterns which will eventually bring us back to a consideration of the symmetry of finite groups of points. 

No centre of symmetry or planes of symmetry have been identified and this alone is sufficient to confirm that molecular species in the point group are chiral. 

The symmetry elements in the hexagonal structure composed of liquid-like columns are shown in fig. 6.4.1. The director n is parallel to the column axis a (or equivalently a or a") is the lattice vector of the two-dimensional hexagonal lattice L, and 6 are two-fold axes of symmetry 3 is a three-fold axis and g a six-fold axis. Any point on or , is a centre of symmetry. Any planes normal to n and the planes ( 7 ) and (Lg, 0 are planes of symmetry. Defects in such a structure have been investigated in detail by Bouligand and by Kleman and Oswald. ... 

I) The segments selected from each parent structure should retain the additional symmetry elements (if there are any) in addition to the minimal symmetry (see 3.3.). These additional elements should be perpendicular to the stacking direction for a linear structure series, perpendicular to the mosaic plane for a two-dimensional series and in the case of a three-dimensional stoicture series the segments should contain the symmetry centre or/and symmetry axes. 

Molecule (13.60) has a three-fold symmetry axis but is devoid of an alternating symmetry axis (i.e. a plane of symmetry or a symmetry centre). It is dissymmetric and therefore optically active. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

The main symmetry elements in SFg can be shown, as in Figure 4.12(b), by considering the sulphur atom at the centre of a cube and a fluorine atom at the centre of each face. The three C4 axes are the three F-S-F directions, the four C3 axes are the body diagonals of the cube, the six C2 axes join the mid-points of diagonally opposite edges, the three df, planes are each halfway between opposite faces, and the six d planes join diagonally opposite edges of the cube. 

Molecules belonging to the 4 point group are very highly symmetrical, having 15 C2 axes, 10 C3 axes, 6 C5 axes, 15 n planes, 10 axes, 6 5io axes and a centre of inversion i. In addition to these symmetry elements are other elements which can be generated from them. 

In terms of the dimensions, a, b and t for the section, several area properties can be found about the x-x and y-y axes, such as the second moment of area, 4, and the product moment of area, 4y. However, because the section has no axes of symmetry, unsymmetrical bending theory must be applied and it is required to find the principal axes, u-u and v-v, about which the second moments of area are a maximum and minimum respectively (Urry and Turner, 1986). The principal axes are again perpendicular and pass through the centre of gravity, but are a displaced angle, a, from x-x as shown in Figure 4.63. The objective is to find the plane in which the principal axes lie and calculate the second moments of area about these axes. The following formulae will be used in the development of the problem. 

The B3Hg ion (p. 166) is a triangular cluster of Cj (rather than C2 ) symmetry (see Fig. 6.15a) the bridging atoms are essentially in the B3 plane with Ht above and below. While it has been conventional to represent the cluster bonding in terms of two BHB and one B-B bond (Fig. 6.15b), recent high-level computations suggest the presence of a 3-centre BBB bond, as depicted approximately in Fig. 6.15c. 

At this point, it is wise to see an example, and I will take H2O with a bond length of 95.6 pm and bond angle of 104.5°. If I take the coordinate system such that the molecule lies in the yz-plane with coordinate origin given by the centre of mass, and the z-axis as the symmetry axis, then the Cartesian coordinates are given in Table 8.4. 

---