Prolate

Moleeules for whieh two of the three prineipal moments of inertia are equal are ealled symmetrie top moleeules. Prolate symmetrie tops have la < Ib = Ic J oblate symmetrie tops have la = Ib < Ic (it is eonvention to order the moments of inertia as la < Ib Ic ) ... 

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

In the symmetric top cases, Hrot can be expressed in terms of J2 and the angular momentum along the axis with the unique moment of inertia (denoted the a-axis for prolate tops and the c-axis of oblate tops) ... 

The ellipsoid of revolution is swept out by rotating an ellipse along its major or minor axis. When the major axis is the axis of rotation, the resulting rodlike figure is said to be prolate when the minor axis is the axis of rotation, the disklike figure is said to be oblate. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

Based on these ideas, the intrinsic viscosity (in 0 concentration units) has been evaluated for ellipsoids of revolution. Figure 9.3 shows [77] versus a/b for oblate and prolate ellipsoids according to the Simha theory. Note that the intrinsic viscosity of serum albumin from Example 9.1-3.7(1.34) = 4.96 in volume fraction units-is also consistent with, say, a nonsolvated oblate ellipsoid of axial ratio about 5. 

Figure 9.3 Intrinsic viscosity according to the Simha theory in terms of the axial ratio for prolate and oblate ellipsoids of revolution.

The intrinsic viscosity of poly(7-benzyl-L-glutamate) (Mq = 219) shows such a strong molecular weight dependence in dimethyl formamide that the polymer was suspected to exist as a helix which approximates a prolate ellipsoid of revolution in its hydrodynamic behaviorf ... 

R/Ro)soiv(f/fo)ellip = n + (mib/m2)(P2/Pi)] (f/fo)eiiip-Briefly justify this expansion of the (f/fo oiv factor. Assuming these particles were solvated to the extent of 0.26 g water (g protein)", calculate (f/fo)eiHp-For prolate ellipsoids of revolution (b/a < 1), Perrin has derived the following expression ... 

A symmetric rotor must have either a C axis with n>2 (see Section 4.1.1) or an 54 axis (see Section 4.1.4). Methyl iodide has a C3 axis and benzene a Ce axis and, therefore, these are symmetric rotors whereas allene, shown in Figure 4.3(d), is also a symmetric rotor since it has an 54 axis which is the a axis allene is a prolate symmetric rotor. 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

Figure 5.6 Rotational energy levels for (a) a prolate and (b) an oblate symmetric rotor...

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

When the effects of centrifugal distortion are included the term values of a prolate symmetric rotor are given by... 

The term —2Djj K J +1) has Ihe efifecl of separating Ihe (J+ 1) componenls of each J + ) J Iransilion wilh dififerenl values of K. This is illuslraled in Figure 5.7, which shows Ihe eighl componenls of Ihe J = 8 — 7 Iransilion of silyl isolhiocyanale (HsSi—N=C=S), which has a linear SiNCS chain and is a prolate symmelric rolor. 

At a simple level, the rotational transitions of near-symmetric rotors (see Equations 5.8 and 5.9) are easier to understand. For a prolate or oblate near-symmetric rotor the rotational term values are given, approximately, by... 

Examples of prolate near-symmetric rotors are the s-trans and s-cis isomers of crotonic acid, shown in Figure 5.8, the a axis straddling a chain of the heavier atoms in both species. The rotational term values for both isomers are given approximately by Equation (5.37) but, because A and B are different for each of them, their rotational transitions are not quite coincident. Figure 5.9 shows a part of a low-resolution microwave spectmm of crotonic acid in which the weaker series of lines is due to the less abundant s-cis isomer and the stronger series is due to the more abundant s-trans isomer. 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

The effect of the AK = 1 selection rule, compared with AK = 0 for an transition, is to spread out the sets of P, Q, and R branches with different values of K. Each Q branch consists, as usual, of closely spaced lines, so as to appear almost line-like, and the separation between adjacent Q branches is approximately 2 A — B ). Figure 6.29 shows such an example, E — A band of the prolate symmetric rotor silyl fluoride (SiH3F) where Vg is the e rocking vibration of the SiH3 group. The Q branches dominate this fairly low resolution specttum, those with AK = - -1 and —1 being on the high and low wavenumber sides, respectively. 

The selection rules are the same for oblate symmetric rotors, and parallel bands appear similar to those of a prolate symmetric rotor. However, perpendicular bands of an oblate symmetric rotor show Q branches with AK = - -1 and — 1 on the low and high wavenumber sides, respectively, since the spacing, 2 C — B ), is negative. 

Whether the molecule is a prolate or an oblate asymmetric rotor, type A, B or C selection mles result in characteristic band shapes. These shapes, or contours, are particularly important in gas-phase infrared spectra of large asymmetric rotors, whose rotational lines are not resolved, for assigning symmetry species to observed fundamentals. 

This general behaviour is characteristic of type A, B and C bands and is further illustrated in Figure 6.34. This shows part of the infrared spectrum of fluorobenzene, a prolate asymmetric rotor. The bands at about 1156 cm, 1067 cm and 893 cm are type A, B and C bands, respectively. They show less resolved rotational stmcture than those of ethylene. The reason for this is that the molecule is much larger, resulting in far greater congestion of rotational transitions. Nevertheless, it is clear that observation of such rotational contours, and the consequent identification of the direction of the vibrational transition moment, is very useful in fhe assignmenf of vibrational modes. 

Diffuorobenzene is a prolate asymmetric rotor and, because the y axis is the b inertial axis, type B rotational selection mles apply. In Figure 7.44(b) is a computer simulation of the... 

Small micelles in dilute solution close to the CMC are generally beheved to be spherical. Under other conditions, micellar materials can assume stmctures such as oblate and prolate spheroids, vesicles (double layers), rods, and lamellae (36,37). AH of these stmctures have been demonstrated under certain conditions, and a single surfactant can assume a number of stmctures, depending on surfactant, salt concentration, and temperature. In mixed surfactant solutions, micelles of each species may coexist, but usually mixed micelles are formed. Anionic-nonionic mixtures are of technical importance and their properties have been studied (38,39). 

Prolate Spheroid (formed by rotating an ellipse about its major... 

---