Characteristic temperature of rotation

With regard to the rotational term, if the temperature is lower than the characteristic temperature of rotation, this mode is not excited and its contribution is zero. For higher temperatures, we must use expression [6.51] ... 

If we sum the three contributions to calculate the value for the heat capacity at constant volume, as the characteristic temperatures of rotation are often lower than the Einstein temperature (see Table 7.3), the variation in the heat capacity, for example of a diatomic molecule, with temperature takes the form of the curve in Figure 7.12(c). At low temperatures, the only contribution is that of translation, given by 3R/2. Then if the temperature increases, the contribution of rotation, is added according to the curve in Figure 7.12(a) until the limiting value of this contribution is reached, then the vibrational contribution is involved until the molecule dissociates which makes the heat capacity become double that of the translational contribution of monoatomic molecules. The limiting value of the vibrational contribution is sometimes never reached. This explains why the values calculated in Table 7.2 are too low if we do not take into account the vibration and too high in the opposite case. 

TABLE 4.2. Typical Values of the Rotational and Vibrational Partition Functions of Diatomic Molecules at 500 K, along with Values of the Characteristic Temperatures of Rotation and Vibration, 0roi and 0vib ... 

Analogous to the above characteristic temperature of vibration it is now sensible to define a characteristic temperature of rotation via... 

The more exoergic reaction Ba + NzO has a smaller reaction cross section ( 90 A2 or 27 A2) [347, 351] and crossed-molecular beams studies [349] show that the BaO product is backward-scattered with a large amount of internal excitation ((Fr) < 0.20). Laser-fluorescence measurements [348] of the BaO(X Z+) product for the reaction in the presence of an argon buffer gas, find population of vibrational states up to v = 32. The relative populations have a characteristic temperature of 600 K for v = 0—4 and 3600 K for v = 5—32 with evidence of non-thermal population of v — 13—16. This study also observes population of A n and a 3II states of BaO with v = 0—4. A molecular beam study of Ba + N20 with laser-induced fluorescence detection indicates that the BaO( X) product is formed with a very high rotational temperature. 

The total heat capacity of a molecular substance is the sum of each contribution (Figure 17.13 of the text). When equipanition is valid (when the temperature is well above the characteristic temperature of the mode T 0m) we can estimate the heat capacity by counting the numbers of inodes that are active. In gases, all three translational modes are always active and contribute 7/ to the molar heat capacity. If we denote the number of active rotational modes by (so for most molecules at normal temperatures vjj = 2 for linear molecules, and 3 for nonlinear molecules), then the rotational contribution is 71 / . If the temperature is high enough for vibrational modes to be active the vibrational contribution to the molar heat capacity is u /f. In most cases vy 0. It follows that the total molar heat capacity is... 

To determine the moment of inertia and the rotational characteristic temperature of HCl from its rotation spectrum in the far infra-red. 

The vibrational wave-numbers , and characteristic temperatures of ethane are given (Hansen and Dennison, J. Chem. Phys. 1952, 20, 317) in table 2. There are six non-degenerate modes including the internal rotation (mode 4) and six pairs of doubly-degenerate modes. The potential energy u restricting the internal rotation has three equal minima and three equal maxima. It may be represented approximately by the form... 

TABLE 28-2 Molecular Parameters and Characteristic Temperatures for Rotation and Vibration of Several Diatomic Molecules... 

It is tempting to identify the break point in Figure 8 as a characteristic temperature of the system. Guillet, in his chapter (18), makes such assignments in several hcmiopolyiiier systons. Indeed one finds in the literatures a maximum at -35 corresponding to the a-methyl group rotation in an nmr relaxation experiment. In this instance the temperature correspondence between the two sets of experiments must be. accidental. 

The phase transition of bilayer lipids is related to the highly ordered arrangement of the lipids inside the vesicle. In the ordered gel state below a characteristic temperature, the lipid hydrocarbon chains are in an all-trans configuration. When the temperature is increased, an endothermic phase transition occurs, during which there is a trans-gauche rotational isomerization along the chains which results in a lateral expansion and decrease in thickness of the bilayer. This so-called gel to liquid-crystalline transition has been demonstrated in many different lipid systems and the relationship of the transition to molecular structure and environmental conditions has been studied extensively. 

How conformation is related with optical rotation is brought out by the fact that with decreasing temperature there is an increase of rotation of conformationally mobile compounds. This is explained that at ordinary temperatures, several conformations occur at equilibrium, the rotation of which may be opposite in sign and so the total rotation is small. With the decreasing temperature the equilibrium is displaced in favour of the most preferred conformation with a characteristic rotation. An example of this is butanol-2. 

Figure B8.2.1 shows the fluorescence spectra of DIPHANT in a polybutadiene matrix. The h/lu ratios turned out to be significantly lower than in solution, which means that the internal rotation of the probe is restricted in such a relatively rigid polymer matrix. The fluorescence intensity of the monomer is approximately constant at temperatures ranging from —100 to —20 °C, which indicates that the probe motions are hindered, and then decreases with a concomitant increase in the excimer fluorescence. The onset of probe mobility, detected by the start of the decrease in the monomer intensity and lifetime occurs at about —20 °C, i.e. well above the low-frequency static reference temperature Tg (glass transition temperature) of the polybutadiene sample, which is —91 °C (measured at 1 Hz). This temperature shift shows the strong dependence of the apparent polymer flexibility on the characteristic frequency of the experimental technique. This frequency is the reciprocal of the monomer excited-state...

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