Rotation, internal entropy

The entropy of a molecule is composed of the sum of its translational, rotational, and internal entropies. The translational and rotational entropies may be precisely calculated for the molecule in the gas phase from its mass and geometry. The entropy of the vibrations may be calculated from their frequencies, and the entropy of the internal rotations from the energy barriers to rotation. 

Stiff vibrations, as found in most covalent bonds, make very low individual contributions to the entropy. Low-frequency vibrations, where the atoms are less constrained, can contribute a few entropy units. Internal rotations have entropies in the range of 13 to 21 J/deg/mol (3 to 5 cal/deg/mol) (Table 2.4). 

The combining of two molecules to form one leads to the loss of one set of rotational and translational entropies. The rotational and translational entropies of the adduct of the two molecules are only slightly larger than those of one of the original molecules, since these entropies increase only slightly with size (Table 2.4). The entropy loss is up to 190 J/deg/mol (45 cal/deg/mol) or 55 to 59 kJ/mol (13 to 14 kcal/mol) at 25°C for the small molecules. This may be offset somewhat by an increase in internal entropy due to new modes of internal rotation and vibration (Figure 2.6). 

For the elements and small, roughly spherical molecules, the entropy of melting is primarily due to expansional and positional entropy. This is because no rotational entropy is gained upon melting and there is no internal entropy contribution. Using Equation (19), with a o value of 100 ... 

Yalkowsky proposed that the entropy of fusion of an organic compound is the sum of translational, rotational, and internal entropy changes when it is released from the crystal lattice (Yalkowsky, 1979) ... 

By "conventional entropy" we mean the sum of all contributions to the entropy from translations, rotations, internal Tbrations and electronic degrees of freedom but excluding nuclear degrees of freedom, in particular nuclear spin, and isotopic mixing. 

Leffek et al. (151) interpret their results in terms of internal entropy, in particular that associated with release of steric hindrance to internal rotation which the two methyl groups exert on one another. They cite Maccoll s (152) report of a not greatly temperature dependent isotope effect on the quasi-heterolytic gas-phase pyrolysis of isopropyl bromide-dt. Maccoll s effect, kn/kn = 2.5, seems too large not to be at least partly primary, and the transition state almast certainly involves a eis- four center configuration with some CH bond rupture. Ip any case the extrapolation from gas-phase pyrolysis at 320°C. to solvol3rsis in water seems a rather long one. 

Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

The steric environment of the atoms in the vicinity of the reaction centre will change in the course of a chemical reaction, and consequently the potential energy due to non-bonded interactions will in general also change and contribute to the free energy of activation. The effect is mainly on the vibrational energy levels, and since they are usually widely spaced, the contribution is to the enthalpy rather than the entropy. When low vibrational frequencies or internal rotations are involved, however, effects on entropy might of course also be expected. In any case, the rather universal non-bonded effects will affect the rates of essentially all chemical reactions, and not only the rates of reactions that are subject to obvious steric effects in the classical sense. 

Overberger, J.E., Steele, W.A., Aston, J.G. (1969) The vapor pressure of hexamethylbenzene. The standard entropy of hexamethyl-benzene vapor and the barrier to internal rotation. J. Chem. Thermodyn. 1, 535-542. 

Pitzer, K.S., Guttman, L., Westrum, Jr., E.F. (1946) The heat capacity, heats of fusion and vaporization, vapor pressure, entropy, vibration frequencies and barrier to internal rotation of styrene. J. Am. Chem. Soc. 68, 2209-2212. 

An important property of chain molecules is that a major contribution to the standard entropy is conformational in nature, i.e. is due to hindered internal rotations around single bonds. This property is most relevant to cyclisation phenomena, since a significant change of conformational entropy is expected to take place upon cyclisation. Pitzer (1940) has estimated that the entropy contribution on one C—C internal rotor amounts to 4.43 e.u, A slightly different estimate, namely, 4.52 e.u. has been reported by Person and Pimentel (1953). Thus, it appears that nearly one-half of the constant CH2 increment of 9.3 e.u. arises from the conformational contribution of the additional C—C internal rotor. 

The last column in Table 19 lists the entropy losses due to reduction of freedom of internal rotations around the single bonds upon cyclisation. It is of interest to note, for example, that the conformational entropy lost upon cyclisation of an 8-rotor chain amounts to f of the corresponding quantity related to cyclisation of a 100-rotor chain. 

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