Transitioning freedom

The freezing point of a pure liquid is the temperature at which the liquid s molecules lose transitional freedom and the solid s molecules... 

In the crystalline solid state, there is little vibrational or transitional freedom, and hence diffusion into a crystalline lattice is slow and difficult. As the temperature of a solid is raised by the input of heat, vibrational and transitional motion increases. At a particular temperature—termed the melting point—this motion overcomes the attractive forces holding the lattice together and the liquid state is produced. The liquid state, on cooling, returns to the solid state as crystallization occurs and heat is released by the formation of strong attractive forces. 

Entropy of polymerization does not vary widely for C=C monomers it is usually in the range of 100-125 J moP for liquid monomers. This is because in each case the dominant factor is the loss of transitional freedom that occurs when a monomer molecule is incorporated into a polymer chain. O Table 18.3 gives some values for entropy of polymerization. 

Shao J, Liao J-L and Poliak E 1998 Quantum transition state theory—perturbation expansion J. Chem. Phys. 108 9711 Liao J-L and Poliak E 1999 A test of quantum transition state theory for a system with two degrees of freedom J. 

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

The classical mechanical RRKM k(E) takes a very simple fonn, if the internal degrees of freedom for the reactant and transition state are assumed to be hamionic oscillators. The classical sum of states for s harmonic oscillators is [16]... 

The reason that non-adiabatic transitions must be included for protons is that fluctuations in the potential for the quantum degrees of freedom due to the environment (e.g. solvent) contain frequencies comparable to the transition frequencies between protonic quantum states. In such cases pure quantum states do not persist. 

Figure 5,30 reprinted from Chemical Physical Letters, 194, Fischer S and M Karplus. Conjugate Peak Refinement An Algorithm for Finding Reaction Paths and Accurate Transition States in Systems with Many Degrees of Freedom. 252-261, 1992, with permission from Elsevier Science. 

Electronic spectroscopy is the study of transitions, in absorption or emission, between electronic states of an atom or molecule. Atoms are unique in this respect as they have only electronic degrees of freedom, apart from translation and nuclear spin, whereas molecules have, in addition, vibrational and rotational degrees of freedom. One result is that electronic spectra of atoms are very much simpler in appearance than those of molecules. 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

In principle, energy landscapes are characterized by their local minima, which correspond to locally stable confonnations, and by the transition regions (barriers) that connect the minima. In small systems, which have only a few minima, it is possible to use a direct approach to identify all the local minima and thus to describe the entire potential energy surface. Such is the case for small reactive systems [9] and for the alanine dipeptide, which has only two significant degrees of freedom [50,51]. The direct approach becomes impractical, however, for larger systems with many degrees of freedom that are characterized by a multitude of local minima. 

Since only the vibrational degrees of freedom take part in a solid-state reaction, the sole reason for this change may be the increase in their frequencies in the transition state... 

Below a temperature of Toi 260 K, the Ceo molecules completely lose two of their three degrees of rotational freedom, and the residual degree of freedom is a ratcheting rotational motion for each of the four molecules within the unit cell about a different (111) axis [43, 45, 46, 47]. The structure of solid Ceo below Tqi becomes simple cubic (space group Tji or PaS) with a lattice constant ao = 14.17A and four Ceo molecules per unit cell, as the four oriented molecules within the fee structure become inequivalent [see Fig. 2(a)] [43, 45]. Supporting evidence for the phase transition at Tqi 260 K is... 

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