Vibrational entropy change

The adaptation is such as to permit the equilibrium microcanonical distribution for the slow coordinate X to be a solution (2.3) when k(X) = 0. The SU(X) in Eq. (2.3) is the vibrational entropy change needed to reach X from... 

The entropy change can be divided into two parts i) a vibrational entropy change, ASyib, reflecting the entropy created by vibrations associated with eaeh new vacancy, and ii) a configurational entropy, ASconb arising from the distribution of n vacancies among N+ny sites. 

The vibrational entropy change is associated with the vibrations in the neighbourhood of each vacancy, and the total vibrational entropy change is proportional to the number of vacancies, nyASvib. This entropy ehange is, as was the enthalpy change, related to a dilute solution of vacaneies. 

The enthalpy and vibrational entropy changes are, as we have discussed, to a first approximation proportional to the number of vacancies formed, while the configurational entropy is a more eomplex function. The change in Gibbs free energy assoeiated with the formation of ny vacaneies may accordingly first be written... 

ASvib in the expressions above represents the vibrational entropy change and AH the enthalpy change per vacancy. If one wants to express the same properties per mole of vacancies, k must be substituted by the gas constant R = NAk where Na is Avogadro s number. Relations corresponding to Eq. 3.10 may be derived for other types of point defects. 

To illustrate the effect of Equations 7.6 and 7.7 on the vacancy concentration, let us consider the case of undoped and doped Si. Here vacancies can occur in three charge states, -1-1, -1, and -2, with energies 1.0 eV below the conduction band, and 0.6 and 1.05 eV above the valence band at 300 K, respectively. The overall energy gap of Si is 1.12 eV. The energy of formation of a neutral vacancy is 2.4 eV, the vibrational entropy change per added vacancy is 1.1 ke, and the site density in the Si lattice is 5x10 2 cm [3]... 

Table III presents integral excess entropies of formation for some solid and liquid solutions obtained by means of equilibrium techniques. Except for the alloys marked by a letter b, the excess entropy can be taken as a measure of the effect of the change of the vibrational spectrum in the formation of the solution. The entropy change associated with the electrons, although a real effect as shown by Rayne s54 measurements of the electronic specific heat of a-brasses, is too small to be of importance in these numbers. Attention is directed to the very appreciable magnitude of the vibrational entropy contribution in many of these alloys, and to the fact that whether the alloy is solid or liquid is not of primary importance. It is difficult to relate even the sign of the excess entropy to the properties of the individual constituents.

Many competing effects can contribute to ligand-receptor binding free energies changes in rotational, translational, conformational, and vibrational entropy of the... 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

Differences in the Debye temperature, or in other words the vibrational character of the two phases, will modify the transitional entropy to some extent. Still, this entropy change is normally not large for transitions where the coordination number is preserved. 

Thus, the steric factor may be explained with the help of entropy change. When two molecules come together to produce the activated complex, the total translational degrees of freedom are reduced (from 6 to 3) and rotational degrees of freedom also diminish. This is compensated by an increase in vibrational degrees of freedom. But the definite orientation in forming the activated complex necessarily reduced the entropy, i.e. AS is negative. This decrease in entropy is small when reaction takes place between simple atoms. The calculated value of kbT/h corresponds to collision frequency... 

While the AH values vary over a wide range for different monomers, the AS values are less sensitive to monomer structure, being relatively constant within the range of 100-120 J K-1 mol-1. The T AS contribution to the AG of polymerization will be small as indicated earlier and will vary only within a narrow range. Thus the variation in the T AS term at 50°C for all monomers is in the narrow range 30-40 kJ mol-1. The entropy changes that occur on polymerization have been analyzed for several monomers [Dainton and Ivin, 1950, 1958], The AS of polymerization arises primarily from the loss of the translational entropy of the monomer. Losses in the rotational and vibrational entropies of the monomer are essentially balanced by gains in the rotational and vibrational entropies of the polymer. Thus AS for polymerization is essentially the translational entropy of the monomer, which is relatively insensitive to the structure of the monomer. 

This result stresses the fact that the heat effects can be detected only at such deformation modes of quasi-isotropic Hookean solids that are accompanied by a change in volume. Thus, the thermal effects are the consequence of the change of the vibrational entropy which in turn is a result of the volume change. It is very important to emphasize now that the internal energy and entropy changes are closely interrelated and their values are of the same order of magnitude. 

This inversion of heat is due to a competition between the increase of the vibrational entropy connected with the volume change at deformation and the decrease of the conformational entropy. The deformation at which a maximum of heat is absorbed at elongation is given by... 

In contrast to intrachain changes of entropy and internal energy, which follow from the statistical theory, interchain changes of the internal energy, vibrational entropy and volume can be predicted by the statistical theory only at small strains (X < 1.3). 

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