Entropy vibration

Vibrational frequencies Rotational enthalpy and entropy Vibrational enthalpy and entropy Translational enthalpy and entropy... 

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. 

Translational entropy Rotational entropy Vibrational entropy Electronic entropy Total... 

Figure 8. Distributions of local K-entropy, vibrational mode-mode coupling, mean kinetic energy and negative curvature of the potential for the three-particle Lennard-Jones cluster simulating Ar3 at constant energy. The energy of the cluster corresponds to a temperature of approximately 30 K. [Reprinted with permission from R. J. Hinde and R. S. Berry, J. Chem. Phys. 99, 2942 (1993). Copyright 1993, American Institute of Physics.]...

Edge-edge ET distance between donor and aceeptor Close contact distance between donor and acceptor Activation entropy Vibrational quantum number Vibrational frequency... 

Real surfaces will generally also have defects analogous to those occurring in the bulk. As the temperature of an initially ideal surface is increased a certain amount of thermal disorder in the form of vacancies, interstitials and self adsorbed ions (and their complexes) will occur due to the balancing effect of the entropy contribution. The formation energies and entropies (vibrational) associated with such defects will be different from the corresponding bulk values and hence their densities will vary with depth into the crystal. Since ionic defects are of prime importance for conduction in many ionic crystals the properties of individual surface point defects are of fundamental interest in assessing surface conductance contributions. 

Figure 20.10 Entropy, vibrational motion, and molecular complexity.

Translational Enthalpy Rotational Enthalpy Vibrational Enthalpy gas constant (RT) Translational Entropy Rotational Entropy Vibrational Entropy ... 

Vibrational energy states are too well separated to contribute much to the entropy or the energy of small molecules at ordinary temperatures, but for higher temperatures this may not be so, and both internal entropy and energy changes may occur due to changes in vibrational levels on adsoiption. From a somewhat different point of view, it is clear that even in physical adsorption, adsorbate molecules should be polarized on the surface (see Section VI-8), and in chemisorption more drastic perturbations should occur. Thus internal bond energies of adsorbed molecules may be affected. 

Thus the entropy of localized adsorption can range widely, depending on whether the site is viewed as equivalent to a strong adsorption bond of negligible entropy or as a potential box plus a weak bond (see Ref. 12). In addition, estimates of AS ds should include possible surface vibrational contributions in the case of mobile adsorption, and all calculations are faced with possible contributions from a loss in rotational entropy on adsorption as well as from change in the adsorbent structure following adsorption (see Section XVI-4B). These uncertainties make it virtually impossible to affirm what the state of an adsorbed film is from entropy measurements alone for this, additional independent information about surface mobility and vibrational surface states is needed. (However, see Ref. 15 for a somewhat more optimistic conclusion.)... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

Molecular Nature of Steam. The molecular stmcture of steam is not as weU known as that of ice or water. During the water—steam phase change, rotation of molecules and vibration of atoms within the water molecules do not change considerably, but translation movement increases, accounting for the volume increase when water is evaporated at subcritical pressures. There are indications that even in the steam phase some H2O molecules are associated in small clusters of two or more molecules (4). Values for the dimerization enthalpy and entropy of water have been deterrnined from measurements of the pressure dependence of the thermal conductivity of water vapor at 358—386 K (85—112°C) and 13.3—133.3 kPa (100—1000 torr). These measurements yield the estimated upper limits of equiUbrium constants, for cluster formation in steam, where n is the number of molecules in a cluster. 

CPR can be used to find continuous paths for complex transitions that might have hundreds of saddle points and need to be described by thousands of path points. Examples of such transitions include the quaternary transition between the R and T states of hemoglobin [57] and the reorganization of the retinoic acid receptor upon substrate entry [58]. Because CPR yields the exact saddle points as part of the path, it can also be used in conjunction with nonnal mode analysis to estimate the vibrational entropy of activation... 

To compute zero-point vibration and thermal energy corrections to total energies as well as other thermodynamic quantities of interest such and the enthalpy and entropy of the system. 

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. 

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.

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... 

Variational principle on the correlation problem, 213, 318, 320, 326 Vibrational electronic modes, 12 Vibrational entropy in dilute solutions, 133... 

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