Entropy electronic contribution

In this section we assume a metallic system where the entropy and interaction energy are those of the ions, and the electrons contribute a constant to e. The mean-field expression, Eqn (7.10), is exact when all the ions interact equally, regardless of the distance between them. Although... 

The final set of thermodynamic quantities to illustrate is the entropy, also listed in Table 8.1. The largest contribution by far is from translation, calculated from Eq. 8.106. The portion of the entropy attributable to rotational and vibrational degrees of freedom are calculated by Eqs. 8.108 and 8.109, respectively. The electronic contribution to S from Eq. 8.101 is large (certainly relative to the role it played for the other thermodynamic functions just considered), 5.763 (= R In 2, from the ground-state degeneracy contribution) J/mole-K. Thus the net value of S at 298.15 K is calculated to be 198.542 J/mole-K, com-... 

Scheme 1.12 The elementary entropy/information contributions to chemical interactions between two different AOs in the minimum basis set z, = 2pz l] of the zr-electron system in benzene.

In these calculations the electronic contributions have been assumed to cancel, and vibrational assignments and internal rotation barriers were calculated according to Pitzer. Other assumptions have been discussed by the authcjrs. The vibrational contributions (not shown in table) nearly cancel each other near 300°K and can be neglected below 500°K in the calculation of AiS° for the reactions, so that the AaS° shown in the last two lines of Table XII.3, aside from symmetry changes, can be equated to the standard entropy of activation. 

Notes See Appendix C, Notes, for units, standard states, and sources of data. See Table D.l for corrections to entropy for symmetry and electronic contributions. Cl and aS° estimated from rule of additivity of bond contributions are good to about 1 cal/mole- K but may be poorer for heavily branched compounds. The values of AH/ are usually within 2 Kcal/mole but may be poorer for heavily branched species. Peroxide values are not certain by much larger amounts. 

The final result for the dimensional model Is an equation relating the entropy (excluding electronic contributions) to the Interatomic distance and atomic masses, m. 

All the data for Th are empirical estimates. An early example using an experimental value for Pu, allowing electronic contributions to the entropy of Rln(2J + 1) and supposing that the other contributions are displaced by 25.9 J moP from the corresponding lanthanide values, give -171.1 J K" mol for 8298° for Th + (18). A more recent estimate gave -S29s°M values of 176.1 and 177.8 J K moP for the trivalent aquated ions of Th and U (19). 

For all these compounds, electronic contributions to the entropy are significant. 

Estimation of the entropy of solvation requires calculation of the entropy of the ion in the gas phase. For a monoatomic ion, the main contribution to the entropy comes from its translational energy. Simple ions formed from the main group elements have the electronic structure of an inert gas and therefore do not have an electronic contribution to the entropy. On the other hand, ions formed from transition metals may have an electronic contribution to the gas phase entropy, which depends on the electronic configuration of the ion s ground state and of any other electronic states which are close in energy to the ground state. The translational entropy is given by the Sackur-Tetrode equation, which is obtained from the solution of the SWE for a particle in a box (see section 2.2)... 

Fig. 6.12. Comparison of theoretical and experimental electronic contributions to the entropy (adapted from Eriksson et at. (1992)). The solid lines are the results of the theoretical analysis and the symbols correspond to experimental values.

As a result of a knowledge of the electronic density of states, it is then possible to assess the electronic contribution to the entropy. An example of these results is given in fig. 6.12 which shows not only the computed electronic entropies for Ti and Zr (the solid lines) but also the experimental values for these entropies as obtained by taking the measured total entropy and subtracting off the contributions due to vibrations. In addition to their ability to shed light on the entropies themselves, these results were also used to examine the relative importance of vibrations and electronic excitations to the hcp-bcc structural transformation in these metals. Unlike in the case of simple metals, it is found that the electronic contribution to the free energy difference is every bit as important as the vibrational terms. In particular, in the case of Ti, it is claimed that the measured entropy change... 

Thermophysical Properties of Materials by G. Grimvall, North-Holland Publishing Company, Amsterdam The Netherlands, 1986. GrimvalTs book is full of insights into the vibrational and electronic contributions to the entropy, in addition to much else. 

E17.12(b) The degeneracy of a species with 5 = is 6. The electronic contribution to molar entropy is... 

There are no experimental data for the heat capacity and therefore entropy of p-ThCk and we have accepted the estimated value of the entropy of Konings [2004KON], based on a systematic set of lattice and electronic contributions to the entropies of the oxides, fluorides and chlorides of the actinide elements, as described in Appendix A. 

Statistical mechanics gives the molar entropy of an ideal gas as the sum of translational, rotational, vibrational, and electronic contributions. (See, for example, Levine, Physical Chemistry, Chapter 22.) The translational contribution depends only on the molar mass of the gas. The rotational contribution depends on the symmetry number and the principal moments of inertia these quantities are readily found from the molecule s equilibrium geometry. The vibrational contribution 5vib depends on the molecular vibrational frequencies, which can usually be rather accurately calculated with the aid of a scale factor. The electronic contribution depends on the electronic... 

Such assumptions have been used successfully to calculate the free energy between metallic crystalline phases. For example, Lam et a/.[60]computed the temperature-pressure phase diagram of beryllium. They predicted the static lattice energy using ab initio pseudopotentials and estimated the phonon energy and entropy from the second order elastic constants. Since the electronic contribution to the entropy varies less for insulators than it does for metals, one would expect a better description of the thermodynamics for the materials here than for beryllium. 

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