Resonance synchronized

A small value of (X - X j-), i.e. X c approaching Xl, will have more chance of a sub-synchronous resonance (SSR) with the rotating machines and a ferro-resonance with the transformers during a switching sequence or line disturbance. 

It is this resonance energy that would be in the main responsible for the difference in energy of the crystal and the gas of diatomic molecules Li2. But the heat of formation of Li2 molecules from atoms is only 6-6 kcal./g.-atom, whereas that of the metal is 39kcal./g.-atom. It seems unlikely, by comparison for example with the analogous case of Kekule-like resonance in aromatic molecules, that the great difference, 32-4 kcal./g.-atom, could result from the synchronized resonance, of type f Li—Li Li Li)... 

The number 3-14 is a measure of the coefficient of the resonance integral for synchronous resonance. 

The energy of the synchronized resonance between structures of this sort would contribute to the stabilization of the crystal, but far greater stabilization would result if there were also unsynchronized resonance to structures such as... 

The equation for synchronized resonance with L = 4 and v - 2 gives R In 3/2 for the residual entropy of ice (14). This value differs by only 1.1% from that given by calculations that do not involve the approximations made in our simple treatment. It is likely that the accuracy of Eq. 4 is also reasonably high. 

The number of structures for synchronized resonance is given by Eq. 2 with n = v ... 

The ratio of the number of structures per atom for unsynchronized resonance to that for synchronized resonance is given by the expression in parentheses in Eq. 4. Its value increases from 2.33 for L = 6, v = 1 to 2.78 for L = 16, v = 8, with average about 2.65. The amount of resonance stabilization for unsynchronized resonance is... 

The calculated number of resonance structures per atom (Eq. 4) is 2.50 for synchronized resonance and 6.25 for unsynchronized resonance. The second number is so much greater than the first that there is no doubt that the structure is one involving unsynchronized resonance, with 28% B+, 44% B°, and 28% B". 

Lithium may serve as an example for illustrating the general picture of resonance structures in metals. Pauling points out that it is unlikely that synchronized resonance, i.e. resonance of two bonds simultaneously, of the type... 

We have applied Pauling s theory to the molecular hydrogen cluster as follows the nonmetallic cluster is well described by the usual Kekule structure (1-2 3-4) (Fig.2), where orbitals 1 and 2 are at one hydrogen molecule and 3 and 4 are at the other one. The synchronized resonance is the mechanism in which the system alternates between structures (1-2 3-4) and (1-4 2-3) (anti-Kekule) (Fig.2), breaking simultaneously the two original covalent bonds and forming two new ones. 

As will be seen in the statistical theory described in the following section, there exist far more unsynchronized resonating structures per atom than there are synchronized resonating structures. Associated with this increase in the number of resonating structures is an increase in stability for the system, with the increased resonance stabilization energy being approximately proportional to the number of additional resonating structures per atom for unsynchronous resonance, less 1. One is consequently led to conclude that unsynchronized resonance of the covalent bonds between the atoms in metallic systems occurs... 

For the case of synchronous resonance, n = v, and the number of resonance structures per atom, vSynch, becomes... 

A comparison of eqns. (5) and (6) reveals that the term in square brackets in eqn. (6) is the ratio of the number of unsynchronized resonance structures per atom to the number of synchronized resonance structures per atom for a hypoelectronic atom. Given the reasonable assumption that the energy corresponding to an unsynchronized resonance structure is the same order of magnitude as that for a synchronized resonance structure, the energy of a crystal composed of hypoelectronic atoms is lowered considerably via unsynchronized resonance. Therefore, one predicts that every element with an extra orbital to serve as the metallic orbital should be a metal. With a single possible exception, namely boron, which will be discussed in a succeeding section, this prediction is borne out. 

This mechanism involves the synchronized resonance of pairs of bonds, but much greater stability will result if we countenance the far more numerous structures arising from unsynchronized resonance between arrangements in which ions are also involved, thus ... 

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