Additivity of Bond Contributions

The next higher order of approximation, the first-order approximation, is obtained by estimating molecular properties by the additivity of bond contributions. In the following, we will concentrate on thermochemical properties only. 

Values for Cp° are now usually within 4.0 J/mol K and rarely as poor as 8.0 J/ mol-K. The values are usually well within 3.0 J/mol K and rarely deviate by more than 6.0 J/mol K. The AHf values are usually within 10.0 kj/mol and seldom deviate by more than 20.0 kj/mol. In many cases, the values for the first member of a series of compounds such as CH+, H2O, or NH3 deviate quite strongly. 

Furthermore, such a scheme cannot distinguish between the values for isomeric hydrocarbons because these compounds have the same number and type of bonds. 

To summarize, such a scheme seems to be sirffident for the estimation of Cp° and S°, but does not give sufficiently accurate values for the heat of formation, AHf. 

In order to develop a quantitative interpretation of the effects contributing to heats of atomization, we will introduce other schemes that have been advocated for estimating heats of formation and heats of atomization. We will discuss two schemes and illustrate them with the example of alkanes. Laidler [11] modified a bond additivity scheme by using different bond contributions for C-H bonds, depending on whether hydrogen is bonded to a primary (F(C-H)p), secondary ( (C-H)g), or tertiary ( (C-H)t) carbon atom. Thus, in effect, Laidler also used four different kinds of structure elements to estimate heats of formation of alkanes, in agreement with the four different groups used by Benson. 

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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. 

In a broad sense, one may include the Free-Wilson equation within the class of linear free energy relationships (LFER). It is also subjected to the assumption of additivity of the contributions to the biological activity by substituent groups at different substitution sites. The assumption requires, for example, that there is no hydrogen bonding interaction between the various substitution groups. 

By contrast, the Dewar resonance energy represents solely the contribution coming from the cyclic electron (bond) delocalization since the model reference structure is represented not by a system of isolated 7r-bonds, but by a hypothetical cyclic polyene with the number of tr- and tr-bonds equal to that in a given molecule. Making use of the additivity of bond energies in acyclic polyenes (65JA692), one may calculate the total energy... 

If the Coulomb interaction between electrons of different pairs is ignored, each localized bond and lone pair contribute independently to the total energy, which implies a perfect additivity of bond energies. In the independent-particle model, the localized bond function can be visualized as a two-center molecular orbital occupied by two electrons. Nevertheless, it is possible to reproduce deviations from additivity rules within this scheme, for instance, by taking into account overlap (for a review, see e.g. 2>). 

ADDITIVITY OF BOND ENERGY CONTRIBUTIONS TO MELTING POINTS... 

In the absence of refractive index data on solutions or on the pure solute the method of addition of bond polarizabilities (Denbigh [51], see LeFevre [52] for a review) can yield quite a good estimate for the magnitude of the electronic contribution to molecular polarizability, with an error of less than 5%. 

The additive contribution of the individual bonds to the anisotropic potential draws attention to the rule of the additivity of bond polarizabilities [122]. This strongly suggests that the average orientation of the substituted benzenes is directly related to their principal polarizabilities and that the anisotropic solute-solvent interaction is determined by London dispersion forces. Considering dispersion forces (in dipolar approximation) one obtains the following expression for the... 

The effects of small Ge additions on physical and mechanical properties were also studied. It was concluded that small additions of Ge contribute to enhance both surface and interface characteristics to improve bonding and bulk materials properties. It was reported that Ge-containing solders have similar or better performance and higher reliability than eutectic Sn-Pb solders. Although these results are very promising, this solder family must be appropriately tested and exercised under actual assembly conditions in manufacturing. The interfacial microstructures and joint rehability require further investigation. 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

Equations (2 -(4) clearly illustrate the increase in distance in the interactions between atoms X and Y in going from the additivity of atomic, to bond, and further to group contributions. 

The accuracy of an additivity scheme can be increased by going from atomic contributions through bond contributions to group contributions. 

The contribution of this polar structure to the bonding lowers the energy of the transition state. This may be viewed as a lower activation energy for the addition step and thus a factor which promotes this particular reaction. The effect is clearly larger the greater the difference in the donor-acceptor properties of X and Y. The transition state for the successive addition of the same monomer (whether X or Y substituted) is structure [V] ... 

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