The Additivity Rule

Davies and Warren have investigated the nitration of naphthalene, ace-naphthene and eight dimethylnaphthalenes in acetic anhydride at o °C. Rates relative to naphthalene were determined by the competition method, and the nitro-isomers formed were separated by chromatographic and identified by spectrophotometric means. The results, which are summarised in the table, were discussed in terms of various steric effects, and the applicability of the additivity rule was examined. For the latter purpose use was made of the data of Alcorn and Wells (table 10.2) relating to the nitration of monomethyl-naphthalenes at 25 °C. The additivity rule was found to have only limited utility, and it was suggested that the discrepancies might be due in part to the... 

After all local slopes of the pile are again less than rjc, one unit of sand is added to a random site i. In terms of the local slope variable, this amounts to applying the addition rule 0add... 

The expectation symbol E obeys the same rules of manipulation, Eq. (3-40), as in the one-dimensional case. The only additional comment needed here is that the addition rule holds even when the two random variables concerned are defined with respect to different sets of r s. The proof of this fact is immediate when the various expectations involved are written as time averages. 

The growing polymer chains have the most probable distribution defined by Equation (13.26). Typically, is large enough that PD 2 for the growing chains. It remains 2 when termination occurs by disproportionation. Example 13.5 shows that the polydispersity drops to 1.5 for termination by pure combination. The addition rules of Section 13.2.2 can be applied to determine 1.5 < PD < 2 for mixed-mode terminations, but disproportionation is the predominant form for commercial polymers. 

The various MO calculations use different basis sets and have different ways of calculating multicenter coulomb and exchange integrals. The current trend in MO is to expand as a linear combination of atomic orbitals (LCAO). The atomic orbitals are represented by Slater functions with expansion in gaussian functions, taking advantage of the additive rule. When the calculation is performed in this... 

When Roman numeral of a lesser value is placed between two greater values, it is first subtracted from the greater numeral placed after it, and then that value is added to the other numeral(s) (i.e., subtraction rule applies first, then the addition rule). 

When mixed phosphorus compounds are considered the sum of the individual contributions of the substituents can be used to calculate the %-value of the ligand as was shown by Tolman. The Rvalue for a single substituent R is simply 1/3 of the %-value of the ligand PR3. In extreme cases the additivity rule may not apply but as a first approximation it remains useful. 

Rule 1 and the additional rules 2, 3, and 4 are displayed (in abridged form) in Tables III and IV (during the exposition of the rules, the reader may benefit by referring to these tables). 

In the context of the MR-type methods, the concept of size-extensivity takes on a much broader meaning. Until recently [65], this fact has not been explicitly pointed out as fas as we know, since most discussions focused solely on the lowest state or, at most, on one state at a time. Thus, when we consider the dissociation process (31), assuming that the subsystems A and B involve Ma and Mb states, respectively, with energies Ei X), i = 1, , Mx, X = A, B, we must generally require that the additivity rule... 

R. Abegg, A. Kanitz, and 0. Pulvermacher have measured the viscosity of soln. of ammonium nitrate, and H. Gorke the reciprocal of the viscosity, i.e. the fluidity at different temp, (water at 25° unity), and his measurements are indicated in Table LXI. A. Kanitz also measured the viscosities of mixtures of ammonium nitrate with potassium, sodinm, or barium nitrate, and found that the results follow the additive rule very closely. W. N. Bond studied the plasticity of crystals of ammonium nitrate. 

Surface Pressure, Potential, and Fluidity Characteristics for Various Interactions in Mixed Monolayers. It is possible to distinguish various types of interactions which occur in mixed monolayers by measuring the surface pressure, surface potential, and surface fluidity of the monolayers. Deviation from the additivity rule of molecular areas indicates either an interaction between components or the intermolecular cavity effect in mixed monolayers. 

Ion-Ion or Ion-Dipole Interaction. Figure 5b, shows the general characteristics of mixed monolayers in which ion-ion or ion-dipole interaction takes place—e.g., alkyl phosphate-alkyl trimethylammonium, or steric acid—octadecanol monolayers. The average area per molecule may or may not show a deviation from the additivity rule line, depending upon whether the two components form expanded or condensed mono-layers. However, surface potential per molecule must show a deviation from the additivity line since ion-ion or ion-dipole interactions reduce the average surface dipole of the molecules in mixed monolayers (31, 42). These interactions result in a negative deviation in the plot of log < vs. mole fraction (6). 

Hydrocarbon-Hydrocarbon Interaction. Figure 5c shows the general characteristics of mixed monolayers in which hydrocarbon-hydrocarbon interaction occurs—e.g., trimyristin-myristic acid monolayers (16). The average area per molecule shows a deviation, whereas the surface potential per molecule follows the additivity rule. Hydrocarbon-hydrocarbon interaction also increases the cohesive force in the lipid layer and therefore reduces the fluidity of the mixed monolayer. It is evident from Figures 3a and 3c that surface fluidity is the only parameter which distinguishes an intermolecular cavity effect from hydrocarbon-hydrocarbon interaction. 

Dipalmitoyl Lecithin—Cholesterol Monolayers. The average area per molecule in dipalmitoyl lecithin-cholesterol monolayers shows deviation at low surface pressures, whereas at 30 dynes per cm. it follows the additivity rule (Figures 8 and 9) (42). The surface pressure—area curve of dipalmitoyl lecithin monolayers is liquid-expanded up to 30 dynes per cm., whereas above this surface pressure it is relatively incompressible (42). Figures 10b and c represent the structures of the dipalmitoyl... 

Van Deenen has reported (48) that the mixed monolayers of dide-canoyl lecithin-cholesterol follow the additivity rule of molecular areas even though this lecithin forms expanded monolayers. This can be explained similarly by an intermolecular cavity of smaller height, which cannot accommodate cholesterol (Figures lOd and 4d). 

Egg Lecithin—Cholesterol Monolayers. The average area per molecule in egg lecithin-cholesterol monolayers shows deviation from the additivity rule at all surface pressures (42). The deviation in this case could be explained by the presence of molecular cavities caused by the kink in the oleoyl chain of egg lecithin, which would reduce the average area per molecule at low as well as high surface pressures (Figure lOg). 

Even though 1,2-dilinoleoyl and l-palmitoyl-2-linolenoyl lecithins form more expanded monolayers than egg lecithin, their mixed mono-layers with cholesterol follow the additivity rule (48). This can be explained as follows. At low surface pressures, these lecithins have greater intermolecular spacing and hence form intermolecular cavities of smaller height which cannot accommodate cholesterol molecules (Figure 4e). At high surface pressure, the linoleoyl and linolenoyl chains, as opposed to oleoyl chains, do not form cavities in the monolayer (Figure lOi). 

Monolayers of dicetyl phosphate-cholesterol follow the additivity rule for average area per molecule, whereas lecithin—cholesterol mono-layers deviate from it. The reverse is true for the additivity rule of average potential per molecule. Thus, the surface potential indicates that there is no interaction (or complex formation) between lecithin and cholesterol, but there is ion-dipole interaction between dicetyl phosphate and cholesterol as well as between phosphatidic acid and cholesterol. 

In contrast to this, the system neutral lipid (2J)/DSPC shows considerably smaller deviations from the additivity rule and the surface pressure/area isotherms indicate two collapse points corresponding to those of the pure components62. Photopolymerization can be carried out down to low monomer concentrations and no rate change is observed. These facts prove that the system (23)/DSPC is immiscible to a great extent. The same holds true for mixed films of diacetylenic lecithin (18, n = 12) with DSPC, as well as for dioleoylphosphatidylcholine (DOPC) as natural component. 

The differences between Ihe observed heals of formation of rix-trans isomers and of branched and unbranched hydrocarbon chains show thal the additivity rule is not strictly rigorous. Various improvements have been suggested. 

Many studies have been devoted to calculation of 5,(0) and L,(0). For example, in Refs. 118 and 119 the authors present the values of 5,(0) and L,(0) for a large number of molecules and discuss their relation to the properties of the molecules. The authors of Ref. 118 have examined the possibility of calculating 5,(0) and L,(0) using the additivity rule for molecules starting from the values of similar quantities for atoms. They have shown that with —2 the values of 5,(0) and L,(0) for molecules calculated this way differ from their exact values by 15-25%, whereas for t = -1,..., 2 the deviation is less. 

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