Energetic equivalence

An estimate of the enthalpy change which conesponds to the activation energy of the collision theory analysis of 167kJmoP may be made by assuming that the formation of tire dimer from two molecules of the monomer is energetically equivalent to tire dipole-dipole and dispersion interactions of two HI molecules. These exothermic sources of interaction are counterbalanced... 

A similar computational exercise was performed by Guirao et al. (1979). They used a code based on the Eulerean FCT approach. Blasts produced by four different, but energetically equivalent, sources ... 

In the first approach, a vapor cloud s potential explosive power is proportionally related to the total quantity of fuel present in the cloud, whether or not it is within flammable limits. This approach is the basis of conventional TNT-equivalency methods, in which the explosive power of a vapor cloud is expressed as an energetically equivalent charge of TNT located in the cloud s center. The value of the proportionality factor, that is, TNT equivalency, is deduced from damage patterns observed in a large number of vapor cloud explosion incidents. Consequently, vapor cloud explosion-blast hazard assessment on the basis of TNT equivalency may have limited utility. 

The assumptions made to derive the Langmuir isotherm (Eq. 2.7) are well known Energetic equivalence of all adsorption sites, and no lateral (attractive or repulsive) interactions between the adsorbate molecules on the surface. This is equivalent to a constant, coverage independent, heat (-AH) of adsorption. 

It should be noted that within the context of the Langmuir isotherm (energetically equivalent adsorption sites, no lateral interactions) Eq. (6.28), which relates two surface properties, i.e. aj and 0j, remains valid even when the surface activity of Sj, aj, is different from the gaseous activity, pJ5 i.e. when Pj(g) Pj(ad). 

X-ray diffraction patterns from dendrimers tend to lack sharp features and are similar to those from amorphous linear polymers. This suggests a molecular arrangement in dendrimers that is fairly disordered. There is also the problem that these molecules can exist in a large number of energetically equivalent conformations and that in solution there can be rapid interchange between these conformations. This contributes to the overall amorphous structure of dendrimers. 

The main problem here is connected with selecting the regularizator R. Since regularity conditions became operator inequalities, it seems reasonable to choose as R operators of the most simplest structures which are energetically equivalent to the operator A. Let, for instance, A and Aq be energetically equivalent operators with constants 7, and 7j, so that... 

Still using the framework of the general theory, it is straightforward to verify that the operator B is self-adjoint (B = B ) with the aid of the relations R = R and T = T,n- We shall need yet the constants 7j and 7, of the energetic equivalence of the operators B and A. Because of (46), we thus have... 

Scheme 14-1. General in-line monoanionic mechanism of phosphodiester cleavage transesterification catalyzed by hairpin ribozyme the first proton transfer (PT1), the nucleophilic attack (Nu), and the exocyclic cleavage (Cl) steps are shown, and the Oip and O2p pathways are indicated by blue and red colored hydrogens, respectively. For the uncatalyzed model reaction in solution, the Ojp and O2p pathways are energetically equivalent...

A number of the assumptions used in the BET theory have been questioned for real samples [6]. One assumption states that all adsorption sites are energetically equivalent, which is not the case for normal samples. The BET model ignores lateral adsorbate interactions on the surface, and it also assumes that the heat of adsorption for the second layer and above is equal to the heat of liquefaction. This assumption is not valid at high pressures and is the reason for using adsorbate pressures less than 0.35. In spite of these concerns, the BET method has proven to be an accurate representation of surface area for the majority of samples [9,10]. 

Another effect of lattice contraction of a boron-implanted layer might be to cause reorientation of the B—H complexes in the layer. Calculations (Denteneer et al., 1989) and measurements (Stavola et al., 1988) show that the B—H complex can readily reorient itself in response to an applied stress at room temperature. Thus, reorientation might occur in a boron-implanted layer and vitiate the analysis of the channeling experiment. In a (100) sample, however, all the (111) directions lie at the same angle to the stress axis if relaxation in the plane of the sample is isotropic, and all orientations of the B—H complex are energetically equivalent. This would not be true in (111) material. 

In liquids all orientations of reactants A and B are energetically equivalent and P = Air) 2 A0aA0b. Therefore, the ratio of rate constants in polymer and liquid is... 

Figure 1.21 Minimum-energy transition states for primary propene insertion on Brookhart-type Ni(II) catalyst for the model including si growing chain (whose last monomeric unit was generated by insertion of -coordinated monomer), (a) There is only minimum-energy transition state for insertion of, v(-propene (like insertion) while (b,c) there are two nearly energetically equivalent minimum-energy transition states for insertion of re-propene (unlike insertion).

We could summarize by saying that thermodynamic work w is energetically equivalent to the lowering or raising of a weight (like the water of the waterfall, above), as discussed below. 

Where ions are disordered over a partially occupied array of energetically equivalent sites, their motion is diffusive. Fig. 3.6 illustrates the variation in ionic potential with intersite position for three possible situations in which a set of energetically equivalent sites are partially occupied by mobile ions. The partially occupied sites may (a) share common faces in a continuously connected network through the structure, (b) be separated from one another by an array of empty sites, or (c) be separated from one another by an array of sites that are occupied by the mobile ions. 

An asymmetric hydrogen bond is common even where a proton coordinates two equivalent anions. The rc-bond repulsive forces between two coordinated anions tend to prohibit a close X-H-X separation, so competition between the two equivalent anions for the shorter X-H bond may set up a double-well potential for the equilibrium proton position between the two coordinated anions. With oxide anions, an O-H-O separation greater than 2.4 A sets up a double-well potential and creates an asymmetric hydrogen bond, which we represent as O-H O. Although displacement toward one anion may be energetically equivalent to a displacement toward the other, one well is made deeper than the other by an amount AH, as a result of the motion of the proton from the centre of the bond. 

The )5-aluminas are described in some detail in Chapter 2, only a few specific features are noted here. In the "-aluminas, the spinel blocks are stacked in such a way that the energetically equivalent sites occupied by Na" ions are ideally just half-filled in the -aluminas the spinel blocks are stacked so as to distinguish two types of Na" -ion sites of different potential energy, the Beevers-Ross (BR) and anti-Beevers-Ross (aBR) sites. In the Na-O planes, the shortest bottleneck distance 2.7 A is just a little greater than the sum of the ionic radii, 2.4 A, at room temperature, so a small value of AH can be anticipated. The discovery of fast Na -ion conductivity in the Na j5-aluminas (Yao and Kummer, 1967 Kummer and Weber, 1967) led to the invention of the Na/S battery that triggered extensive interest in the solid-electrolyte problem. 

Since the two structures are chemically and energetically equivalent, it is possible that in long chains one part tends to assume one version of the structure and another part the other version with a transition zone in between. This zone may extend over a few carbon atoms and one may assume that from one side the double bond gradually become longer and the single bond accordingly shorter, in order progressively to reverse the alternation of the bond sequence ... 

If, however, site interactions of atoms on energetically equivalent sites are equal and the standard molar volumes of mixing components are not dissimilar (i.e., within 5 to 10% difference), equation 3.157 may be simplified and the activity of any component i in the mixture (a,) may be expressed in a generalized fashion as shown by Helgeson et al. (1978) ... 

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