Principle of additivity

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

Rheological Classification of Drilling Fluids 829. Flow Regimes 830. Principle of Additive Pressures 834. Friction Pressure Loss Calculations 836. Pressure Loss Through Bit Nozzles 839. 

Equation 4-104 expresses the principle of additive pressures. In addition to Equation 4-104, there is the equation of slate for the drilling fluid. 

The method used for predicting the different physical and mechanical characteristics of crystalline or glassy polymeric composites is somewhat different. Most frequently it has been proposed (cf., e.g. [118]) to introduce an extra term into the relationship between some characteristic and the composition of the material based, more often than not, on the principle of additivity of the filler and matrix characteristics and taking into account their relative volumes in the composite. This extra term is the product of the interphase volume by a characteristic other than the characteristics of either the matrix or the filler. 

Brubaker and Sincius have reported the exchange reaction as not occurring in sulphate media ([H ], 3 to 12 M) unless chloride ions are present. McKay plots deviating from linearity were observed for added chloride ion 0.4 to 6.0 M, and complicated dependencies on the ions H , and Sb(V) were also found. For mixtures of Sb(lll) and Sb(V), the principle of additivity of absorbance was obeyed in sulphate media. 

Scheme 6b can be further condensed into one line (Scheme 6c) according to the principle of addition of parallel branches. The rate constants k2 and / -3[Q] are multiplied by the fractional concentration of the enzyme species within the equilibrium segment that participates in that pathway, /eab and /e. 

The principles of additivity of residence times and additivity of variances can be extended to several reaction vessels in series (Fig. 2.19). Thus, for vessels A, B and C, if tA, 1b, tc are the residence times and a, a, passing through each of the vessels in turn, then the overall residence time will be tA + 7 -1- tc and the overall increase in the variance of a pulse will be a -(- + dispersion number for the vessel must... 

This question could be answered more easily if we knew that the C6C3 units conformed with the principle of additivity. This principle can be formulated as follows. If the introduction of each of two substituents alters the free energy of activation at a particular position by amounts x and y, the presence of both substituents would alter the free energy of activation by an amount (x + y). If this relationship holds, it allows one to predict the reactivities of the individual positions in disubstituted benzene derivatives from the rate data obtained for the corresponding monosubstituted ones. The partial rate factor for a given position of a... 

If the protodedeuteration rates were governed strictly by the principle of additivity, the partial rate factors would be independent of the degree,of substitution, except for the rate factors for ortho positions flanked by two substituent groups. Examples of such factors are/0Me0 for 4-hydroxy-3,5-dimethoxytoluene and /0Me for the position 2 of 4-hydroxy-3-methoxy-... 

The data in Table IV demonstrate both reasonable agreement and substantial deviations from the principle of additivity discussed in detail below. 

C. Application of the Principle of Additivity of Free Energies of Activa-... 

For p-xylene, mesitylene, and durene the rate constants can be calculated from the following equations which follow from the principle of additivity ... 

Especially by the possibility of these two refinements, the principle of additivity becomes even more useful for practice and permits us to estimate physical quantities with an accuracy, which could hardly be expected. 

The analysis of various physical-chemical macro- and microprocesses results in the conclusion that in many cases the inverse values of kinetic or energy parameters of subsystems are added when estimating the resulting interaction of atom-molecular structures. Therefore tabulated (initial) values of spatial-energy parameters can be calculated based on the principle of addition of inverse values of energy components of free atom systems [3] ... 

Another similar route using the same principle of addition reaction was studied by two independent groups81 92. ... 

For the computation of the adsorbent-adsorbate energies, it has been customary to adopt the principle of additivity of the pairwise interactions. Thus, each atom or ion in the surface layers of the solid is regarded as a force centre and the total interaction experienced by the adsorbate molecule is given by the summation... 

Using the principle of additivity, we calculate the approximate value of standard entropy ... 

On the Additivity of the Kerr Constant. The principle of additivity states that the molecular Kerr constant Km of a system is the sum of the molecular Kerr constants of its components. For a two-component system, this principle yields ... 

In the solid state, tri-O-acetyl-p-D-xylopyranosyl halides adopt the all-equatorial 4Ci conformation, -whereas tri-O-benzoyl-p-D-xylopyranosyl halides crystallize in the all-axial 1C4 conformation, except for tri-O-benzqyl-p-D-xylopyranosyl chloride, which adopts the (skew) conformation. The variations of bond lengths to the anomeric carbon atom are discussed with regard to the anomeric effect. By comparative studies of the conformational equilibria of acetylated and benzoylated 1,5-anhydropentitols on the one hand, and pentopyranoses exhibiting the anomeric effect on the other, data for the non-bonding interactions may be derived. Based on the principle of additivity of these factors, the conformational distribution of structurally related compound may be estimated. In connection with these results, it is noted that two benzoyl groups show a smaller 1,3-diaxial interaction than two acetyl groups the difference between these interactions is about... 

The systematic names of coordination entities are derived by following the principles of additive nomenclature, as outlined in Chapter IR-7. Thus, the groups that surround the central atom or structure must be identified in the name. They are listed as prefixes to the name of the central atom (see Section IR-9.2.2.1) along with any appropriate multipliers (see Section IR-9.2.2.2). These prefixes are usually derived in a simple way from the ligand names (see Section IR-9.2.2.3). Names of anionic coordination entities are furthermore given the ending ate . 

Using the principle of additivity of extrathermodynamic parameters (see section A), we can sum the same parameters for various substituents at different position and write... 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Next we shall verify that, when the entropic parameter z is an extensive variable of state in the thermodynamic limit, the ideal gas is in accordance with the principle of additivity. Suppose that the system is divided into two subsystems (1 and 2). Then the extensive variables of state of the canonical ensemble are additive... 

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics in the microcanonical ensemble. Equation (140) proves the zero law of thermodynamics for the microcanonical ensemble [6]. 

A Weight-of-Evidence (WOE) Method To provide further guidance on the evaluation of chemical interactions and their impact on risk values calculated from data not reflective of the mixture, a weight-of-evidence (WOE) method was developed [25,26], The assessment of WOE for interactions enables assessors to judge if the interactions influence the overall toxicity of the mixture and if the anticipated joint toxicity will be greater than or less than expected based on the principle of additivity. The WOE method yields a composite representation of all the evidence on toxicologic interactions from human studies to animal bioassay data relevance of route, duration, and sequence and the significance of interactions. The method consists of a classification... 

No less important for the understanding of the hydrophobic properties of chemical compounds is the matter of additivity of hydrophobic increments of substituents. On the one hand, many examples of successful employment of the additivity approach in calculating the relative hydrophobicity of various chemical compounds are reported 1-310). On the other hand, quite a few examples of significant deviations of the experimental data from the calculated values are known (see, e.g., Ref.112) and ll3)). The results obtained in the study of the relative hydrophobicity of a number of peptides 11,ln) have shown that the above additivity principle is often invalid for amino acid residues under the conditions employed. Comparison of the estimates of the relative hydrophobicity of such mononucleotides as AMP, ATP, and c-AMP 110) indicates that formation of the intramolecular covalent bond affects the relative hydrophobicity of AMP much more than introduction of two additional phosphate groups. Such a result would be quite unpredictable on the basis of the principle of additivity of hydrophobic increments of substituents. The infringement of the... 

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