Additivity hypothesis

The deviation of the heat capacity of a solid solution from the heat capacity calculated by the additivity hypothesis (Kopp-Neumann rule), a quantity of importance for the evaluation of the... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

To evaluate averages with respect to the hard coordinates, we must introduce an additional hypothesis of local equilibrium for the distribution of values for these... 

Although there is some experimental justification for the conception of a saturation concentration yet this equation containing but one variable was found by Schmidt to conform to the experimental results but very indifferently. Schmidt Zeit. PJiys. Chem. Lxxviii. 667,1912) made the additional hypothesis that the adsorption coefficient k decreases in magnitude with the amount of gas adsorbed in an exponential manner, or... 

Example 5.5. In particular we have shown that Ms G) is a finite locally free Os -module in these cases (without additional hypothesis on 5). However, this is not true in general. 

An additional hypothesis that is consistent with and helps explain the role played by the 16 Hz amplitude modulation is as follows the nonlinear medium demodulates the carrier releasing a 16-Hz signal within the brain tissue, and it is the 16-Hz signal that causes changes in the calcium binding. 

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation ... 

A comment needs to be made about the state of the platinum surface during these experiments. The solutions that were used contained 0.05 mol dm 3 HC104 to curtail iron(III) hydrolysis and most of the work was carried out at 5°C to decrease the contribution of the homogeneous reaction. Under these conditions the surface of the platinum at the catalytic potential was always in the reduced state, irrespective of its preconditioning treatment. As mentioned in Sect. 4.4, such a surface specifically adsorbs iodide ions [239, 264, 265]. In order to allow for their effect, the current-potential curves for the Fe(III)/Fe(II) couple had been determined in the presence of a small concentration (5 x 10 6 mol dm 3) of potassium iodide. It is interesting that this treatment was sufficient to maintain the validity of the additivity hypothesis, in contrast to the results obtained for the Fe(CN)g" +1 system on cathodically pretreated platinum surfaces (Fig. 25). 

Under the previous assumptions and the additional hypothesis that k,j = Ujis independent of r (that is, all particles have the same Q ), equation (37) becomes... 

Genuine false equilibria exist. Investigations of H. Pflabon on the formation of sulphuretted hydrogen.—The false equilibria studied in the preceding chapter are apparent false equilibria they are nowise in disaccord with the principles of thermodynamics they contradict merely an additional hypothesis which represents, in certain cases, a sufficient approximation and which, in other cases, cannot be conserved. 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

Optical Rotatory Dispersion and Circular Dichroism. O.r.d. data for several carotenoids have been tabulated.C.d. data have been recorded for astaxanthin (27 and 3S), /8,jS-carotene-2,3r3 -triol (6) and jS,/S-carotene-2,3,2, 3 -tetrol (7), decaprenoxanthin (37), the synthetic 2,2-dimethyl-e,e-carotenes (60) and (61)/ and the novel monocyclic C50 carotenoid (12). In the last case only very rough agreement was obtained between the observed spectrum and that calculated on the basis of the additivity hypothesis. ... 

Phillips, O. and Gentry, A. H. (1993b). The useful plants of Tambopata, Peru II. Additional hypothesis testing in quantitative ethnobotany. Economic Botany, 47 (1) 33-43. 

In any experimental setup that constitutes an open system, what is measured is the heat exchange with the surrounding media, which includes the exchange of matter and a certain amount of work. In this case, one is tempted to introduce thermodynamic potentials, i.e., to introduce entropy. Nevertheless, we feel it more prudent to find a way to relate the experimental data with the internal energy without introducing any additional hypothesis. 

However, an additional hypothesis makes it possible to solve this problem. If we assume that the deformation distribution is a Gaussian function, we can write ... 

Thus interaction of hydralazine or a metabolite with macromolecules may underlie the immune response. An alternative or additional hypothesis involves inhibition of the complement system. The complement system helps remove immune complexes by solubilization, but if it is inhibited deposition and accumulation of such complexes would be increased. Hydralazine and some of its metabolites interfere with part of the complement system, inhibiting the covalent binding of complement C4 by reaction with the thioester of activated C4. However, the conentrations required are high relative to the normal therapeutic concentration. More recently it has been shown that hydralazine inhibits DNA methylation in the T-cell. The inhibition of DNA methyl transferase may initiate immune reactions via activation of genes as a result of this interference with DNA methylation. However, although the mechanism of hydralazine-induced LE is not yet understood, it is an important example of drug-induced toxicity for two reasons ... 

Here we suggest that ions or molecules temporarily bound to the membrane surface may have their transmembrane movement enhanced by pore formation and that this possible mechanism also has catalytic features. This additional hypothesis envisions that local membrane conformational changes can result from both the local transmembrane voltage and the surface binding of a transported molecule (S). That is, a pore-substrate complex is formed. One possible outcome is transmembrane transport in which S is temporarily bound to the inner surface of a pore, with subsequent electrical lateral motion (relative to the pore inner surface) by diffusion or lateral drift to the other side. Alternatively, as a pore shrinks and closes, S is presented to the other side of the membrane. In either case, upon dissociation, transport of S will have been accomplished. 

The overall mechanism summarises many elementary reactions. The method explained has the advantage that it is free of any additional hypothesis. Its disadvantage is, that it cannot be applied in general. This fact will be discussed for the reaction A B —> C. 

Equation 8 predicts a reaction rate order between 1 and 2. The fitting of the turnover frequencies data for the system Me2Si(2-Me-Benz[e]ind)2ZrCl2/]VIAO with eq 8 is adequate only for certain reciprocal values of the parameters, which turned out to be inconsistent with the kinetic scheme hence, additional hypothesis and calculations will have to be worked out. 

In the nonrelativistic quantum mechanics to which we are confining ourselves, electron spin must be introduced as an additional hypothesis. We have learned that each physical property has its corresponding linear Hermitian operator in quantum mechanics. For such properties as orbital angular momentum, we can construct the quantum-mechanical operator from the classical expression by replacing p Py,Pz by the appropriate operators. Hie inherent spin angular momentum of a microscopic particle has no analog in classical mechanics, so we cannot use this method to construct operators for spin. For our purposes, we shall simply use symbols for the spin operators, without giving an explicit form for them. 

In electrochemical kinetics, this model corresponds to the Butler-Vohner equation widely used for the electrode reaction rate. The latter postulates an exponential (Tafel) dependence of both partial faradaic currents, anodic and cathodic, on the overall interfacial potential difference. This assumption can be rationalized if the electron transfer (ET) takes place between the electrode and the reactant separated by the above-mentioned compact layer, that is, across the whole area of the potential variation within the framework of the Helmholtz model. An additional hypothesis is the absence of a strong variation of the electronic transmission coefficient", for example, in the case of adiabatic reactions. 

Using the above Iq, the additivity hypothesis kon (the sum being made over all double-bonded C-atoms) was verified for the 28 alkenes and nonconjugated dialkenes for which data are available. All predicted oh aro within 20 % of the experimental values, and the standard deviation is only 12.6 % (excepting a-pinene, for which the deviation is 30 %, probably due to steric hindrance caused by the bridged 6-cycle) [19],... 

The most comprehensive attack on the theoretical problems posed by the existence of liquid semiconductors has been made by Mott (1966, 1967, 1970, 1971). TTie essential features of his model are shown in Figure 7.53 and have been described in detail by Cohen, Fritzsche and Ovshinsky (1969) and by Davies and Mott (1970). and separate the extended from the localized states (shown shaded) and mark the energy at which a substantial drop in the mobility occurs. An additional hypothesis is that the energy range of localised states is somewhat greater in the conduction band than in the valence band, thereby explaining the tendency for liquid semiconductors to exhibit positive thermoelectric powers. All the liquids... 

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