The Additivity Principle

Department of Chemistry, Imperial College of Science, Technology and Medicine, London SW7 2AY United Kingdom 

An electrode in a system containing two couples automatically adopts a mixed potential (or more correctly, a mbcture potential) At this 

Modem Aspects of Electrochemistry, Number 34, edited by John O M. Bockris et al Kluwer Academic / Plenum Publishers, New York, 2001. 

The anodic current due to couple 1 and the cathodic current due to couple 2 (-/mix) exactly balance at E, as illustrated in Fig.l. Each of these mixture currents also represents the rate v at which the reaction between the couples 

Provided the additivity principle holds, the catalytic rate Vcat of reaction (3) at the surface should therefore be the same as the value Vmx predicted from the current-potential curves of the eouples involved. Moreover, the measured potential Eau of the catalyzed mixture should be 

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The phenomenon was established firmly by determining the rates of reaction in 68-3 % sulphuric acid and 61-05 % perchloric acid of a series of compounds which, from their behaviour in other reactions, and from predictions made using the additivity principle ( 9.2), might be expected to be very reactive in nitration. The second-order rate coefficients for nitration of these compounds, their rates relative to that of benzene and, where possible, an estimate of their expected relative rates are listed in table 2.6. 

The isomer proportions for the nitration of the chlorotoluenes, to be expected from the additivity principle, have been calculated from the partial rate factors for the nitration of toluene and chlorobenzene and compared with experimental results for nitration with nitric acid at o °C. The calculated values are indicated in brackets beside the experimental values on the following structural formulae. In general, it can be... 

The suggestion outlined above about the way in which through-conjugation influences the nitration of p-chloronitrobenzene is relevant to the observed reactivities (ortho > meta > para) of the isomeric chloronitrobenzenes. Application of the additivity principle to the... 

The lack of steric hindrance is also shown by the kinetic data for p-xylene, mesitylene, and durene, the observed reactivities being close to those calculated by the additivity principle. The additivity principle has also been tested for the last seven compounds in Table 177, and for the first five of these it holds very well. If one assumes a value for/3Me0 of ca. 4.0 and takes the average of the values listed in the table for the methyl substituent partial rate factors, then the observed calculated reactivity ratios are 1.6, 0.85, 0.75, 1.4 and 1.0. For the last two compounds in the table the ratios are 5.3 and 4.1, the reason for this being unknown. 

For the additivity principle to hold, steps (8) and (9), the anodic processes taking place under nitrogen, would be coupled only with the cathodic reduction of oxygen in step (11). It is step (10), the attack by the constituents of the oxygen couple on the intermediate Cu, which lies outside the scope of the principle and which explains the observed findings. 

Andersen et al. predicted that similar results would be expected for the corrosion of other multivalent metals oxidizing via lower oxidation states. They also pointed out that their interpretation was consistent with the kinetics of the corrosion of copper in oxygenated HCl solutions. Here the final product is Cu and thus there is no vulnerable intermediate. In consequence, the rate of copper dissolution from either Nj-saturated or 02-saturated HCl solutions was the same at a given potential in conformity with the additivity principle. 

The underlying problem in testing the validity of the additivity principle in corrosion, mineral extraction, and electroless plating is that the electrode metal itself forms part of one of the half-reactions involved, e.g., zinc in equation (5) and copper in equations (8) and (12). A much better test system is provided by the interaction of two couples at an inert metal electrode that does not form a chemical part of either couple. A good example is the heterogeneous catalysis by platinum or a similar inert metal of the reaction... 

Experiments by Freund and Spiro/ with the ferricyanide-iodide system showed that the additivity principle held within experimental error for both the catalytic rate and potential when the platinum disk had been anodically preconditioned, but not when it had been preconditioned cathodically. In the latter case the catalytic rate was ca 25% less than the value predicted from adding the current-potential curves of reactions (15) and (16). This difference in behavior was traced to the fact that iodide ions chemisorb only on reduced platinum surfaces. Small amounts of adsorbed iodide were found to decrease the currents of cathodic Fe(CN)6 voltam-mograms over a wide potential range. The presence of the iodine couple (16) therefore affected the electrochemical behavior of the hexacyanofer-rate (II, III) couple (15). 

The additivity principle was well obeyed on adding the voltammograms of the two redox couples involved even though the initially reduced platinum surface had become covered by a small number of underpotential-deposited mercury monolayers. With an initially anodized platinum disk the catalytic rates were much smaller, although the decrease was less if the Hg(I) solution had been added to the reaction vessel before the Ce(lV) solution. The reason was partial reduction by Hg(l) of the ox-ide/hydroxide layer, so partly converting the surface to the reduced state on which catalysis was greater. 

A Critique of the Additivity Principle for Mixed Couples Spiro, M. 34... 

Martin was the first to point out that a substituent changes the partition coefficient of a substance by a given factor that depends on the nature of the substituent and the two phases employed, but not on the rest of the molecule. Martin s treatment assumes that for any stated solvent system, the change in retention (ARmin TLC) caused by the introduction of group X into a parent structure is of constant value, providing that its substitution into the parent structure does not result in any intramolecular interactions with other functions in the structure. On the other hand, it can be appreciated that if the introduction of a group into a structure causes a breakdown in the additivity principle, then intra- or intermolecular effects are likely to be more significant within the substituted structure. These effects are as follows ... 

J Kovacs, R Cover, G Jham, Y Hsieh, T Kalas. Application of the additivity principle for prediction of rate constants in peptide chemistry. Further studies on the problem of racemization of peptide active esters, in R Walter, J Meienhofer, eds. Peptides Chemistry, Structure and Biology. Ann Arbor, MI, 1975, pp 317-324. 

From equations 8.104 and 8.105, applying the electroneutrality condition, it follows that the additivity principle is also valid for conventional properties—i.e.,... 

Equations 8.107 and 8.108, referring to absolute properties, are equally valid for conventional properties based on equations 8.105 and 8.106. Moreover, the additivity principle is applied to all partial contributions thus, for a generic electrolyte K, we have ... 

DilP recently discussed the merits and limitations of models that assume thermodynamic additivity and independence (of energy types, of neighbor interactions, of conformational freedom, of monomer contact pairing frequencies, etc.). He states that biological molecules may achieve stability in the face of thermal uncertainty, as polymers do, by compounding many small interactions this summing can stump modelers because application of the additivity principle leads to accumulated error. Entropies and free energy may not be additive to describe weak interactions that are ensembles of states. He concludes that additivity principles appear to be few and limited in scope in biochemistry. 

The amount of heat generated by a reaction is the same whether the reaction takes place in one step or in several steps. Hence, A//values (and, thus, AG values) are additive. This law, also known as the Law of Constant Heat Summation, was the earliest example of the Additivity Principle, which also states that AG values are additive. In biochemical processes, especially when dealing with multiple steps as in protein folding, application of the Additivity Principle can give spurious results if the accuracy and precision of the thermodynamic parameters is insufficient. 

Expressions for the optical anisotropy AT of Kuhn s random link (an equivalent to the stress-optical coefficient) of stereo-irregular and multirepeat polymers are derived on the basis of the additivity principle of bond polarizabilities and the RIS approximation for rotations about skeletal bonds. Expressions for the unperturbed mean-square end-to-end distance , which are required in the calculation of Ar, are also obtained. 

Since linear variation of hardness is not always the case, equation (5.7) is approximate. But Glazov and Vigdorovich consider that the production of many very complex solid solution systems does require some method if only rough, for hardness analysis of such systems. They formulate the additivity principle for multicomponent systems as follows the numerical increase in hardness of multi-component solid solutions equals the sum of hardness increments in bi-component solutions... 

The validity of the additivity principle has been tested so far in relatively few cases, but it has been notably successful in predicting correctly the rates of halogenation (72) and hydrogen exchange (9) reactions of polymethylbenzenes. 

In order to determine the applicability of the additivity principle for polysubstituted models, the rate data in Table III have been expressed in terms of partial rate factors of individual substituent groups in Table IV. The partial rate factors for polysubstituted compounds are calculated on the same basis as those of monosubstituted ones. For instance, the partial rate factors in guaiacol refer to the factors by which the rate constants of phenol are modified in positions 3, 4, 5, and 6 by introducing a methoxyl group in position 2. Likewise, the partial rate factors for methoxyl in 4-hydroxy-3,5-dimethoxytoluene indicate how the protodedeuteration rates of 4-hydroxy-3-methoxytoluene at positions 2 and 6 are modified by inserting a methoxyl group at position 5. 

A comparison of/0OMe and/p0Me values of different compounds shows that 1,2-disubstitution in veratrole causes a distinct but minor lowering of these rate factors. A further lowering is observable in the 1,2,3-substituted 2,6-dimethoxyphenol, although the deviation even in this case is not very large. The conformity with the additivity principle is, as a matter of fact, much better than that estimated by Satchell (77), who based his conclusions on the reactivity of 3,5-ditritio-2,6-dimethoxyphenol. 

Partial Rate Factors of a Methyl Group. Comparison of the f0Me and /mMe values of toluene with those of 4-hydroxy-3-methoxytoluene does not reveal significant differences. The second /0Me value that was determined for position 2 in the latter compound is lower than that for position 6 owing to steric inhibition effects caused by two ortho substituents. Consequently, the additivity principle appears valid for the 1,3,4-trisubstitution pattern. 

The two remaining aromatic positions in 4-hydroxy-3,5-dimethoxy-toluene are analogous to the 2-position in 4-hydroxy-3-methoxytoluene, and the additivity principle would predict/0Me values of equal magnitude. The observed values differ, however, by a factor of 10. The lower reactivity of the former compound is again related to the sterically crowded substitution pattern. 

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