The Theta-Rule

It has been mentioned above that with nonuniform catalyst surfaces, as they mostly occur in practice, the above equations give merely an upper limit of the velocity although in such catalysts the true surface would be expected to be greater than the geometrical one. In nearly all the cases where a reaction has been measured on different catalysts, it has been found that fco is by no means a universal constant as would have to be expected from the simple theory. There is a regular connection between fco and the activation energy, of the general form  

A study of the relationship between the rates of catalyzed reactions and the concentrations of reactants may clear up the intermolecular mechanism and such studies may help in finding out which molecules are adsorbed, whether the adsorption is weak or strong, and which fraction of the surface is active. Every theory of the intermolecular mechanism can and must be checked by a calculation of the absolute velocity. 

We shall now go a step farther and consider the intramolecular mechanism by asking what changes are effected within the molecules during the reaction as a consequence of their adsorption by the catalyst. The study of temperature coefficients is useful for this purpose. This yields the energy of activation, q, according to 

Polanyi (27) as early as 1921 suggested that the activated state consists of free atoms which are adsorbed or bound by large affinities (heat of adsorption or diminution of the homogeneous activation energy) to the catalyst. A more quantitative treatment became possible after the theory of wave mechanics presented equations for the mutual interaction of different covalent bonds. London (28) showed that the total interaction energy of three atoms X, Y, and Z is 

B and C being the Coulomb energy functions, a, 0, and y the resonance energy functions over the distances XY, YZ and XZ. Hence, if these functions are known (a usual approximation is to simply subtract 10 % of the total energies for the Coulomb interaction), the saddle energy may be calculated for certain distances as being the activation energy of the reaction 

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Another old mystery in heterogeneous catalysis, dating back to F. H. Constable (1925) and G.-M. Schwab (1929), was the compensation effect or theta rule. The Arrhenius plots for similar reactants on the same catalyst or for the same reactant on similar catalysts would differ in slope across a common point of intersection. Approximately 70 years later, a First Workshop on the Compensation Effect was organized (DECHEMA, Berlin, 1997) to debate this enduring mystery. Werner Haag demonstrated that such an effect must necessarily result from the temperature dependence of reactant adsorption (and hence its site concentration) and that of the reaction rate of the adsorbed species, operating in opposite directions. A second workshop may never follow ... 

This relation is called an isokinetic relation, compensation law, or theta rule. The last name was sometimes applied for this phenomenon in the field of heterogeneous catalysis (cf. Cremer, 6) where it was observed mainly with a series of different catalysts. 

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