Polanyi parameter

The relationship between thermodynamics and kinetics in chemical reactions is usually expressed by the Bronsted equation (eq. 3.52 in chapter 3.4) k = gKa, where k is the rate constant, K is the equilibrium constant of the elementary stage, and g and a (Polanyi parameter) are constant values for a serious of reactions. These constants are determined by parameters characterizing the elementary mechanism (composition and structure of the activated complexes, etc.) thus allowing for the existence of an optimum catalyst, on which the rate of catalytic reaction per unit of surface has a maximum value. Equations of the type (3.52) were used for the explanation of "volcano-curves", when catalytic activity as a function of thermodynamic characteristics follows a curve with a maximum. An example for a volcano curve in methanation of CO is given in Figure 7.6. 

If the values of the Polanyi parameters are equal for both steps ai=oc2=a in the steady-state the reaction rate is given by... 

Here a is the constant of proportionality between the effective charge in the transition state of step 1 and adsorbed species and has the meaning of Polanyi parameter. In a similar way the... 

If one assumes that the values of Polanyi parameters are equal... 

It clearly follows from eq.(7.183), that when the values of the Polanyi parameters and effective charges are equal for both elementary reactions 2 and 3 , the selectivity would be exactly the same for uniform and nonuniform surfaces. However, if molecule A has two functional groups, which are different in their abilities of react with B (for example, olefinic and carbonyl groups in unsaturated aldehydes), the charge transfer toward catalyst surface is different for two adsorbed configurations (e.g.r A2 T AJ) and the selectivity will differ for an energetically nonuniform surface in comparison with a uniform surface. 

If the values of Polanyi parameter are equal to each other =a2 = a, then... 

Further, simplifying the assumption that the values of Polanyi parameters are the same and have a value of 0.5, typical for heterogeneous catalysis, results in an expression providing a possibihty to evaluate dependence of selectivity on the particle size... 

The same thermodynamic approach can be used for nonunifbrm surfaces giving instead of Eq. (7.81) for the two-step sequence and equal values of Polanyi parameter... 

Analysis ofEq. (7.407) indicates that the cluster size at maximum depends on the difference between Gibbs energy of adsorption on edges and terraces, the Polanyi parameter as well as the parameter p2> which is a function of the partial pressures (or concentrations in the case of reactions in the liquid phase) as well as temperature. 

The maximum TOF in the case ofEq. (7.411) has a rather comphcated behavior the analysis could be, however, simpfrfied for Ja=Zb and a typical value of Polanyi parameter a = 0.5 resulting after straightforward manipulations in... 

Assuming the same values of Polanyi parameters and irreversible poisoning, this expression can be essentially simplified, giving activity changes with time-on-stream... 

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