Rate volcano plot

Figure 6.3. Examples for the four types of global electrochemical promotion behaviour (a) electrophobic, (b) electrophilic, (c) volcano-type, (d) inverted volcano-type, (a) Effect of catalyst potential and work function change (vs I = 0) for high (20 1) and (40 1) CH4 to 02 feed ratios, Pt/YSZH (b) Effect of catalyst potential on the rate enhancement ratio for the rate of NO reduction by C2H4 consumption on Pt/YSZ15 (c) NEMCA generated volcano plots during CO oxidation on Pt/YSZ16 (d) Effect of dimensionless catalyst potential on the rate constant of H2CO formation, Pt/YSZ.17 n=FUWR/RT (=A(D/kbT).

A very useful analysis of catalytic reactions is provided for by the construction of so-caUed volcano plots (Figure 1.2). In a volcano plot, the catalytic rate of a reaction normahzed per unit reactive surface area is plotted as a function of the adsorption energy of the reactant, product molecule, or reaction intermediates. 

A volcano plot correlates a kinetic parameter, such as the activation energy, with a thermodynamic parameter, such as the adsorption energy. The maximum in the volcano plot corresponds to the Sabatier principle maximum, where the rate of activation of reactant molecules and the desorption of product molecules balance. 

Figure 1.2 Volcano plot illustrating the Sabatier principle. Catalytic rate is maximum at optimum adsorption strength. On the left of the Sabatier maximum, rate has a positive order in reactant concentration, and on the right of Sabatier maximum the rate has a negative order.

Sabatier s Principle is illustrated in Fig. 6.40 where the ammonia rate is plotted for similar conditions versus the type of transition metals supported on graphite. The theory outlined so far readily explains the observed trends metals to the left of the periodic table are perfectly capable of dissociating N2 but the resulting N atoms will be bound very strongly and are therefore less reactive. The metals to the right are unable to dissociate the N2 molecule. This leads to an optimum for metals such as Fe, Ru, and Os. This type of plot is common in catalysis and is usually referred to as a volcano plot. 

The critical role of the M/M—OH redox system in determining the population of the surface active metal sites is, with high probability, the actual reason for the strong predictive power of the M—Ox bond strength with regard to the relative rates of ORR at different metal surfaces. In fact, a better presentation of the volcano plot would be obtained by using, for the ordinate of the plot the value (1 /Z + 1) exp(— /RT),... 

For the ascending branch of the volcano plot, the term (1/Z + 1) could serve by itself as an effective ORR activity predictor, whereas, for the descending branch, (1/Z + 1) becomes close to unity at 0.85 V, and the exponential factor exp(—A//, /R70, then determines the ORR rate based on the residual interaction of dioxygen with the (excessively) noble metal catalyst surface. 

The expression (1/Z+ 1)] exp[— AHl /RT] at 0.85 V, better reflects the reality of a partially oxidized Pt surface and the critical effect of active site availability on the rate of the ORR. Effects of site availability were not considered in the calculations in Nprskov et al. [2004] of ORR activity for various metals. The expression used to calculate activity defined the ordinate parameter in the ORR volcano plots presented. This parameter was defined in Nprskov et al. [2004] as kT min,- log(k,/ko). 

FIGURE9.t. Volcano Plot of formic acid decomposition. Abscissa Calculated A HadsofHCOOH Ordinate Temperature at which rate of HCOOH decomposition reaches the same value for all metals. 

Several important conclusions can be drawn from Figure 4.38. It appears that in general a simple catalytic reaction, which includes the dissociation of a diatomic molecule, will have this dissociation as the rate-determining step, when the reaction takes place under conditions close to equilibrium. This agrees well with the ammonia synthesis being dissociation rate-determined, as this process is the prototype of an equilibrium-limited reaction [128]. When the reaction is taking place far from equilibrium, the actual approach to equilibrium becomes unimportant, and the volcano plot very closely follows the volcano defined by the minimum value among the maximal possible rates for all reaction steps. 

Figure 2.15 Examples of volcano plots, describing the reaction rate as a function ofthe heat of adsorption (left), and the activity of the second-row and third-row transition metal sulfides in the hydrodesulfurization of dibenzothiophene (right).

Thus the reaction rate as a function of the temperature will present a typical volcano plot as seen on Fig. 14. 

Figure 1.9 Volcano plot showing dependence of rate on strength of adsorption the upper part shows the corresponding variation of surface coverage 6.

In the above we have deliberately stayed close to Vannice s model, and changed only the nature of the rate-determining step from H-assisted C-O bond breaking (15) into (irreversible) hydrogenation of surface carbon (28). Consequently, the overall rate equation (30) differs from Eq. (17) only in that it contains the term 0c instead of Ochoh- When assuming further that Cads instead of CHOHajs is the most abundant surface intermediate, the model can be made formally identical to that of Vannice, including the explanations it offers for the volcano plot and the compensation effect. 

Even more common appears to be the case of "inverted-volcano" plots (Figures 26 to 28) where the rate goes through a minimum with eO. As shown in these figures, the rate minimum frequently coincides with open-circuit, i.e., unpromoted, operation (AeO=0). 

These considerations can also be applied to other metals. Thus die (100) planes of metals with larger atomic spacings than nickel (e.g., Pd, Pt, and Fe) should exhibit weaker chemisorption, and the same should also be true of metals with shorter interatomic distances such as tantalmn. Figure 5-19 shows the rate of ethylene hydrogenation as function of metal-metal distance (volcano plot). 

In the following example we shall examine the hydrogenation of CO on various metal catalysts. A clear dependence of reaction rate on d-band filling is observed (Fig. 5-25). Thus the familiar volcano plots can also be explained by an electronic factor [38]. 

In long-term studies, however, if a drug is effective, people may stay on it longer than they remain on the control arm. In such cases, comparing annualized rates relative to time of exposure may be more relevant. In case where the adverse events occur most frequently at the beginning of exposure, piecewise rates (e.g., with the first 30 days and after 30 days) might be more informative. Depending on the type of metric chosen, the volcano plots may present not risk difference on the x-axis, but another measure such as hazard ratio, risk ratio, or difference between annualized event rates. 

As a result, it is to be expected that the maximum rate of hydrogen evolution will occur at intermediate values of AG ds which lead to a significant but not monolayer coverage by adsorbed hydrogen atoms. This is indeed observed and Fig. 1.16 shows a volcano plot of exchange current density vs AG ds for a series of metal cathodes. Similar dependences of rate parameters on the free energy of adsorption of an intermediate are common in gas-phase catalysis. 

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