Sabatier analysis

For the general case, the limiting Sabatier Volcano-Curve can be defined as  

Here max Rt is the maximal rate of reaction step i, which is calculated by assuming optimal coverages for that reaction step. This (usually multi-dimensional) volcano-curve we shall refer to as the Sabatier volcano-curve, as it is intimately linked to the original Sabatier principle [132,133]. This principle states that desorption from a reactive metal catalyst is slow and will increase on less reactive metals. On very noble metals the large energy barrier for dissociation will, however decrease the dissociation rate. The best catalyst must be a compromise between the two extremes. As has been shown above, this does not necessarily mean that the optimal compromise is obtained exactly where the maximal desorption and dissociation rates are competing. That is only the case far from equilibrium. Close to equilibrium the maximum will often be attained while dissociation is the rate-determining step, and the maximum of the volcano-curve will then be reached due to a lack of free sites to dissociate into. 

We will refer to the method of using the maximal possible reaction rates for all the reaction steps of a heterogeneously catalyzed reaction as Sabatier Analysis . This analytical method might prove useful for various purposes in the future. Here its 

This Sabatier Analysis shows that the assumption of rate-determining dissociation was valid to a large extent We would have to modify the results obtained from the simple analysis based on a rate-determining step only at relatively small approaches to equilibrium, extremely low pressures or quite strongly bound molecular precursor states. 

We now introduce a method that provides the simplest possible conceptual framework for analyzing microkinetic models of heterogeneous reactions, the so-called Sabatier analysis. We call it so because it brings out the qualitative reasoning behind the Sabatier principle in a quantitative form. 

Consider again Reaction (7.1), but let us now relax the assumption that the activation of A is rate determining. The approach to equilibrium for the fuU reaction is shown in Equation (7.5), and we can write the equilibrium constant in terms of equilibrium constants for the two elementary steps  

We will now focus on the net reactions proceeding in the forward direction so that 

We will start by analyzing the reaction on the surface of a catalyst that bonds intermediates too strongly (to the left of the maximum in the activity map, Fig. 7.2). The surface coverage will be high (0 1) and desorption of AB will be the rate-determining step. This means that the first reaction step (dissociative adsorption of A, Eq. 7.2) is in equilibrium and hence that /j 1 and V/ The TOP, can now be approximated via the second reaction step, r  

For too noble surfaces (to the right of the maximum in the activity map, Fig. 7.2), where dissociation of A is rate determining, the coverage of free sites wiU be approximately 1 (d, 1), and we have the second step in equilibrium so that and y, y. For such catalysts, we can approximate the TOF as 

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Determine the values of the descriptors from step 1 that yield optimal catalytic activity. This determination can, again, be made empirically, via microkinetic modeling, or via Sabatier analysis. 

A simple tool is described, which provides a conceptual framework for analyzing microkinetic models of heterogeneous reactions. We refer to this tool as the Sabatier Analysis . The Sabatier Analysis of the microkinetic models developed in this section suggests that the clustering of good catalysts can be explained by the combination of the universal BEP-relation and activated re-adsorption of synthesis products onto the catalyst. 

Figure 7.6 shows the Sabatier map in comparison to the full solution from the microkinetic model. The Sabatier map gives an excellent description for small approaches to equilibrium (y-> 0). There is some discrepancy at the maximum where the Sabatier analysis predicts too high rates. This can be attributed to a failure of describing coverages that are in between 0 and 1, since these are the limiting cases... 

The Sabatier analysis can in principle be performed for any reaction. In case that there are more than two reaction steps, the Sabatier volcano could be constructed in analogy to Equation (7.16) by assuming that all intermediates that go in the forward direction have optimal coverages and by calculating the approach to equihbrium for each forward rate from the given approach to equilibrium for the overall reaction (as done in Eq. 7.8 for the reaction earher) under the assumption that all other partial reactions are in equihbrium. This wiU give a first approximation to each forward rate of the reaction. The Sabatier volcano provides an upper limit to the total rate by setting this rate equal to the minimum of all forward rates ... 

The former is most facile when the S-surface bond is strong, while that makes the latter more difficult. A Sabatier analysis using the adsorption energy of SH as descriptor describes the known trends quite well (see Fig. 7.11). RuS and Co-and Ni-promoted MoS are found to be closest to the top, but there may be room for improvement. 

Comparison of the volcano curves obtained from a Sabatier analysis and full microkinetic models at different approaches to equilibrium y. Reprinted from T. Bligaard, et al., J. Catal., 2004, 224, 206-217 with permission from Elsevier. ... 

The Sabatier analysis for surface reactions was first introduced by Bligaard et al and uses a Sabatier rate that is obtained by following this fairly simple recipe ... 

The simplified NH3 synthesis mechanism is almost identical to the generic mechanism used in the introduction of the Sabatier analysis in Section 1.4. If we substitute A2=N2 and B = 3/2H2, we can directly use the previously obtained results and apply them to the current problem. This gives us the following two maximum rates for each step, where the stoichiometric factor of 3/2 for H2 is included as the exponent to Pyi in the expression for... 

In summary, this example of computational screening for an NH3 synthesis catalyst has demonstrated how BEP relations are applied to reduce the number of parameters to a single descriptor, AEn, and how to perform a Sabatier analysis. As a result we obtained the Sabatier volcano shown in Figure 1.12... 

We have already performed a preliminary Sabatier analysis of the CO oxidation reaction in Section 1.5.2, and derived an analytic solution under the assumption that the adsorption of CO and O2 are quasi-equilibrated in Section 1.6. Now we will formulate a numerical solution to the complete microkinetic model as a function of the descriptors AEco and AEq- We will analyze the reaction mechanism in terms of rate and catalyst control, and at the end of this section, the effect of high surface coverages on the volcano curve will also be briefly addressed. 

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