Inverted behaviour

Both indirect and direct evidence supports this prediction. The first observation of an inverted region in semiconductor ET was made by Lu et al. (1993), in a study of back electron transfer from the conduction band of colloidal TiOj into Fe" (CN)5L" , where L is a series of substituted pyridines. Dang and Hupp (1999) also confirmed inverted behaviour in similar experiments on back ET from SnOj nanoparticles into a series of Ru"L3 " complexes. 

In experimental studies of photoinduced electron transfer reactions, the free energy dependence of the quenching of a particular metal complex (e.g, Ru(bpy)3 + ) by a series of structurally related quenchers of known properties is used to confirm the calculated potential of the excited state couple and to estimate its exchange rate. For a given quencher a graded series of polypyridine complexes with different substituted bpy/phen ligands can also be used equivalently. These provide a set of experimentally measured data to check the observation of normal and inverted behaviour predicted by Marcus theory under certain conditions. Rate constants for back electron transfer of photoredox products have been measured in a similar manner in some cases and these were also subjected to analysis. 

The main difference between the three functions is in the repulsive part at short distances the Lennard-Jones potential is much too hard, and the Exp.-6 also tends to overestimate the repulsion. It furthermore has the problem of inverting at short distances. For chemical purposes these problems are irrelevant, energies in excess of lOOkcal/mol are sufficient to break most bonds, and will never be sampled in actual calculations. The behaviour in the attractive part of the potential, which is essential for intermolecular interactions, is very similar for the three functions, as shown in... 

At the TS the energy along the reaction path is a maximum, but it is a minimum in the perpendicular direction(s). A one-dimensional cut through the (0,0) and (1,1) comers for path A in Figure 15.30 thus corresponds to Figure 15.28. A similar cut through the (0,1) and (1,0) comers will display a normal (as opposed to inverted) parabolic behaviour. 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

Electrochemical promotion of the unpromoted Rh/YSZ film, via application of 1 or -1 V, leads to significant rate enhancement (tenfold increase in rCo2> four fold increase in rN2 (filled circles and diamonds in Fig. 2.3). This is a catalytic system which as we will see in Chapters 4 and 8 exhibits inverted volcano behaviour, i.e. the catalytic rate is enhanced both with positive and with negative potential. 

Many reactions exhibit both electrophobic and electrophilic behaviour over different UWr and O ranges leading to volcano-type51 (Fig. 4.16) or inverted-volcano-type (Fig. 4.25) behaviour.70... 

Figure 4.33. Inverted volcano behaviour. Effect of catalyst potential and work function on the rate of C2H6 oxidation on Pt/YSZ. po2=107 kPa, pc2H6 65 kPa T=500°C , T=460°C , T=420°C.24 Reprinted with permission from Academic Press.

A fourth important case of r vs O dependence is the inverted volcano behaviour depicted in Figure 4.33 for the case of C2H6 oxidation on Pt/YSZ.24 The rate is enhanced by a factor of 7 for negative potentials and by a factor of 20 for positive ones. 

Figure 6.2. (Top) Definitions of local electrophobic and local electrophilic behaviour for two reactions exhibiting global volcano-type behaviour (a) and global inverted-volcano-type behaviour (b). (Bottom) Corresponding variations in surface coverages of adsorbed electron donor (D) and electron acceptor (A) reactants. As shown in this chapter volcano-type behaviour corresponds in general to high reactant coverages, inverted-volcano-type behaviour corresponds in general to low reactant coverages.

Inverted volcano (or V-type) behaviour, i.e. electrophilic behaviour followed by electrophobic one. 

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).

Rule G4 A reaction exhibits inverted volcano (minimum rate) type behaviour when the kinetics are positive order in both the electron acceptor (A) and electron donor (D) reactant. 

NBMCA behaviour Inverted Volcano type behaviour ... 

Figure 6.18. Model predicted electrochemical promotion behaviour (a) electrophobic, (b) electrophilic, (c) volcano-type, (d) inverted volcano-type.

Figure 6.18 shows how the model predicts the four main types of r vs O global behaviour (electrophobic, electrophilic, volcano, inverted volcano) for fixed XD and IA, Pd and pA, by just varying the adsorption equilibrium constants kD and kA. Note that in Figure 6.18 and till the end of this chapter we omit the units of Pd and pA (e.g. kPa) and kD,kA (e.g. kPa 1), unless we refer to experimental data. This is because one is free to use any consistent set of units, since only the dimensionless products kApA and kDpD enter the calculations. 

Effect of partial electron transfer parameter Figure 6.23 depicts the effect of the value of the partial charge transfer parameter A,d for fixed XA(= 0.15) on the rate enhancement ratio p(=r/r0) for the four main types of promotional behaviour, i.e., electrophobic, electrophilic, volcano and inverted volcano. The main feature of the Figure is that it confirms in general the global mle... 

G5 and G7. Regarding global rule G5 it can be seen in Figs. 6.23a,b and c that as long as A,D > XA rules G1 to G3 remain valid regardless of the sign of A.D with some deviations predicted only for rule G4 in the case of positive n (in this case a shift from inverted volcano to electrophilic behaviour is predicted when both X,A and kD are negative (Fig. 6.23d)). 

For strong adsorption of D (Fig. 6.23b), p is again an increasing function of Xu and for negative XD values a transition to inverted-volcano type behaviour is predicted. 

Rules G5 and G7 are also predicted for the case of strong adsorption of both D and A (Fig. 6.23c), i.e. for the case of volcano behaviour. In the case of weak adsorption of D and A (Fig. 6.23d) a transition from inverted volcano to purely electrophilic behaviour is predicted when XD is negative as already noted. 

The excellent prediction by the model of all global promotion rules is not only qualitative. The predicted p values ( 102 for IV variation in UWR, Fig. 6.18a) is in excellent agreement with experiment (e.g. Fig. 4.24). Also the pmax values ( 10-20) predicted for volcano and inverted volcano behaviour are in very good agreement with experiment. Finally the Xd, Xa which are used ( 0.15) are physically very reasonable. For example for Uwr-1 V at 673 K it is EM7, thus the XD and X.A values used in the simulations, which are physically very reasonable, give exp(Xjn), and thus p, values between 10 2 and 102 in good qualitative agreement with experiment. 

Figure 6.24. Experimentally observed (bottom) and model predicted (top) transition from inverted volcano to electrophobic behaviour upon increasing the 02 to ethylene (i.e. A/D) ratio by a factor of 10, C2H4 oxidation on Pt/Ti02.28 Reprinted with permission from Academic Press.

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