The General Rate Equation

Agitated slurry reactor (suspended fine solids) 

Three-phase fluidized bed. these vary widely in G/L ratio from G bubbling through an L slurry to a G fluidized bed of damp particles. 

Gas reactant must first dissolve in the L, then both reactants must diffuse or move to the catalyst surface for reaction to occur. Thus the resistance to transfer across the G/L interface and then to the surface of solid both enter the general rate expression. 

To develop the rate equation, let us draw on the two-film theory, and let us use the following nomenclature  

Graphically we show the resistances as in Fig. 22.2. We can then write the following general rate equations  

---

If a pH-rate curve does not exhibit an inflection, then very probably the substrate does not undergo an ionization in this pH range. The kinds of substrates that often lead to such simple curves are nonionizable compounds subject to hydrolysis, such as esters and amides. Reactions other than hydrolysis may be characterized by similar behavior if catalyzed by H or OH . The general rate equation is... 

The general rate equation for the first-order rate constants, k, is given by Eq. (4) ... 

The general rate equation for an ordered, single-displacement reaction where A is the leading substrate is... 

Before showing an example where the chain-branching rearrangement and not the carbonylation is rate-determining, the general rate equations for these processes will be discussed. 

The order of a reaction is the same as the number of concentration terms in the rate expression. Consider the general rate equation ... 

Most oxides show, in accordance with the general rate equation (5.16) the same trend with regard to the rate dependence on pH a decrease in pH in the acid range and an increase in pH in the alkaline range. 

The general rate equation resulting from the steady-state hypothesis is... 

We discuss these special cases and present their particular rate equations later, after we present the general rate equation. 

Since the last resistance term in the general rate equation, Eq. 5, is negligible (large k), the rate form for this case is... 

In other cases (and this is what we usually find) the general rate equation is easier to use. ... 

If we assume a constant fractional yield and no slowing of the rate as a result of product poisoning or increase in cell crowding in the broth, then the general rate equation of the previous chapter, Eq. 28.8, reduces to the well-known Mo-nod equation... 

By the above results and by the general rate equations for initiation, propagation, and termination, Minoura concluded that the polymerization rate is proportional to the monomer concentration (M) and to the square root of the dif-... 

The general rate equation for the oxidation of ammonia to nitrogen is of the redox type (161—163). 

Since the ozonation of a compound M involves both the direct and indirect reaction pathways, the general rate equation 4-2 has to be modified to include both reactions ... 

The general rate equation valid for this proposed mechanism can be calculated and verified by experimental evidence. It is clear that if there is one contradiction between calculated results and experimental evidence, the proposed mechanism cannot be valid. Step 3 of the mechanism is proposed as RDS because it is the only electrochemical step in the reaction scheme. Its rate equation can be written as ... 

After substitution of the coverage fraction of S02 into the general rate equation (6.15), the following equation is obtained ... 

All three reactions are represented by the general rate equation... 

This tendency to react with a range of nucleophiles is reflected in the general rate equation for reactions of this type, as seen for the hydrolysis of [Co(en)2(H2NCH2C02 Pr)]3+. Typically, a three-term rate equation is obtained. This is indicative of a process in which at least three parallel reaction pathways are being followed. In the case of the hydrolysis of [Co(en)2(H2NCH2C02 Pr)]3+, the kx term refers to attack of the chelated ester by water, the k2 term to attack by hydroxide and the /% term to general base attack by any other nucleophile which is present in solution. The rate is defined in terms of the loss of the starting complex cation, rather than the formation of any specific product of the reaction. 

This adjustment process results in a change in the concentrations of some or all of the species. The rate of the adjustment to new equilibrium conditions or the rate of chemical relaxation is determined by the rate of the reactions that make up the equilibrium. By measuring the relaxation rate, one can obtain information that can be used to determine ki and k 1. Assume that the general rate equation... 

These observations can be represented as a special case of the general rate equation derived by the application of order-disorder theory to diffusionless transitions in solids.3 According to this equation, the shape of the rate curve is determined by the relative numerical values of zkp/kn and of c. The larger the factor is relative to c, the more sigmoidal the curves become. This is understandable since the propagation effect which is responsible for the autocatalytic character of the transformation becomes more noticeable when kPlkn is large and c small. Under these conditions some time elapses before a sufficient number of nucleation sites are formed then the... 

In the general rate equation, the power a is called the order of the reaction with respect to reactant A, and b is the order of the reaction with respect to B. The sum of these powers, (a + b), is called the overall order of the reaction. 

Taking the inverse for the general rate equation and dividing both sides by kR kmQ one obtains... 

Similarly for the reverse reaction of denitrosation of the nitrosourea derivative, the general rate equation is given by (14) since, in the presence of the nitrous acid trap hydrazine sulphate, step (o) can be neglected. Again if -i[X-] ki no catalysis by X is predicted. The reversibility of the reaction was tested quantitatively by noting the variation of the observed rate constant... 

The active hydroformylation catalyst is believed to be the square-planar 16e complex HPt(SnCl3)(CO)(PPh3) and one proposed mechanism is shown in Scheme 13. Although the general mechanism closely follows the fundamental steps seen for Rh/PPhs catalysts (see Scheme 7), the general rate equation (equation 6) indicates that there are certainly some differences. The SnCl2 can react with one of the chlorides to form a... 

For the general rate equation 7.9 to reduce to a one-plus form rather than a power law, the denominator must have at least two non-zero terms. However, only the first row of the matrix contains no reverse coefficients, so the first term alone would survive if all reverse coefficients were zero. 

For pathway I above, the general rate equation reduces to... 

Moreover, as the matrix 7.10 or 6.7 shows, the first denominator term of the general rate equation contains only forward X coefficients, and must therefore be negligible to produce the order minus one in CO. Without its first term, the denominator contains neither nor X12 (first matrix row now negligible). But to produce the observed order less than one in H2, the denominator must contain the H2 pressure as co-factor in at least one additive term. Accordingly, no pathway in which H2 reacts in either the first or second step can be correct. 

Rigorous rate equations for multistep catalytic reactions in terms of total amount of catalyst material are enormously cumbersome. Just the reduction to the level complexity of the Christiansen formula calls for the Bodenstein approximation of quasi-stationary behavior of the intermediates, requiring these to remain at trace concentrations, and that formula still entails a lot more algebra than does the general rate equation for noncatalytic simple pathways For a reaction with three intermediates, the Christiansen denominator contains sixteen terms instead of four for a reaction with six intermediates, forty-nine instead of seven Although the mathematics is simple and easy to program for modeling purposes and, usually, some... 

The general rate equation for the cycle converting A to P (with m members) can be written... 

Derivation ofeqn 8.81. In an ordinary, single, k-membered cycle, eqn 8.32 can be used to replace Clou in the general rate equation 8.30 to give ... 

---