Generalized rate equations

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

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. 

Because of the variety of chain reaction mechanisms that exist, it is difficult to develop completely general rate equations for these processes. However, some generalizations with regard to the overall rate expression can be made for certain classes of reactions. 

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. 

Blum and Lasaga (1987) and Lasaga (1990) propose the very general rate equation... 

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

Substituting F into a general rate equation produces ... 

A general rate equation has been derived for reactions with a single relaxation time. Consider ... 

General Rate Equation. The differential equation for a bimolecular rate process can be written as ... 

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

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... 

Generalized Rate Equations. This last reaction is typical of a large class of reactions which may proceed simultaneously by several parallel paths. The rate can be expressed as the sum of several contributions from the uncatalyzed and catalyzed paths. The overall rate for the hydrolysis of an amino acid ester in the presence of a metal salt may be expressed as... 

Grambow, B. 1985. A general rate equation for nuclear waste glass corrosion. In Jantzen, C. M., Stone, J. A. Ewing, R. C. (eds) Scientific Basis for Nuclear Waste Management VIII. Materials Research Society Symposia Proceedings, 44, 16-27. 

Grambow, B. 1985. A general rate equation for nuclear waste glass corrosion. Materials Research Society Symposium Proceedings, 44, 15-27. 

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 ... 

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... 

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