Specific reaction rates

The object of the mathematical analysis is to determine the type of reaction and to evaluate the constant, k. This may be done from two experimental observations with the help of the integrated formulas. Methods will be developed later for calculating k at various temperatures, and sometimes k can be estimated even without experimental determinations. The determination of the numerical value of k is the practical goal from which useful, quantitative predictions can be made. For example, when k is known for simple reactions, the rate of reaction at any time, or the per cent conversion of material after any length of time can be calculated. 

Chemical kinetics is intimately bound up with chemical equilibrium as given in the simple formula for the reaction aA + bB + = gG+hH +  

I The effective concentration at equilibrium of the products divided by those of the reactants is, by definition, the equilibrium constant K. If the reaction happens to be first order in one direction and second order in the reverse, the relation still holds because the effective concentrations occur with the same exponent (e.g. a, b, g, h, etc. in equation (13)) in both rate and equilibrium formulas. 

This formula is of great importance in kinetics, for frequently it is not possible to determine the rate of a given chemical change because it is too fast, or too slow, or because the measurements of concentrations can not be determined by direct experimental methods yet devised. The determination of the equilibrium constant K is usually comparatively simple and a knowledge of it and one of the velocity constants permits a ready calculation of the other by equation (13). It is surprising to realize that, except possibly in one or two cases, this important formula has not been subjected to complete experimental test, by measuring all three quantities. There is, however, no question as to the validity of the relation if the reaction proceeds in the manner indicated. 

It should be emphasized that it is possible to calculate K by thermodynamic methods but that thermodynamics can not give k it gives only the ratio of the two k s. Thermodynamics has nothing to do with time for it is concerned with systems in a state of equilibrium and therefore can never predict reaction rates. On the other hand when chemical kinetics reaches such a state of perfection that the two k s can be predicted, then K can be calculated easily and we will have all that chemical thermodynamics now gives and, in addition, we will have the time factor. 

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Reaction and Transport Interactions. The importance of the various design and operating variables largely depends on relative rates of reaction and transport of reactants to the reaction sites. If transport rates to and from reaction sites are substantially greater than the specific reaction rate at meso-scale reactant concentrations, the overall reaction rate is uncoupled from the transport rates and increasing reactor size has no effect on the apparent reaction rate, the macro-scale reaction rate. When these rates are comparable, they are coupled, that is they affect each other. In these situations, increasing reactor size alters mass- and heat-transport rates and changes the apparent reaction rate. Conversions are underestimated in small reactors and selectivity is affected. Selectivity does not exhibit such consistent impacts and any effects of size on selectivity must be deterrnined experimentally. 

Scale-Up Principles. Key factors affecting scale-up of reactor performance are nature of reaction zones, specific reaction rates, and mass- and heat-transport rates to and from reaction sites. Where considerable uncertainties exist or large quantities of products are needed for market evaluations, intermediate-sized demonstration units between pilot and industrial plants are usehil. Matching overall fluid flow characteristics within the reactor might determine the operative criteria. Ideally, the smaller reactor acts as a volume segment of the larger one. Elow distributions are not markedly influenced by... 

Several products resulted when 4-HMP was allowed to condense under the same conditions. These are provided in Scheme 6. These products indicate that a p-hydroxymethyl group can react with either an unsubstituted ortho position or through an ipso-attack on a hydroxymethyl-substituted para position. The two condensation products were formed at approximately equal rates. Since there were two unoccupied orthos, it seems that the specific reaction rate for the ortho site is about half that of the occupied para position in this situation. The condensation rate for 4-HMP was about 5 times the rate for 2-HMP overall. 

The batch reactor initially contains 227 kg of acetyiated castor and die initial temperature is 613 K. Complete hydrolysis yields 0.156 kg acetic acid per kg of ester. Eor diis reaction, die specific reaction rate constant k is... 

A straight line is produced when the logarithm of a specific reaction rate is plotted against the reciprocal of the absolute temperature. Temperature has a marked influence on the reaction rates, but the range between reactions that are too slow or too fast to measure is really quite narrow. 

Transfer constants for polystyrene chain radicals at 60° and 100°C, obtained from the slopes of these plots and others like them, are given in the second and third columns of Table XIII. Almost any solvent is susceptible to attack by the propagating free radical. Even cyclohexane and benzene enter into chain transfer, although to a comparatively small extent only. The specific reaction rate at 100°C for transfer with either of these solvents is less than two ten-thousandths of the rate for the addition of the chain radical to styrene monomer. A fifteenfold dilution with benzene was required to halve the molecular weight, i.e., to double l/xn from its value (l/ rjo for pure styrene (see Fig. 16). Other hydrocarbons are more effective in lowering the degree of polymerization through chain transfer. 

The termination constants kt found previously (see Table XVII, p. 158) are of the order of 3 X10 1. mole sec. Conversion to the specific reaction rate constant expressed in units of cc. molecule" sec. yields A f=5X10". At the radical concentration calculated above, 10 per cc., the rate of termination should therefore be only 10 radicals cc. sec., which is many orders of magnitude less than the rate of generation of radicals. Hence termination in the aqueous phase is utterly negligible, and it may be assumed with confidence that virtually every primary radical enters a polymer particle (or micelle). Moreover the average lifetime of a chain radical in the aqueous phase (i.e., 10 sec.) is too short for an appreciable expectation of addition of a dissolved monomer molecule by the primary radical prior to its entrance into a polymer particle. 

Since the total amount of substance being converted is proportional to the amount of charge, the specific reaction rate Vj, which is the amount of substance j converted in unit time per unit surface area of the electrode, is proportional to the current density i ... 

For higher efficiency in catalytic action and smaller quantitative needs, the catalysts often are used in a highly disperse state. An important practical criterion for such catalysts is the specific reaction rate [i.e., the reaction rate per unit mass of the catalyst... 

Rate equations are differential equations of the general form dcjdt = kf (Cj, c2,... cn) = kf (.c), where i is the particular product or reactant, and C is its molar concentration (NJV). The constant k goes by a number of names such as velocity coefficient, velocity constant specific reaction rate, rate constant, etc., of the particular reaction. Physically, it stands for the rate of the reaction when the concentrations of all the reactants are unity. The function fc) and the rate constant k are determined from experimental data. 

The initial state-specific reaction rate constant for both diatom-diatom and atom-triatom reactions is calculated by averaging the corresponding cross-section over a Boltzmann distribution of translational energy ... 

Figure 4 Correlation between the specific reaction rate measured on Remodified 0.3Sn0.3Pt(b) catalysts and the concentration of promoters (Sn+Re).

Figure 5 shows the specific reaction rate has a maximum at Re/(Re+Sn+Pt) concentration ratio of 0.5. The trimetallic catalysts at concentration ratio of 0.4-0.6 are almost twice as active than the 0.3Sn0.3Pt(b) catalyst and the... 

The right side of this expression is identical with the rate of production of species B and C. Hence the maximum production rate for a fixed reactor volume occurs when the reactor contents have a composition that maximizes the specific reaction rate. Now, in terms of the fraction conversion,... 

The three models used are described by Eq. (6-8) below. The Eqn. (6) is the first-order model based on Michaelis-Menten model, Eqn. (7) is the second-order model, and the Eqn. (8) is the competitive-substrate model. Rso represents the initial specific reaction rate for the substrate S. 

These advances in catalyst preparation techniques have certainly stimulated the already growing interest in the relations between the catalytic and sorptive properties of catalysts and their mode of preparation. Many authors have studied the dependence of specific reaction rate upon particle size, mainly in hydrogenation, dehydrogenation, and hydrogenolysis reactions. The results of this work have recently been compiled by Schlosser (6). 

In quite a number of cases the particle size was not found to have an effect on the specific reaction rate, whereas in some others this effect was clearly observed. Boudart et al. (7) coined the term facile for reactions in which the specific activity does not depend on the particle size, using the term demanding in referring to those cases where such a dependence exists. 

FromEqs. (2.60), (2.61), one can deduce an approximate power-law dependence of specific reaction rates on temperature, Tv, since... 

The rate constant fe in equation 4.1-3 is sometimes more fully referred to as the specific reaction rate constant, since r = k( when c,- = 1 (i = 1,2,.. . , N). The units of Inland of A) dependon the overall order of reaction, n, rewritten from equation 3.1-3 as... 

Temperature is a direct measure of the heat energy available at release (Edwards and Lawrence, 1993). Temperature is the most important factor influencing reaction rate as shown in the Arrhenius equation. In practice an increase in temperature of 10°C will increase a specific reaction rate by two to four times depending on the energy of activation (CCPS, 1995a). 

Alkylation of toluene and acetylene in the presence of sulfuric acid is accomplished in the four-stage reactor of the sketch. Retention time in each stage is 10 min, temperature is 41 F and pressure is 50 psig. On the assumption that the liquid always is saturated with acetylene, the reaction is first order with respect to toluene. At the conditions shown, the reaction is estimated 95% complete. Find the specific reaction rate. 

Hydrocarbon cracking is a first-order reaction. Neglecting pressure drop, find the specific reaction rate. 

Much of the material presented in previous chapters relating to compositional and enzymic differences has a bearing on this problem, and it tends to support the concept that these urinary patterns arise, in part at least, because of the differences in the fundamental body chemistry (differing specific reaction rates) of each individual concerned. 

In summary, we may say that excretion studies point to the existence of marked differences in specific reaction rates in different individuals and to the probability that each individual exhibits a highly distinctive pattern with respect to his internal chemistry. 

Thus, the important conclusion is that the specific reaction rate constant k is dependent on temperature alone and is independent of concentration. Actually, when complex molecules are reacting, not every collision has the proper steric orientation for the specific reaction to take place. To include the steric probability, one writes k as... 

FIGURE 2.2 Arrhenius plot of the specific reaction rate constant as a function of the reciprocal temperature. 

For a temperature of 1000 K, calculate the pre-exponential factor in the specific reaction rate constant for (a) any simple bimolecular reaction and (b) any simple unimolecular decomposition reaction following transition state theory. 

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