Reaction arbitrary rate

The rate of a chemical reaction can be described in any of several different ways. The most commonly used definition involves the time rate of change in tlie amount of one of the components participating in tlie reaction tliis rate is usually based on some arbitrary factor related to tlie reacting system size or geometry, such as volume, mass, or interfacial area. Tlie definition shown in Eq. (4.6.7), wliich applies to homogeneous reactions, is a convenient one from an engineering point of view. 

In more general terms for an arbitrary reaction and rate expression the orders of the forward and reverse reactions must obey the relation... 

For the more general case of arbitrary rate constants, the analysis is more complex. Various approximate techniques that are applicable to the analysis of reactions 5.4.1 and 5.4.2 have been described in the literature, and Frost and Pearson s text (11) treats some of these. One useful general approach to this problem is that of Frost and Schwemer (12-13). It may be regarded as an extension of the time-ratio method discussed in Section 5.3.2. The analysis is predicated on a specific choice of initial reactant concentrations. One uses equivalent amounts of reactants A and B (A0 = 2B0) instead of equi-molal quantities. 

The rational design of a reaction system to produce a polymer with desired molecular parameters is more feasible today by virtue of mathematical tools which permit prediction of product distribution. New analytical tools such as gel permeation chromatography are being used to check theoretical predictions and to help define molecular parameters as they affect product properties. There is a laudable trend away from arbitrary rate constants, but systems other than styrene need to be treated in depth. A critical review of available rate constants would be useful. Theory might be applied more broadly if it were more generally recognized that molecular weight distributions as well as rates can be calculated from combinations of constants based on the pseudo-steady-st te assumption. These are more easily determined than the individual constants in chain reactions. 

The reaction characteristic of the present system are best performed in a semicontinuous reactor in which the solid is stationary, as described in the previous section. This easily permits the two steps. In general, however, continuous reactors in which both the gas and solid phases move continuously are more important. We therefore briefly consider in this section the mathematical basis for the design of such a reactor. The chief reactor and operating parameters are gas and solids feed rates, product size distribution, bed size, and so on, and procedures for determining them are described. With a size distribution o(R), an elutriation stream and an arbitrary rate law for the changing particle size, a material balance on solids of size between R and R + dR yields... 

Particles, Drops, and Bubbles. Arbitrary Rate of Reaction... 

For an arbitrary rate of volume chemical reaction, the mean Sherwood number at high Peclet numbers can be calculated according to the approximate formula... 

Finally, we will take a brief look at the relation between the order of a reaction and the molecularity mentioned above. Reaction orders are experimentally determined quantities while molecularity (of a reaction step) is a theoretical quantity essential for the elucidation of a reactitMi mechanism. In single-step reactions, molecularity and reaction order (as well as conversion number sum) agree with each other because all the particles react simultaneously with each other according to their appearance in the conversion formula. Conversely, it is not necessarily possible to infer the molecularity of an arbitrary reactimi from its order. This is because in complex reaction processes made up of several single-step reactions, simple rate laws might still be vaUd. 

The couple numbers assigned are completely arbitrary, depending principally on the order in which we studied them. Hydrazine couples span an astonishing range of intrinsic reactivity, 16.6 kcal/mol, which is 83% of the total ACj, (fit) range observed for all compounds studied. The very unreactive, high AG (fit) hydrazines are essential to study the intrinsically most reactive couples. Only by having slow couples can the cross-reaction ET rate constant be kept small relative to diffusion rates, so the... 

For a single step electrochemical reaction of arbitrary rate... 

After that, the user has to decide how the results should be visualized. It is possible to print the answer in the form of individual values of the desired function, an array, etc. However, the most visual output form is the graphical one. The plotting of the results is provided by the command odeplot from the graphical library plots. Figures 3.11 and 3.12 show a solution of the differential equation set, which describes the kinetics of the first-order reversible reaction B with arbitrary rate constant values. 

This is the situation exploited by the so-called isolation method to detennine the order of the reaction with respect to each species (see chapter B2.1). It should be stressed that the rate coefficient k in (A3,4,10) depends upon the definition of the in the stoichiometric equation. It is a conventionally defined quantity to within multiplication of the stoichiometric equation by an arbitrary factor (similar to reaction enthalpy). 

Although the reaction rate of ethylene and various copolymers differs substantially, the reaction constants can be estabUshed by using an arbitrary value of 1 for ethylene (5). Thus, a value of 0.1 would indicate that the comonomer reacts at 10 times the rate of ethylene. However, the wide range of reaction rates can present problems not only in determining the comonomer content of the final product but also in producing a homogeneous product (4,6). 

In batch classification, the removal of fines (particles less than any arbitrary size) can be correlated by treating as a second-order reaction K = (F/Q)[l/x(x — F)], where K = rate constant, F = fines removed in time 0, and x = original concentration of fines. 

Reaction quotient (Q) An expression with the same form as Kbut involving arbitrary rather than equilibrium partial pressures, 333-334 Reaction rate The ratio of the change in concentration of a species divided by the time interval over which the change occurs, 285 catalysis for, 305-307 collision model, 298-300 concentration and, 287-292,314q constant, 288 enzymes, 306-307 egression, 288... 

The time required to produce a 50% reduction in properties is selected as an arbitrary failure point. These times can be gathered and used to make a linear Arrhenius plot of log time versus the reciprocal of the absolute exposure temperature. An Arrhenius relationship is a rate equation followed by many chemical reactions. A linear Arrhenius plot is extrapolated from this equation to predict the temperature at which failure is to be expected at an arbitrary time that depends on the plastic s heat-aging behavior, which... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

Fig. 4. Dependence of relative concentrationa nj/nt of reaction components A, B, and C on time variable r (arbitrary units) in the case of consecutive (— — ) reactions according to scheme (Ha) or parallel (C ) reactions according to scheme (lib). Ads X, Ads A, Des Y denotes that the rate determining step in the overall transformation is adsorption or desorption of the respective substance Des (B + C) denotes that the overall rate is determined by simultaneous desorption of the substance B and C. Ki/Ki = 0.5 for consecutive, and Ki /Ki — 0.5 for parallel reactions, b nxVn. 0 = 2.5 for consecutive reactions Kt = 0.5, and for parallel reactions Ki/Ki — 0.5. c nxVnA0 = 2.5 fcdesBKi Ky/fcdesoXj Kx = 10 [cf. (53)]. d Ki = 1.75 for consecutive, and Ki/Ki = 1.75 for parallel reactions.

Fig. 5. Dependences of relative concentrations Cj on time variable r (arbitrary units) for consecutive catalytic reactions according to scheme (III) for various values of rate constants of the adsorption k,(ub and desorption fcduB of the intermediate B. Left-hand column (fcdesB/fcs = 0.1) desorption of B is slower than its surface transformation. Middle column (fcde.B/fcs = 1) equal rates of desorption of B and of its surface transformation. Right-hand column (fcdesB/fcj = 10) desorption of B is faster than its surface transformation. From G. Thomas, R. Montarnal, and P. Boutry, C.R. Acad. Sri., Ser. C 269, 283 (1969).

There are two uses for Equation (2.36). The first is to calculate the concentration of components at the end of a batch reaction cycle or at the outlet of a flow reactor. These equations are used for components that do not affect the reaction rate. They are valid for batch and flow systems of arbitrary complexity if the circumflexes in Equation (2.36) are retained. Whether or not there are spatial variations within the reactor makes no difference when d and b are averages over the entire reactor or over the exiting flow stream. All reactors satisfy global stoichiometry. 

Suppose a gradientless reactor is used to obtain intrinsic rate data for a catalytic reaction. Gas-phase concentrations are measured, and the data are fit to a rate expression using the methods of Chapter 7. The rate expression can be arbitrary ... 

Furthermore, reaction (27) of Westheimer s scheme seems to be rather arbitrary. If chromium(II) were really formed, the rate of oxidation of alcohol would certainly be influenced by oxygen. However, careful experiments show that there is no oxygen effect at all . 

Electrochemical reactions differ fundamentally from chemical reactions in that the kinetic parameters are not constant (i.e., they are not rate constants ) but depend on the electrode potential. In the typical case this dependence is described by Eq. (6.33). This dependence has an important consequence At given arbitrary values of the concentrations d c, an equilibrium potential Eq exists in the case of electrochemical reactions which is the potential at which substances A and D are in equilibrium with each other. At this point (Eq) the intermediate B is in common equilibrium with substances A and D. For this equilibrium concentration we obtain from Eqs. (13.9) and (13.11),... 

The reaction vial (see Fig. 6.1) was changed in order to make the distance between sensor and tantalum filament (generator of ethyl radicals) equal to the distance between filament (radical source) and selenium film as well as to the distance between the sensor and selenium filament. Dimensions of pipes linking them were also the same. Then, measuring the initial rate of the change in electric conductivity of the sensor during generation of radicals one can assess in arbitrary units the concentration of radicals incident on the surface of the sensor. Due to... 

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