Overall rate

The slopes of the fimctions shown provide the reaction rates according to the various definitions under the reaction conditions specified in the figure caption. These slopes are similar, but not identical (nor exactly proportional), in this simple case. In more complex cases, such as oscillatory reactions (chapter A3.14 and chapter C3.6). the simple definition of an overall rate law tluough equation (A3.4.6) loses its usefiilness, whereas equation (A3.4.1) could still be used for an isolated system. 

The Landolt reaction (iodate + reductant) is prototypical of an autocatalytic clock reaction. During the induction period, the absence of the feedback species (Irere iodide ion, assumed to have virtually zero initial concentration and fomred from the reactant iodate only via very slow initiation steps) causes the reaction mixture to become kinetically frozen . There is reaction, but the intemiediate species evolve on concentration scales many orders of magnitude less than those of the reactant. The induction period depends on the initial concentrations of the major reactants in a maimer predicted by integrating the overall rate cubic autocatalytic rate law, given in section A3.14.1.1. 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

Within this approximation, the overall rate of transitions is given by ... 

Before elosing this ehapter, it is important to emphasize the eontext in whieh the transition rate expressions obtained here are most eommonly used. The perturbative approaeh used in the above development gives rise to various eontributions to the overall rate eoeffieient for transitions from an initial state i to a final state f, these eontributions inelude the eleetrie dipole, magnetie dipole, and eleetrie quadrupole first order terms as well eontributions arising from seeond (and higher) order terms in the perturbation solution. 

In summary, solvents can influence Diels-Alder reactions through a multitude of different interactions, of which the contributions to fire overall rate uniquely depend on the particular solvent-diene-dienophile combination. Scientists usually feel uncomfortable about such a situation and try to extract generalities. When limited to the most extensively studied type A Diels-Alder reactions this approach seems feasible. These Diels-Alder reactions are dominated by hydrogen bonding interactions in combination with solvophobic interactions. This observation predicts a very special role of water as a solvent for type A Diels-Alder reactions, which is described in Section 1.4. 

Assuming complete binding of the dienophile to the micelle and making use of the pseudophase model, an expression can be derived relating the observed pseudo-first-order rate constant koi . to the concentration of surfactant, [S]. Assumirg a negligible contribution of the reaction in the aqueous phase to the overall rate, the second-order rate constant in the micellar pseudophase lq is given by ... 

Applying the Arrhenius equation to Eq. (6.116) shows that the apparent activation energy for the overall rate of polymerization is given by... 

As illustrated ia Figure 6, a porous adsorbent ia contact with a fluid phase offers at least two and often three distinct resistances to mass transfer external film resistance and iatraparticle diffusional resistance. When the pore size distribution has a well-defined bimodal form, the latter may be divided iato macropore and micropore diffusional resistances. Depending on the particular system and the conditions, any one of these resistances maybe dominant or the overall rate of mass transfer may be determined by the combiaed effects of more than one resistance. 

The term dqljdt represents the overall rate of mass transfer for component / (at time t and distance averaged over a particle. This is governed by a mass transfer rate expression which may be thought of as a general functional relationship of the form... 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

Any one of the five basic processes may be responsible for limiting the extraction rate. The rate of transfer of solvent from the bulk solution to the soHd surface and the rate into the soHd are usually rapid and are not rate-limiting steps, and the dissolution is usually so rapid that it has only a small effect on the overall rate. However, knowledge of dissolution rates is sparse and the mechanism may be different in each soHd (1). 

An example of a commercial semibatch polymerization process is the early Union Carbide process for Dynel, one of the first flame-retardant modacryhc fibers (23,24). Dynel, a staple fiber that was wet spun from acetone, was introduced in 1951. The polymer is made up of 40% acrylonitrile and 60% vinyl chloride. The reactivity ratios for this monomer pair are 3.7 and 0.074 for acrylonitrile and vinyl chloride in solution at 60°C. Thus acrylonitrile is much more reactive than vinyl chloride in this copolymerization. In addition, vinyl chloride is a strong chain-transfer agent. To make the Dynel composition of 60% vinyl chloride, the monomer composition must be maintained at 82% vinyl chloride. Since acrylonitrile is consumed much more rapidly than vinyl chloride, if no control is exercised over the monomer composition, the acrylonitrile content of the monomer decreases to approximately 1% after only 25% conversion. The low acrylonitrile content of the monomer required for this process introduces yet another problem. That is, with an acrylonitrile weight fraction of only 0.18 in the unreacted monomer mixture, the low concentration of acrylonitrile becomes a rate-limiting reaction step. Therefore, the overall rate of chain growth is low and under normal conditions, with chain transfer and radical recombination, the molecular weight of the polymer is very low. 

The productivity of DR processes depeads oa chemical kinetics, as weU as mass and heat transport factors that combine to estabhsh the overall rate and extent of reduction of the charged ore. The rates of the reduction reactions are a function of the temperature and pressure ia the reductioa beds, the porosity and size distribution of the ore, the composition of the reduciag gases, and the effectiveness of gas—sohd contact ia the reductioa beds. The reductioa rate geaerahy iacreases with increasing temperature and pressure up to about 507 kPa (5 atm). 

Simultaneous reactions. The overall rate is the algebraic sum of the rates of the individual reactions. For example, take the three reactions ... 

Performance data on some typical tray and compartment diyers are tabulated in Table 12-10. These indicate that an overall rate of evaporation of 0.0025 to 0.025 kg water/(s m") of tray area may be expected from tray and tray-truck diyers. The thermal efficiency of this type of diyer will vary from 20 to 50 percent, depending on the diying temperature used and the humidity of the exhaust air. In diying to very low moisture contents under temperature restrictions, the thermal efficiency may be in the order of 10 percent. The major operating cost for a tray diyer is the labor involved in loading and unloading the trays. About two labor-hours are required to load and unload a standard two-truck tray diyer. In addition, about one-third to one-fifth of a... 

Coalescence Coalescence is the most difficult mechanism to model. It is easiest to write the population balance (Eq. 20-71) in terms of number distribution by volume n v) because granule volume is conserved in a coalescence event. The key parameter is the coalescence kernel or rate constant P(ti,i ). The kernel dictates the overall rate of coalescence, as well as the effect of granule size on coalescence... 

Overall Rate Equations with Diffusional Resistances. 23-52... 

OVERALL RATE EQUATIONS WITH DIFFUSIONAL RESISTANCES... 

The individual steps in chain reactions involving radicals are characteristically of small activation energy, between about 10 and 50kJmol and so these reactions should occur at an immeasurably high rate at temperatures above 500 K (see Table 2.1), which is a low temperature for a useful combustion process. The overall rate of the process will tlrerefore depend mainly on the concentrations of tire radicals. 

The overall rate for tire formation of ammonia must tlrerefore be a balance between the formation and tire decomposition of the product species. Experimental data suggest tlrat tlris balance can be represented by the equation... 

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