Reaction rates independent

Alternatively, we may assume that there exists some (but possibly limited) knowledge about the typical concentrations involved. For each metabolite, we can then specify an interval St < S1- <. S ) that defines a physiologically feasible range of the respective concentration. Furthermore, the steady-state flux vector v° is subject to the mass-balance constraint Nv° = 0, leaving only r — rank(N) independent reaction rates. Again, an interval v(. < v9 < v+ can be specified for all independent reaction rates, defining a physiologically admissible flux space. 

The difference in the kinetics for two limiting cases Da = 0 and Da = Db becomes more obvious in terms of the current critical exponents defined earlier, equation (4.1.68). It yields the slope of decay curves shown in Fig. 6.39. The conclusion can be drawn from Fig. 6.40 that in the symmetric case we indeed observe well-known algebraic decay kinetics with a(oo) = 1 corresponding to time-independent reaction rate. However, in the asymmetric case the critical exponent increases in time thus indicating the peculiarity of the kinetics as we qualitatively estimated in the beginning of this Section 6.4. 

As will be shown in detail in Section IV, the large-barrier case leads to well-behaved time-independent reaction rates k = const, but for the no-barrier case, time-dependent reaction rates k = k t) are predicted. These can also appear in other processes, for example, in Forster energy transfer, where k t) is proportional to t112.59... 

Not all strained compounds are necessarily more reactive than less strained analogs. Reactivity will always depend on the type of reaction under scrutiny, and if the rate determining step of a given reaction is not accelerated by strain, the rate of reaction of strained and unstrained compounds will be similar. One example of such strain-independent reaction rates is the hydrolysis of lactams under basic reaction conditions (Scheme 3.8). Although /3-lactams are more strained than six-membered lactams, both are hydrolyzed at approximately the same rate, presumably because the rate determining step is the addition of hydroxide to the amide bond, and not... 

A small hold-up volume can be provided between main reservoir and the condenser, allowing small liquid samples to be taken, an analysis of which will yield independent reaction rates. The recycling of even small concentration percentages of liquid products can be eliminated by interposing several reservoirs containing about two or three times the amount of liquid circulated in an individual experiment. The gas evolution rates are measured by determining the rate of pressure rise with a manometer when the system is temporarily closed off from the atmosphere. 

Ballantine (4) observed that the y-induced emulsion polymerization of styrene is about 100 times faster and yields higher molecular weights (up to 2 X 10 ) than the y-induced bulk polymerization. He explains the large difference in reaction rates by the high radical yield (G/ value) of water, as compared with the G/j value of styrene. An over-all activation energy of 3.7 kcal. per mole was calculated from the temperature dependence of the reaction. Allen et al. (1) prepared and grafted polystyrene and poly (vinyl acetate) dispersions under the influence of y-radiation. Mezhirova et al. (28) found a temperature-independent reaction rate of the y-induced emulsion polymerization of styrene. 

The same expression was obtained for non-catalytic reactions, i.e., the catalyst does not inaease the maximum rate but merely reduces the induction period. This seemingly paradoxical conclusion is based on the fact that in both cases the maximum rate is determined by the equality of catalyst-independent reaction rates of chain propagation which, in the maximum, are equal to the rate of chain branching and termination. The existence of a maximum rate was quantitatively confirmed for the oxidation of some hydrocarbons (tetralin, ethylbenzene, diphenylmethane, etc.) [14a]. 

The first solution supposes size-independent reaction rates. Von Smoluchowski gave the solution for constant rates (kij = k for all i and j) ... 

Mathematical representation of simple and complex reactions Independent reactions Rate equations... 

The temperature-independent reaction rate observed sometimes and recognized by the temperature coefficient of unity indicates that the reaction mechanism is simple. Indeed, turning to the above reaction mechanism of hydrogen iodide decomposition, we find for the stationary reaction rate w = —d[HI]/dt = 2d[l2]/dt. 

A particularly important result is that the matrix of phenomenological resistances (linear combinations of the Rj) which appears in the equations of motion relating the independent reaction rates to their conjugate thermodynamic forces in the stationary state is symmetric. However, the symmetry is of an algebraic nature the more stringent condition of differential symmetry does not hold in general. 

It is clear from figure A3.4.3 that the second-order law is well followed. Flowever, in particular for recombination reactions at low pressures, a transition to a third-order rate law (second order in the recombining species and first order in some collision partner) must be considered. If the non-reactive collision partner M is present in excess and its concentration [M] is time-independent, the rate law still is pseudo-second order with an effective second-order rate coefficient proportional to [Mj. 

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

To conclude this section it should be pointed out again that the friction coefficient has been considered to be frequency independent as implied in assuming a Markov process, and that zero-frequency friction as represented by solvent viscosity is an adequate parameter to describe the effect of friction on observed reaction rates. 

According to this mechanism, the reaction rate is proportional to the concentration of hydronium ion and is independent of the associated anion, ie, rate = / [CH3Hg][H3 0 ]. However, the acid anion may play a marked role in hydration rate, eg, phosphomolybdate and phosphotungstate anions exhibit hydration rates two or three times that of sulfate or phosphate (78). Association of the polyacid anion with the propyl carbonium ion is suggested. Protonation of propylene occurs more readily than that of ethylene as a result of the formation of a more stable secondary carbonium ion. Thus higher conversions are achieved in propylene hydration. 

Model Reactions. Independent measurements of interfacial areas are difficult to obtain in Hquid—gas, Hquid—Hquid, and Hquid—soHd—gas systems. Correlations developed from studies of nonreacting systems maybe satisfactory. Comparisons of reaction rates in reactors of known small interfacial areas, such as falling-film reactors, with the reaction rates in reactors of large but undefined areas can provide an effective measure of such surface areas. Another method is substitution of a model reaction whose kinetics are well estabUshed and where the physical and chemical properties of reactants are similar and limiting mechanisms are comparable. The main advantage of employing a model reaction is the use of easily processed reactants, less severe operating conditions, and simpler equipment. 

When the overflow clarity is independent of overflow rate and depends only on detention time, as in the case for high soHds removal from a flocculating suspension, the required time is deterrnined by simple laboratory testing of residual soHd concentrations in the supernatant versus detention time under the conditions of mild shear. This deterrnination is sometimes called the second-order test procedure because the flocculation process foUows a second-order reaction rate. 

Equation 20 is the rate-controlling step. The reaction rate of the hydrophobes decreases in the order primary alcohols > phenols > carboxylic acids (84). With alkylphenols and carboxylates, buildup of polyadducts begins after the starting material has been completely converted to the monoadduct, reflecting the increased acid strengths of these hydrophobes over the alcohols. Polymerization continues until all ethylene oxide has reacted. Beyond formation of the monoadduct, reactivity is essentially independent of chain length. The effectiveness of ethoxylation catalysts increases with base strength. In practice, ratios of 0.005—0.05 1 mol of NaOH, KOH, or NaOCH to alcohol are frequendy used. 

The stoichiometric relationship between chlorine dioxide added and color removed during bleaching is nonlinear, but it is independent of temperature, pH, and pulp concentration under conditions normally used. Models used to explain the kinetics and stoichiometry show a strong dependence on chromophore concentration that probably results from differences in the reaction rates of the various chromophores present in the pulps (80). 

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

Diethyl ether is the principal by-product of the reaction of ethyl alcohol with diethyl sulfate. Various methods have been proposed to diminish its formation (70—72), including separation of diethyl sulfate from the reaction product. Diethyl sulfate not only causes an increase in ether formation but is also more difficult to hydroly2e to alcohol than is ethyl hydrogen sulfate. The equiUbrium constant for the hydrolysis of ethyl hydrogen sulfate is independent of temperature, and the reaction rate is proportional to the hydrogen ion concentration (73—75). 

For many laboratoiy studies, a suitable reactor is a cell with independent agitation of each phase and an undisturbed interface of known area, like the item shown in Fig. 23-29d, Whether a rate process is controlled by a mass-transfer rate or a chemical reaction rate sometimes can be identified by simple parameters. When agitation is sufficient to produce a homogeneous dispersion and the rate varies with further increases of agitation, mass-transfer rates are likely to be significant. The effect of change in temperature is a major criterion-, a rise of 10°C (18°F) normally raises the rate of a chemical reaction by a factor of 2 to 3, but the mass-transfer rate by much less. There may be instances, however, where the combined effect on chemical equilibrium, diffusivity, viscosity, and surface tension also may give a comparable enhancement. 

The laboratory studies utilized small-scale (1-5-L) reactors. These are satisfactoiy because the reaction rates observed are independent of reac tor size. Several reac tors are operated in parallel on the waste, each at a different BSRT When steady state is reached after several weeks, data on the biomass level (X) in the system and the untreated waste level in the effluent (usually in terms of BOD or COD) are collected. These data can be plotted for equation forms that will yield linear plots on rec tangular coordinates. From the intercepts and the slope or the hnes, it is possible to determine values of the four pseudo constants. Table 25-42 presents some available data from the literature on these pseudo constants. Figure 25-53 illustrates the procedure for their determination from the laboratory studies discussed previously. 

The role that acid and base catalysts play can be quantitatively studied by kinetic techniques. It is possible to recognize several distinct types of catalysis by acids and bases. The term specie acid catalysis is used when the reaction rate is dependent on the equilibrium for protonation of the reactant. This type of catalysis is independent of the concentration and specific structure of the various proton donors present in solution. Specific acid catalysis is governed by the hydrogen-ion concentration (pH) of the solution. For example, for a series of reactions in an aqueous buffer system, flie rate of flie reaction would be a fimetion of the pH, but not of the concentration or identity of the acidic and basic components of the buffer. The kinetic expression for any such reaction will include a term for hydrogen-ion concentration, [H+]. The term general acid catalysis is used when the nature and concentration of proton donors present in solution affect the reaction rate. The kinetic expression for such a reaction will include a term for each of the potential proton donors that acts as a catalyst. The terms specific base catalysis and general base catalysis apply in the same way to base-catalyzed reactions. 

Equation 1-106 predicts that the initial rate will be proportional to the initial enzyme concentration, if the initial substrate concentration is held constant. If the initial enzyme concentration is held constant, then the initial rate will be proportional to the substrate concentration at low substrate concentrations and independent of the substrate concentration at high substrate levels. The maximum reaction rate for a given total enzyme concentration is... 

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

As in the acid-catalyzed halogenation of aldehydes and ketones, the reaction rate is independent of the concentration of the halogen chlorination, bromination, and iodination all occur at the same rate. Fomnation of the enolate is rate-detemnining, and, once fomned, the enolate ion reacts rapidly with the halogen. 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

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