Rate equation mixed order

If one were to mix fixed concentrations E and / at time zero and then measure the concentration of El complex as a function of time after mixing, the data would appear to be described by the pseudo-first-order rate equation ... 

Styrene (A) and Butadiene (B) are to be polymerized in a series of mixed-flow reactors, each of volume 25 m3. The rate equation is first order with respect to A and B ... 

Thus for hydrolysis in 50% aqueous acetone, a mixed second and first order rate equation is observed for phenylchloromethane (benzyl chloride, 10)—moving over almost completely to the SV1 mode in water alone. Diphenylchloromethane (11) is found to follow a first order rate equation, with a very large increase in total rate, while with triphenylchloromethane (trityl chloride, 12) the ionisation is so pronounced that the compound exhibits electrical conductivity when dissolved in liquid S02. The main reason for the greater promotion of ionisation—with consequent earlier changeover to the SW1 pathway in this series—is the considerable stabilisation of the carbocation, by delocalisation of its positive charge, that is now possible ... 

Reference has already been made (p. 82) to the fact that the reactions of some substrates, e.g. secondary halides, may follow a mixed first/second order rate equation. The question then arises whether such a reaction is proceeding via both SN2 and SN1 pathways simultaneously (their relative proportions depending on the solvent, etc.) or whether it is proceeding via some specific, in between mechanistic pathway. 

Theoretical studies are primarily concentrated on the treatment of flame blow-off phenomenon and the prediction of flame spreading rates. Dunskii [12] is apparently the first to put forward the phenomenological theory of flame stabilization. The theory is based on the characteristic residence and combustion times in adjoining elementary volumes of fresh mixture and combustion products in the recirculation zone. It was shown in [13] that the criteria of [1, 2, 5] reduce to Dunskii s criterion. Longwell et al. [14] suggested the theory of bluff-body stabilized flames assuming that the recirculation zone in the wake of the baffle is so intensely mixed that it becomes homogeneous. The combustion is described by a second-order rate equation for the reaction of fuel and air. 

The functional nature of some rate equations requires that rate constants occasionally can be of a mixed order, (i.e., they can have nonintegral dimensions with respect to reactant molarity). Whether the reaction order is an integer or a fractional value, its value n is obtained from the slope of a plot of reaction rate versus logio [Reactant]. 

For a given duty the ratio of sizes of mixed and plug flow reactors will depend on the extent of reaction, the stoichiometry, and the form of the rate equation. For the general case, a comparison of Eqs. 5.11 and 5.17 will give this size ratio. Let us make this comparison for the large class of reactions approximated by the simple nth-order rate law... 

Mixing is a rate-related parameter, in that most rates of reaction or transport are dependent on mixing in environmental systems. When mixing is dominant (the slowest process), the first-order rate equation can be described as... 

A perfectly mixed vessel therefore can be considered as a batch vessel during the time required for the bubbles to sweep the vessel contents exactly once (i.c., up to /, = l/ 2 ). The definition of the singlebubble collection efficiency ( ) requires that the observed removal efficiency ( ) should equal after one bubble sweep. This assumes that all the oil drops that collide with a bubble stick to it. The solution of the First-order rate equation for a batch vessel for /j l/ 2 and ( = R yields ... 

Substrate concentration is yet another variable that must be clearly defined. The hyperbolic relationship between substrate concentration ([S ) and reaction velocity, for simple enzyme-based systems, is well known (Figure C1.1.1). At very low substrate concentrations ([S] ATm), there is a linear first-order dependence of reaction velocity on substrate concentration. At very high substrate concentrations ([S] A m), the reaction velocity is essentially independent of substrate concentration. Reaction velocities at intermediate substrate concentrations ([S] A"m) are mixed-order with respect to the concentration of substrate. If an assay is based on initial velocity measurements, then the defined substrate concentration may fall within any of these ranges and still provide a quantitative estimate of total enzyme activity (see Equation Cl. 1.5). The essential point is that a single substrate concentration must be used for all calibration and test-sample assays. In most cases, assays are designed such that [S] A m, where small deviations in substrate concentration will have a minimal effect on reaction rate, and where accurate initial velocity measurements are typically easier to obtain. 

The field of predominantly kinetic influence (base of the voltammogram, BV relation valid) and the held of a mixed influence of kinetics and transport are suitable to determine parameters such as rate constant, reaction order and transfer coefficients. The held controlled by transport (Equation 1.51 valid, in practice usually with c0" or cR =0) can lead to the diffusion coefficient. 

If each type of sulfur compound is removed by a reaction that was first-order with respect to sulfur concentration, the first-order reaction rate would gradually, and continually, decrease as the more reactive sulfur compounds in the mix became depleted. The more stable sulfur species would remain and the residuum would contain the more difficult-to-remove sulfur compounds. This sequence of events will, presumably lead to an apparent second-order rate equation which is, in fact, a compilation of many consecutive first-order reactions of continually decreasing rate constant. Indeed, the desulfurization of model sulfur-containing compounds exhibits first-order kinetics, and the concept that the residuum consists of a series of first-order reactions of decreasing rate constant leading to an overall second-order effect has been found to be acceptable. 

For the former case (Equation (3)), which is environmentally more relevant for low contamination situations, the rate obeys first-order kinetics with respect to substrate and biomass (second-order overall), whereas in the latter case (Equation (4)), the kinetics have a first-order relationship to biomass but are independent of substrate concentration. Methods for measuring of biomass, B, have varied widely, and, for studies involving mixed populations, in which only a fraction of the organisms can degrade the substrate, a means for quantifying the responsible fraction is not available. 

Under most conditions the initial rate, Uo, of the reaction is directly proportional to enzyme concentration (8). In assays to determine the amount of enzyme in a sample the initial substrate concentration should be at least 10 times so that the reaction is zero order with respect to substrate concentration (Equation 3). At substrate concentrations less than 0.1 Km the reaction follows a first-order rate process with the rate directly proportional to substrate concentration. Enzyme rate assays to determine the amount of a compound as substrate in a sample should be run under these conditions. At substrate concentrations greater than 0.1 Km and less than 10 Km the reaction follows a mixed-order process intermediate between first and zero order. 

Most elementary reactions involve either one or two reactants. Elementaiy reactions involving three species are infrequent, because the likelihood of simultaneous three-body encounter is small. In closed, well-mixed chemical systems, the integration of rate equations is straightforward. Results of integration for some important rate laws are listed in Table 2.7, which gives the concentration of reactant A as a function of time. First-order reactions are particularly simple the rate constant k has units of s , and its reciprocal value (1/k) provides a measure of a characteristic time for reaction. It is common to speak in terms of the half-life ( 1/2) for reaction, the time required for 50% of the reactant to be consumed. When... 

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