First-order reactions, substrate

The reaction is now first-order in substrate, and the rate of the reaction can be used to determine the substrate s concentration by a fixed-time method. 

By using the product ratio, the overall rate can be dissected into the individual rates for formation of 2 and 3. These rates are found to be substituent/dependent for formation of 2 (p = +1.4) but substituent independent for formation of 3 (p = —0.1 0.1). The reactions are both second-order, first-order in base and first-order in substrate. 

Table 8.1 presents the results of the ICR retention time studies, sugar concentration (dual substrate) studies and cell density. The kinetic model for ICR was derived on the basis of a first order reaction, plug flow and steady-state behaviour. 

Once formed by this process, the carbene may undergo any of the normal carbene reactions (see p. 250). When the net result is substitution, this mechanism has been called the SnIcB (for conjugate base) mechanism. Though the slow step is an SnI step, the reaction is second order first order in substrate and first order in base. 

The first step, as we have already seen (12-3), actually consists of two steps. The second step is very similar to the first step in electrophilic addition to double bonds (p. 970). There is a great deal of evidence for this mechanism (1) the rate is first order in substrate (2) bromine does not appear in the rate expression at all, ° a fact consistent with a rate-determining first step (3) the reaction rate is the same for bromination, chlorination, and iodination under the same conditions (4) the reaction shows an isotope effect and (5) the rate of the step 2-step 3 sequence has been independently measured (by starting with the enol) and found to be very fast. With basic catalysts the mechanism may be the same as that given above (since bases also catalyze formation of the enol), or the reaction may go directly through the enolate ion without formation of the enol ... 

ArNHNHAr and Ar NHNHAr give no molecules containing both Ar and Ar An important discovery was the fact that, although the reaction is always first order in substrate, it can be either first or second order in With some substrates the reaction is entirely first order in [H" ], while with others it is entirely second order in regardless of the acidity. With still other substrates, the reaction is... 

Kemp and Waters also found the oxidations of cyclohexanone and of mandelic, malonic and a-hydroxyisobutyric acids by Cr(VI) to be Mn(II)-catalysed. In these cases, as with oxalic acid, the [Cr(VI)] versus time plots are almost linear and the reaction becomes first order in substrate (or involves Michaelis-Menten kinetics), and, except at lowest catalyst concentrations, approximately first order in [Mn(II)]. Detailed examination of the initial rate of oxidation of a-hydroxyrobutyric acid as a function of oxidant concentration revealed, however, that the dependence is... 

The data are summarised in Table 9. At high concentrations of formic acid the reaction becomes less than first-order in substrate this indicates the possibility of complex-formation, but a medium effect may also be influential in the vicinity of 1 Af formic acid. Complex-formation affects the kinetics of the Tl(rrr) oxidation at all but the lowest reactant concentrations " . 

It is well known that the base hydrolysis of polyacrylamide is catalyzed by OH ions (first order reaction) and obeys autoretarded kinetics due to the electrostatic repulsion between the anionic reagent and the polymeric substrate(3-5). In the range of slightly acid pH (3 < pH < 5), Smets and Hesbain(6) have demonstrated a... 

First-order rate constants are used to describe reactions of the type A — B. In the simple mechanism for enzyme catalysis, the reactions leading away from ES in both directions are of this type. The velocity of ES disappearance by any single pathway (such as the ones labeled k2 and k3) depends on the fraction of ES molecules that have sufficient energy to get across the specific activation barrier (hump) and decompose along a specific route. ES gets this energy from collision with solvent and from thermal motions in ES itself. The velocity of a first-order reaction depends linearly on the amount of ES left at any time. Since velocity has units of molar per minute (M/min) and ES has units of molar (M), the little k (first-order rate constant) must have units of reciprocal minutes (1/min, or min ). Since only one molecule of ES is involved in the reaction, this case is called first-order kinetics. The velocity depends on the substrate concentration raised to the first power (v = /c[A]). 

FIRST-ORDER REACTIONS The velocity, or rate, of a reaction is the change in substrate concentration per unit time. For a simple reaction of the type A — P, the velocity of the formation of P or the disappearance of A is found (usually) to be proportional to the concentration of A that is present at the time the velocity is measured ... 

For a FIRST-ORDER REACTION, the velocity decreases as the concentration of substrate decreases as it is converted to product. As a result, a plot of substrate concentration against time is a curved line. 

One consequence of a first-order reaction is that it takes a constant amount of time for half the remaining substrate to be converted to product—regardless of how much of the reactant is present. It takes the same amount of time to convert 100,000 A molecules to 50,000 P molecules as it takes to convert 10 A molecules to 5 P s. A first-order reaction has a constant half-time t1/2. 

A mechanism for a pseudo-first-order reaction involving the hydrolysis of substrate S catalyzed by acid HA that is consistent with the observed rate law rs = kohscs, is as follows ... 

There are two limiting cases of Michaelis-Menten kinetics. Beginning from Eq. (1) at high substrate excesses (or very small Michaelis constants) Eq. (4 a) results. This corresponds to a zero-order reaction with respect to the substrate, the rate of product formation being independent of the substrate concentration. In contrast, very low substrate concentrations [26] (or large Michaelis constants) give the limiting case of first-order reactions with respect to the substrate, Eq. (4b) ... 

In Figure 10.1, it can be seen that even with substrate excesses of [S] = 20 KM> the saturation range is not yet reached. Conversely, the data in Figure 10.2 indicate that even for very small substrate concentrations ([S]=0.05 KM) the limiting case for the first-order reaction - when the rate is directly proportional to the substrate concentration - is not identical with the values from Eq. (1). 

Fig. 10.2 Comparison of Eq. (1) (upper line) with the limiting case of a first-order reaction Eq. (4b) (lower line) for very low substrate concentrations.

In homogeneous catalysis, the quantification of catalyst activities is commonly carried out by way of TOF or half-life. From a kinetic point of view, the comparison of different catalyst systems is only reasonable if, by giving a TOF, the reaction is zero order or, by giving a half-time, it is a first-order reaction. Only in those cases is the quantification of activity independent of the substrate concentration utilized ... 

The results of the kinetic analysis for the investigated systems are summarized in Table 10.2, the substrate concentration used being the same for all trials. In the case of methyl- and cyclohexyl-substituted ligands the Michaelis constant is smaller than the initial substrate concentration of [S]o=0.06666 mol L-1 (Table 10.2). However, a description of the hydrogenations with other catalyst ligands as first-order reactions shows that in each of these cases the Michaelis constant must be much greater than the experimentally chosen substrate concentration. 

Interpretation of the reciprocals of the Michaelis constants allows the following conclusions to be made regarding hydrogenations under specified experimental conditions. In the case of the methyl and cyclohexyl ligand, the prevailing form of the catalyst in solution is the catalyst-substrate complex. However, for the other examples of first-order reactions, large Michaelis constants (or very... 

As explained earlier, the pre-equilibria are characterized by the limiting values of Michaelis-Menten kinetics. In the case of first-order reactions with respect to the substrate, we have Kfvl [S]0. Since the pre-equilibria are shifted to the side of educts during hydrogenation, only the solvent complex is detectable. In contrast, in the case of zero-order reactions only catalyst-substrate complexes are expected under stationary hydrogenation conditions in solution. These consequences resulting from Michaelis-Menten kinetics can easily be proven by var-... 

Thus, if information is being sought about intermediates for this type of catalysis, it does not make sense to analyze systems that lead to first-order reactions Rather, systems in which the hydrogenation rate is independent of the substrate concentration would be more appropriate. Indeed, for both catalytic systems shown in Figure 10.21, in each case one of the catalyst-substrate complexes could be isolated and characterized by crystal structure analysis (Fig. 10.23). 

The kinetics of hydrogenation transfer is covered by the use of an exchange superoperator assuming a pseudo first-order reaction. Thereby, competing hydrogenations of the substrate to more than one product can also be accommodated. In addition, the consequences of relaxation effects or NOEs can be included into the simulations if desired. Furthermore, it is possible to simulate the consequences of different types of pulse sequences, such as PH-INEPT or INEPT+, which have previously been developed for the transfer of polarization from the parahydrogen-derived protons to heteronuclei such as 13C or 15N. The... 

The kinetic investigation of this reaction reveals the reaction is first-order in substrate, catalyst and hydrogen concentration, and thus yields the rate law r=kCat[Os][alkyne][H2]. The proposed mechanism as given in Scheme 14.6 is based on the rate law and the coordination chemistry observed with these osmium complexes. 

Kinetics of the photooxidation of organic water impurities on illuminated titania surfaces has been generally regarded to be based on the Langmuir-Hinshelwood equation with first-order reaction kinetics vs. initial substrate concentration was established univocally by many authors... 

With first-order reaction kinetics, the rate of metabolism is proportional to the substrate the following relation can be expressed ... 

Two possible approaches are indicated in Schemes 4 and 5. In the first, a reactive radical R> is spin-trapped in competition with its pseudo-first order reaction with a substrate SH, which occurs at a known rate to give RH and S. The growth of both spin-adducts (ST—R ) and (ST—S ) is monitored, and simple analysis leads to the trapping rate constant kT. In the second approach, R-does not react with a substrate, but undergoes unimolecular rearrangement or fragmentation at a known rate to give a new species R. This latter procedure... 

In almost all kinetic investigations it is found that hydroformylation is first order in substrate and hydrogen concentration. This suggests slow steps in the reaction cycle involving olefin and hydrogen, and the reaction rate ta becomes... 

The rearrangements are first order in substrate and in acid and are mainly intramolecular in nature, although a small intermolecular component has been identified. Shine and coworkers investigated this reaction to try to prove that the process occurred by the nonconcerted (the favored) pathway (Scheme 10) rather than the concerted pathway (Scheme 11). 

The reaction was second order in acid and first order in substrate, so both rearrangements and the disproportionation reaction proceed via the doubly-protonated hydrazobenzene intermediate formed in a rapid pre-equilibrium step. The nitrogen and carbon-13 kinetic isotope effects were measured to learn whether the slow step of each reaction was concerted or stepwise. The nitrogen and carbon-13 kinetic isotope effects were measured using whole-molecule isotope ratio mass spectrometry of the trifluoroacetyl derivatives of the amine products and by isotope ratio mass spectrometry on the nitrogen and carbon dioxide gases produced from the products. The carbon-12/carbon-14 isotope... 

The yield (y) of a biomass production process is defined as the moles of biomass formed per mole of substrate consumed. Aerobic conditions are more conducive to higher biomass formation (and therefore also to biofilm formation) than anaerobic conditions. Empirically, under aerobic conditions, a yield of 0.05 - 0.6mol biomass/mol carbon can be obtained, while under anaerobic conditions the attainable yield falls to 0.04 -0.083mol. The reaction kinetics of biodegradation processes can be approximated by the first-order reaction rate constant k as follows ... 

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