First order rate constants reversible reactions, 55-7 rapid

The reactions obeyed pseudo-first-order kinetics consistent with a rapid reversible protonation of the substrate, S, at the ester carbonyl followed by a rate-determining decomposition to acetic acid and nitrenium ion according to Scheme 19. In accordance with equation 13, the pseudo-first-order rate constant, k, was shown to be proportional to acid concentration and inversely proportional to the activity of the water/acetonitrile solvent . 

The catalytic reaction is divided into two processes. The enzyme and the substrate first combine to give an enzyme-substrate complex, ES. This step is assumed to be rapid and reversible with no chemical changes taking place the enzyme and the substrate are held together by noncovalent interactions. The chemical processes then occur in a second step with a first-order rate constant kc.dl (the turnover number). The rate equations are solved in the following manner. 

Even in the absence of a catalyst, this hydration reaction proceeds at a moderate pace. At 37°C near neutral pH, the second-order rate constant k j is 0.0027 M s f This corresponds to an effective first-order rate constant of 0.15 s i in water ([H2O] = 55.5 M). Similarly, the reverse reaction, the dehydration of bicarbonate, is relatively rapid, with a rate constant of A . j = 50 s f These rate constants correspond to an equilibrium constant of. Si j = 5.4 x 10 5 and a ratio of [CO2] to [H2CO3] of 340 1. 

Interpretation of the results from the product studies and nanosecond laser photolysis experiments led to the rate constants for each step in the overall dimerization reaction summarized in Scheme 4. The addition step is quite rapid, taking place with a rate constant of 1.5 x 10 M" s" , but formation of the cyclobutane radical cation is reversible, with a calculated rate constant of 8 x 10 s" for cycloreversion to regenerate the 4-methoxystyrene radical cation and neutral 4-methoxystyrene. The two other processes available to the intermediate cyclobutane radical cation are rearrangement to the hexalriene radical cation with a first-order rate constant of 2.5 x 10 s". and electron transfer with neutral 4-methoxy- styrene with a rate constant of 1.5 X 10 M s to generate the neutral cyelobutane and to regenerate the 4-methoxystyrene radical cation. 

The inactivation is normally a first-order process, provided that the inhibitor is in large excess over the enzyme and is not depleted by spontaneous or enzyme-catalyzed side-reactions. The observed rate-constant for loss of activity in the presence of inhibitor at concentration [I] follows Michaelis-Menten kinetics and is given by kj(obs) = ki(max) [I]/(Ki + [1]), where Kj is the dissociation constant of an initially formed, non-covalent, enzyme-inhibitor complex which is converted into the covalent reaction product with the rate constant kj(max). For rapidly reacting inhibitors, it may not be possible to work at inhibitor concentrations near Kj. In this case, only the second-order rate-constant kj(max)/Kj can be obtained from the experiment. Evidence for a reaction of the inhibitor at the active site can be obtained from protection experiments with substrate [S] or a reversible, competitive inhibitor [I(rev)]. In the presence of these compounds, the inactivation rate Kj(obs) should be diminished by an increase of Kj by the factor (1 + [S]/K, ) or (1 + [I(rev)]/I (rev)). From the dependence of kj(obs) on the inhibitor concentration [I] in the presence of a protecting agent, it may sometimes be possible to determine Kj for inhibitors that react too rapidly in the accessible range of concentration. ... 

As noted earlier, some carbonic anhydrases can hydrate carbon dioxide at rates as high as a million times a second (10 s ). The magnitude of this can be understood from the following observations. In the first step of a carbon dioxide hydration reaction, the zinc-bound water molecule must lose a proton to regenerate the active form of the enzyme (Figure 9.27). The rate of the reverse reaction, the protonation of the zinc-bound hydroxide ion. is limited by the rate of proton diffusion. Protons diffuse very rapidly with second-order rate constants near 10 M. Thus, the backward rale constant i must be less than 10 s F Because the equilibrium... 

This describes a first order reaction with the observed rate constant cat/ M- Remembering that has the dimensions of a dissociation constant, we can compare the above equation to that derived for two step ligand binding, when the second step is preceded by a rapid pre-equilibrium (see p. 66). The constant k JK can be determined from the analysis of the record as a first order reaction. Although, even at such low initial substrate concentrations, the effects of reversibility or product inhibition may perturb the later parts of the reaction. The more usual procedure found in the literature is to plot v/ce(0) against Cs(0) from a set of initial rate measurements and to take k JK as the initial slope. As pointed out in section 3.2, one obtains an apparent second order rate constant. 

They argued that pre-equilibria to form Cl+ or S02C1+ may be ruled out, since these equilibria would be reversed by an increase in the chloride ion concentration of the system whereas rates remained constant to at least 70 % conversion during which time a considerable increase in the chloride ion concentration (the byproduct of reaction) would have occurred. Likewise, a pre-equilibrium to form Cl2 may be ruled out since no change in rate resulted from addition of S02 (which would reverse the equilibrium if it is reversible). If this equilibrium is not reversible, then since chlorine reacts very rapidly with anisole under the reaction condition, kinetics zeroth-order in aromatic and first-order in sulphur chloride should result contrary to observation. The electrophile must, therefore, be Cli+. .. S02CI4- and the polar and non-homolytic character of the transition state is indicated by the data in Table 68 a cyclic structure (VII) for the transition state was considered as fairly probable. 

In certain situations, a chemical of interest may be involved in a rapid reversible transformation in the water phase. Such a reaction would affect the concentration in the boundary zone and thus would alter the transfer rate. The reaction time tr (defined by the inverse of the first-order reaction rate constant, tr =k7x) determines whether air-water exchange is influenced by the reaction. Three cases can be distinguished. 

Wilson and Cannan (18) reported detailed observations on the equilibrium and velocity constants in the glutamic acid—pyrrolidone carboxylic acid system in dilute aqueous solution. They found that the conversion of glutamic acid to pyrrolidone carboxylic acid follows the equation for a reversible first-order reaction. The equilibrium constant and the rate at which the equilibrium is achieved depend on the pH of the solution and the temperature. In neutral solutions, the equilibrium favors almost complete conversion of glutamic acid to pyrrolidone carboxylic acid however, the rate of the reaction is very slow and thus only 1% conversion occurs after 2-3 hr at 100°. In weakly acid (pH 4) and alkaline (pH 10) solutions, the conversion of glutamic acid to pyrrolidone carboxylic acid is much faster and about 98% conversion occurs in less than 60 hr. In strong acid (2 N HC1) and base (0.5 N NaOH) the conversion of pyrrolidone carboxylic acid to glutamic acid proceeds rapidly and virtually to completion. Other studies have shown that the conversion of glutamic acid to pyrrolidone carboxylic acid can be carried out within 2 hr at 142° with little alteration of optical rotation (80). 

With faster scan cyclic voltammetry, a new two-electron anodic peak was detected, at more negative potentials, for the first stage of the oxidation process, with an accompanying cathodic peak on the reverse scan (11). The ratio of the forward to the reverse peak currents increased towards unity as the scan rate was raised to —200 V s 1 (Fig. 15). This behavior was attributed to the initial two-electron process being accompanied by a fairly rapid follow-up chemical reaction and was successfully analyzed in terms of an EqCi process (quasi-reversible electron transfer followed by a first-order irreversible chemical process), with a rate constant for the chemical step, k, = 250 s 1. 

The rate of a-chymotrypsin-catalyzed hydrolysis as a function of overall GPANA concentration in CTAB reversed micelles and in aqueous solution are shown in Figure 5. It is apparent that the reaction rate in the reversed micellar solution is on the order of 50 times more rapid than in the aqueous system. Furthermore, in the reversed micellar system there is no indication of enzyme saturation as the reaction is first order in substrate concentration. As enzyme saturation kinetics are not observed, it is impossible to differentiate between the parameters kcat and Kg. Instead a second order bimolecular rate constant for both the micelle interior ( micelle) and for what is experimentally observed ( observed) is defined. 

Studies of single channels formed in lipid bilayers by Staphylococcus aureus alpha toxin showed that fluctuations in the open-channel current are pH-dependent (47). The phenomenon was attributed to conductance noise that arises from reversible ionization of residues in the channel-forming molecule. The pH-dependent spectral density of the noise, shown in Figure 6, is well described by a simple model based on a first-order ionization reaction that permits evaluation of the reaction parameters. This study demonstrates the use of noise analysis to measure the rate constants of rapid and reversible reactions that occur within the lumen of an ion channel. 

Table II shows the average fraction of exchangeable- Cs in sediments after 3 sequential 24-hr Nlij-extractions and the additional fraction mobilized after the fourth, long-term (400-842 days) extraction. Assuming (1) that all (rapidly) exchangeable radiocesium had been removed by the three prior extractions, and (2) a first order remobilization process, we can roughly calculate the reverse rate constant that describes the slow remobilization of Cs from the sediments. Table II indicates that the half-life of this reaction, which is interpreted as slow release from the edge-...

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