Apparent first-order rate constant

Such a reaction is described as first order and the proportionality constant k is known as the rate constant. Such first-order kinetics is observed for unimolecular processes in which a molecule of A is converted into product P in a given time interval with a probability that does not depend on interaction with another molecule. An example is radioactive decay. Enzyme-substrate complexes often react by unimolecular processes. In other cases, a reaction is pseudo-first order compound A actually reacts with a second molecule such as water, which is present in such excess that its concentration does not change during the experiment. Consequently, the velocity is apparently proportional only to [A]. 

The other experiments, in which the concentrations of the components in the aqueous phase are fixed, were carried out by Wu. The volume of organic phase is also fixed at 50 mL, in which the amount of catalyst and phosphazene are also fixed. A generalized apparent rate constant (pseudo-first-order rate law) k j pp is defined as... 

We have next to consider the measurement of the relaxation times. Each t is the reciprocal of an apparent first-order rate constant, so the problem is identical with problems considered in Chapters 2 and 3. If the system possesses a single relaxation time, a semilogarithmic first-order plot suffices to estimate t. The analytical response is often solution absorbance, or an electrical signal proportional to absorbance or to another physical property. As shown in Section 2.3 (Treatment of Instrument Response Data), the appropriate plotting function is In (A, - Aa=), where A, is the... 

When acetone is treated with hydroxylamine in aqueous solution near neutral pH, the carbonyl UV absorption intensity decreases very rapidly this fast spectral change is followed by a much slower absorption increase that is due to the appearance of the oxime product. This suggests that, at such pH values, the initial addition is very rapid and the second step, dehydration of the carbinolamine, is the rds. Figure 5-12 is a plot of the apparent first-order rate constant against pH for this reaction. As the pH is decreased from neutrality, the rate increases, indicating that the rds... 

The isomerization of A to B yielded kinetic data that conformed to a first-order rate law. but the apparent first-order rate constant depended on the initial concentration of A. The authors propose competing unimolecular and bimolecular processes, and they show that the system reduces to a first-order expression when the equilibrium constant K is unity that is,... 

Composite temperature dependence. Show that a plot of the apparent first-order rate constant of Eq. (7-34) as ln(/tapp/r) versus lIT will always be curved downward, corresponding to an apparent AW that decreases with increasing temperature. What will the shape of the plot be if by chance AW = AHif ... 

A Hammett plot (a) of the apparent first-order rate constants for semicarbazone formation at pH 3.9, where a change in the RCS occurs, and (b) of the denitrosation of A-methyl-A-nitrosoanilines, where the mechanism changes. Data are from Refs. 9 and 10. 

Assuming plug flow of both phases in the trickle bed, a volumetric mass transfer coefficient, kL a, was calculated from the measurements. The same plug flow model was then used to estimate bed depth necessary for 95% S02 removal from the simulated stack gas. Conversion to sulfuric acid was handled in the same way, by calculating an apparent first-order rate constant and then estimating conversion to acid at the bed depth needed for 95% S02 removal. Pressure drop was predicted for this bed depth by multiplying... 

Apparent first-order absorption rate constant (min 1) ... 

This technique is readily adaptable for use with the generalized additive physical approach discussed in Section 3.3.3.2. It is applicable to systems that give apparent first-order rate constants. These include not only simple first-order irreversible reactions but also irreversible first-order reactions in parallel and reversible reactions that are first-order in both the forward and reverse directions. The technique provides an example of the advantages that can be obtained by careful planning of kinetics experiments instead of allowing the experimental design to be dictated entirely by laboratory convention and experimental convenience. 

Bronsted and Guggenheim [J. /Am. Chem. Soc., 49 (2554), 1927] have studied the mutaro-tation of glucose as catalyzed by acids and bases. The reaction takes place slowly in pure water, is weakly catalyzed by hydrogen ions, and is strongly catalyzed by hydroxide ions. When strong acids and bases are employed as catalysts, the apparent first-order rate constants can be written as... 

This value is considerably higher than the experimental value (0.17) obtained from rate measurements on different size particles, but several factors may be invoked to explain the inconsistency. There will be a distribution of both pore radii and pore lengths present in the actual catalyst rather than uniquely specified values. Alumina catalysts often have a bimodal pore-size distribution. Our estimate of an apparent first-order rate constant using the method outlined above will be somewhat in error. The catalyst surface may not be equally active throughout if selective deactivation has taken place and the peripheral region is less active than the catalyst core. Other sources of error are the... 

The ortho-para conversion of molecular hydrogen is catalyzed by NiO. A supported catalyst is available with a specific surface area of 305 m2/g and a void volume of 0.484 cm3/g. A spherical catalyst pellet has an apparent density of 1.33 g/cm3 and a diameter of 0.5 cm. If the system is not far from equilibrium, an apparent first-order rate constant (kr) can be defined in the following manner. 

A catalyst for cracking cumene is available commercially in the form of 0.35 cm diameter pellets. These pellets have a specific surface area of 420 m2/g and a void volume of 0.42 cm3/g. If the apparent first-order rate constant for this reaction is 1.49 cm3/sec-g catalyst at 412 °C, determine the effectiveness factor of the catalyst. 

Various rate constants which enter into the expression for k i, Equation 14.14, have now been discussed. kunj as defined in Equation 14.13 has the appearance of a first order rate constant for the disappearance of A molecules but it is actually only a pseudo first order rate constant since it explicitly depends on the concentration of M, the species involved in the activation and deactivation of A molecules. In the limit of high concentration, [M] oo, kuni reduces to an apparent first order process, lim (kuni j oo) = ka(E)(8ki(E)/k2)[A] = kt(Apparent)[A], while at low concentration the reduction is to an apparent second order process, lim(kunij[M]->.o) = 8k (E)[A][M] = k2 (Apparent) [A] [M],... 

The control strategies for determining the feed policies were decided on the basis of a numerical solution of the terpolymerisations described by equations 1 - 3 using a microcomputer and a general purpose simulation package, BEEBSOC (10). Where necessary, these data were acquired in the course of this study, otherwise literature values were used. The apparent first order rate constants in terpolymerisations have been shown to be composition dependent. The variation in rate constants with... 

Actin filaments grow rapidly within cells, and the clearest evidence of this rapid growth is the ability of the cell s leading edge to move at rates of 0.5 to 1 micrometer per second. Likewise, actin-based motility of Listeria and Shigella can attain rates of nearly 0.5 micrometers per second. Because microfilaments contain about 360 actin monomers per micrometer of length, a motility rate of 0.5 to 1 micrometer per second corresponds to an apparent first-order rate constant (/.e., / apparent = on [Actin-ATP]) of about 180-360 s . The bimolecular rate constant for actin-ATP addition to the barbed end has a nominal value of 2-3 X 10 s . Therefore, one can estimate... 

The effect of steam treatment of ZSM-5 on its cracking activity and selectivity was measured with experiments using n-hexadecane feed. With the thermally treated ZSM-5 catalyst, concentration of the unconverted n-hexadecane in the product was not measurable while 507e of the feed was unconverted with the steam treated ZSM-5 catalyst (Table II). The lower limit of conversion with the thermally treated catalyst corresponding to detection limit of n-hexadecane is 99.99%. This lower limit suggests at least an order of magnitude reduction in apparent first order rate constant of the ZSM-5 catalyst upon steam treatment. The small reduction in crystallinity upon steaming cannot fully explain the dramatic activity loss. Loss of active sites due to dealumination of ZSM-5 can be postulated to explain the reduction in activity. 

K = Apparent First Order Rate Constant (hours ) (Phosphate, acetate and borate buffers adjusted to 0.1 ionic strength)... 

The optical rotation of the mixture approaches zero (a racemic mixture) over time, with apparent first-order kinetics. This observation was supported by the semi-log plot [ln(a°D/ aD) vs time], which is linear (Figure 1). It has been shown that this racemization process does in fact follow a true pseudo-first-order rate equation, the details of which have been described by Eliel.t30 Therefore, these processes can be described by the first-order rate constant associated with them, which reflects precisely the intrinsic rate of racemization. Comparison of the half-lives for racemization under conditions of varying amino acid side chain, base, and solvent is the basis for this new general method. 

The rate constants which have just been defined differ from the rate constants of an ordinary chemical reaction. They are constant only for the course of an exchange reaction with a single mixture of reacting gases, but they are dependent on pressure and assume different values if the initial pressures of the reactants are altered. The true kineties of an exchange reaction can be determined only by means of a series of experiments with different mixtures of the reactants because the course of a reaction with a single mixture follows the apparent first-order Equation (5). 

Rate of biological inactivation of nitric oxide. The apparent first-order rate constants were calculated as the natural logarithm of two divided by the half-life of nitric oxide measured in perfusion cascades. The half-lives were 30 sec with room air and 6-8 sec with 95% oxygen (Furchgott and Vanhoutte, 1989). Under anaerobic conditions, nitric oxide is indefinitely stable. The slope of the line give a second-order rate constant for the inactivation of nitric oxide by oxygen as approximately 100 M" sec". ... 

Apparent first-order rate constants for peroxynitrite decomposition in various buffers versus pH. When peroxynitrite is fully protonated at acidic pH, the decomposition rate is constant. The breakpoint in the curve identifies the pK, of peroxynitrite since a larger fraction present as an anion slows the rate of decomposition. In 50 mM potassium phosphate, the apparent pK, is at 6.8 and is not affected by temperatute (Koppenol, 1993). The rate of decomposition is not affected hy DMSO, mannitol, or ethanol. As shown in Fig. 28, many buffers can slightly accelerate the decomposition of peroxynitrite and the rate of decomposition reaches a maximum at high buffer concentrations. When these maximal rates are plotted as a function of pH, peroxynitrite exhibits a second pK, of approximately 8.0. 

The observed apparent first-order rate constants are listed in Table I. From the constants at different temperatures but at the same initiator concentration, it is possible to calculate an apparent activation energy for the propagation reaction in each case. This has been done, and the Arrhenius plot is shown in Figure 5. It is interesting to note the similarity of these... 

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