Half-life of the reaction

First, let us consider batch mixing processes, as exemplified by ordinaiy laboratory practice in solution kinetics. A portion of one solution (say, of the substrate) is added by pipet to a second solution (containing the reagent) in a flask, the flask is shaken to achieve homogeneity, and then samples are withdrawn at known times for analysis, or the solution is subjected to continuous observation as a function of time, for example, by spectrophotometry. For reactions on a time scale (measured by the half-life) of hours or even several minutes, the time consumed in these operations is a negligible portion of the reaction time, but as the half-life of the reaction decreases, it becomes necessary to consider these preliminary steps. Let us distinguish three stages ... 

Three ranges of values of n were considered, >1, 0.7—1.0 and <0.7. When n> 1, and particularly when 3 < n < 4, the Weibull distribution readily reduces to a normal distribution if the Erofe ev function is symmetrical about a = 0.5. [The Weibull distribution is symmetrical for n = 3.26, i.e. (1 — In 2)-1, and the inflection point varies only slowly with n.] Thus, under these conditions (3 < n < 4 and symmetry about a = 0.5), we may derive the parameters of the corresponding normal distribution (where p defines the half-life of the reaction and the dispersion parameter, a, is a measure of the lack of homogeneity of the surface centres), viz. 

The half-time (or half-life) of the reaction is independent of [A]o. The reciprocal of the rate constant, t = l/k, is referred to as the lifetime or the mean reaction time. In that time [A] falls to l/e of its initial value. The pharmaceutical industry refers to the shelf life or t90, the time at which [A]/[A]o reaches 0.90. Both t and t90 are also independent of [A]o. 

The first-order decomposition of compound X, a gas, is carried out and the data are represented in the following pictures. The green spheres represent the compound the decomposition products are not shown. The times at which the images were taken are shown below each flask, (a) Determine the half-life of the reaction, (b) Draw the appearance of the molecular image at 8 s. 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

It is often found that the ratio R (measured, for instance, by gas adsorption methods) of actual metal surface area accessible to the gas phase, to the geometric film area, exceeds unity. This arises from nonplanarity of the outermost film surface both on an atomic and a more macroscopic scale, and from porosity of the film due to gaps between the crystals. These gags are typically up to about 20 A wide. However, for film thicknesses >500 A, this gap structure is never such as completely to isolate metal crystals one from the other, and almost all of the substrate is, in fact, covered by metal. In practice, catalytic work mostly uses thick films in the thickness range 500-2000 A, and it is easily shown (7) that intercrystal gaps in these films will not influence catalytic reaction kinetics provided the half-life of the reaction exceeds about 10-20 sec, which will usually be the case. 

The half-life of the reaction depends on the concentration of A and, thus, this reaction cannot be first-order. For a second-order reaction, the half-life varies inversely with the... 

A second order reaction, A + B 4- Products, is found to be 25% complete in 50 minutes when both starting concentrations are 0.2 mol/liter. Find the specific rate and the half life of the reaction. 

The main extra species that forms upon acidification of paratungstate was identified as a -[(H)Wi204o]7, which is internally protonated and forms at higher pH than a -[(H2)Wi204o]6 (141). This monoproto-nated species, a-[(H)Wi204o]7, was previously only obtained by reduction and reoxidation (154). It is slowly reprotonated to o -[(H2) Wi2O40]6-, the half-life of the reaction being 1 day at room tempera-... 

The conductivity remained constant between about 15 s after the initiation and the end of the first half-life of the reactions and was approximately equal to the conductivity of the monomer solution alone (ca. 10"8 Q 1 cm 1) and variable from one experiment to another the reaction mixtures were completely colourless. At or shortly after the end of the first half-life, the conductivity began to increase slowly at first when the reaction was almost complete, it rose quickly and then became constant. At this stage the solution was yellow. 

Problem 1.21 The data of a chemical reaction is plotted as l/[reactant] vs time and the plot is a straight line with intercept 4.0 x 102 mol 1 dm3 and slope 4.0 mol dm3 s 1 as shown in the figure. Calculate the half-life of the reaction. 

With the presence of two methyl substituents at the allenic terminus of 20a, the a,3-didehydrotoluene biradical 21 having a tertiary benzylic radical center was generated after cycloaromatization (Eq. 20.1). As a result, the half-life of the reaction is only -70 min at 37 °C, which is significantly shorter than that of 8. 

O O A first-order decomposition reaction has a rate constant of 2.34 X 10 year . What is the half-life of the reaction Express your answer in years and in seconds. 

The influence of the nature of the anion on the intercalation process was also studied. The intercalation of 5 M solutions of liX (with X = Br, NO3, OH and ISO4) were followed at 120 °C. The extent of reaction plots vary greatly between the different salts (Fig. 9). The plots shown in Fig. 8 are reduced time plots, in which the time is divided by the half-life of the reaction. 

Important quantities characteristic of a first-order reaction are the half-life of the reaction, which is the value of t when [A], = [A](,/2, and t, the relaxation time, or mean lifetime, defined as k. ... 

In this case, the rate constant has units of r and is usually termed half-life of the reaction (ti/2), the amount of time required to reduce the initial concentration c , to half ... 

Defining the half-life of the reaction, ty2, as the time needed for the concentration of reactants to drop to one-half the original value, we obtain... 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

The time that it takes to mix reactants or to bring them to a specified temperature may be significant in comparison to the half-life of the reaction. 

Another important concept is the half-life of the reaction x (sometimes tv2), the time needed to decrease the concentration to one half of the initial value c(A )0. It is related to the order of reaction and the rate constant through the following equations ... 

Both [A] and [B] follow simple exponentials. It should be noted that the half-life of the reaction, tm, [A]0/2, is given by where [A] = [B] =... 

The reaction A — C is first-order in the reactant A and is known to go to completion. The product C is colored and absorbs light strongly at 550 nm, while the reactant and intermediates are colorless. A solution of A was prepared, and the absorbance of C at 550 nm was measured as a function of time. (Note that the absorbance of C is directly proportional to its concentration.) Use the following data to determine the half-life of the reaction ... 

The equilibrium is concentration- and dilution-dependent, i.e., the trimeric species will be preferred in dilute solution and high temperatures, and the tetramer in the concentrated ones and at low temperatures. Rate studies of the interconversion of the tetramer into the trimer indicate that the half-life of the reaction is 30 minutes when diluted with 58 parts of carbon tetrachloride. 

To follow the reaction over time, a 2 /iL aliquot is removed as soon as possible (e.g., 15 s) and loaded in the first well of the gel. With the current running (do not touch the gel), reaction aliquots are carefully loaded in adjacent wells at different times, with intervals chosen to span the expected half-life of the reaction. It is recommended that one also prepare samples of unfolded and fully renatured RNA as controls. 

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