Half-time first-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... 

Here, as in all first rate kinetics, a plot of the natural logarithm of the count rate vs. the time results in a straight line whose slope is proportional to the rate constant and whose intercept is In N. Another reaction rate characteristic, called the half-life, t. is the time required for the initial reactant concentration to be reduced to one half. For first order reactions, the t, independent of concentration, = 0.693/k. 

A systematic formal kinetic analysis starts with measured concentrationtime curves (e.g., in batch processes, as illustrated in Fig. 2.4 for substrate concentrations). From these data a reaction scheme can be extracted. At this point a clear differentiation must be made between reaction scheme and reaction mechanism. Due to the fictitious character of a mechanism, it may be disproven but never proven. A reaction scheme, on the other hand, can be more or less definitely established and may be extended later only if there is evidence of additional steps. From the shape of the concentration-time curves several conclusions can be made (Moser, 1983b) concerning the interpretation of apparent reaction orders n. Linearity can be a sign for transport limitation or can indicate the presence of a biosorption effect resulting in a reaction order of zero. Half- and first-order reaction can be interpreted as internal transport... 

An important characteristic property of a radioactive isotope is its half-life, fj/2, which is the amount of time required for half of the radioactive atoms to disintegrate. For first-order kinetics the half-life is independent of concentration and is given as... 

The unit of the veloeity eonstant k is see Many reaetions follow first order kineties or pseudo-first order kineties over eertain ranges of experimental eonditions. Examples are the eraeking of butane, many pyrolysis reaetions, the deeomposition of nitrogen pentoxide (NjOj), and the radioaetive disintegration of unstable nuelei. Instead of the veloeity eonstant, a quantity referred to as the half-life iyj is often used. The half-life is the time required for the eoneentration of the reaetant to drop to one-half of its initial value. Substitution of the appropriate numerieal values into Equation 3-33 gives... 

The half-life method requires data from several experiments, eaeh at different initial eoneentration. The method shows that the fraetional eonversion in a given time rises with inereased eoneentration for orders greater than one, drops with inereased eoneentration for orders less than one, and is independent of the initial eoneentration for reaetions of first order. This also applies to the reaetion A -i- B —> produets when... 

In addition to the elimination rate constant, the half-life (T/i) another important parameter that characterizes the time-course of chemical compounds in the body. The elimination half-life (t-1/2) is the time to reduce the concentration of a chemical in plasma to half of its original level. The relationship of half-life to the elimination rate constant is ti/2 = 0.693/ki,i and, therefore, the half-life of a chemical compound can be determined after the determination of k j from the slope of the line. The half-life can also be determined through visual inspection from the log C versus time plot (Fig. 5.40). For compounds that are eliminated through first-order kinetics, the time required for the plasma concentration to be decreased by one half is constant. It is impottant to understand that the half-life of chemicals that are eliminated by first-order kinetics is independent of dose. ... 

The half-life tvi is defined to be the time required for the reactant concentration to decay to one-half its initial value. To find tvi for a first-order reaction we use Eq. (2-6) with the substitutions Ca = c°/2 and t = finding... 

Figure 2-1 is a plot of Eq. (2-10) from n = 0 to = 4. Note that equal time irrcrements result in equal fractional decreases in reactant concentration thus in the first half-life decreases from 1.0 to 0.50 in the second half-life it decreases from 0.50 to 0.25 in the third half-life, from 0.25 to 0.125 and so on. This behavior is implicit in the earlier observation that a first-order half-life is independent of concentration. 

Evidently the reaction shown in Fig. 2-8 is first-order (or pseudo first-order) over the period of time (about 2.5 half-lives) plotted in the figure. From the slope of the line the rate constant is calculated to be 4.90 x 10 s . ... 

Evidently the measurement of should be accomplished with at least the same level of accuracy as the measurement of the A, values, so the question arises When does t = 00 That is, when is the reaction essentially complete For a first-order reaction, we calculate, with Eq. (2-10), that reaction is 99.9% complete after the lapse of 10 half-lives. This would ordinarily be considered an acceptable time for the measurement of... 

The only kinetic data reported are in a Ph.D. thesis (41). Integral order kinetics were usually not obtained for the reaction of a number of ketones with piperidine and a number of secondary amines with cyclohexanone. A few of the combinations studied (cyclopentanone plus piperidine, pyrrolidine, and 4-methylpiperidine, and N-methylpiperazine plus cyclohexanone) gave reactions which were close to first-order in each reactant. Relative rates were based on the time at which a 50% yield of water was evolved. For the cyclohexanone-piperidine system the half-time (txn) for the 3 1 ratio was 124 min and for the 1 3 ratio 121 min. It appears that an... 

FIGURE 14.4 Plot of the course of a first-order reaction. The half-time, <1/9, is the time for one-half of the starting amonnt of A to disappear. 

Another way to describe reaction rates is by half-life, t/, the amount of time it takes for the reactant concentration to drop to one half of its original value. When the reaction follows a first-order rate law, rate = -krxn[reactant], ti is given by ... 

The analysis of Example 11.3c reveals an important feature of a first-order reaction The time required for one half of a reactant to decompose via a first-order reaction has a fixed value, independent of concentration. This quantity, called the half-life, is given by the expression... 

Half life The time required to convert half of the original amount of reactant to product, 294 first-order, 294 second-order, 296... 

Drug elimination may not be first order at high doses due to saturation of the capacity of the elimination processes. When this occurs, a reduction in the slope of the elimination curve is observed since elimination is governed by the relationship Vmax/(Km- -[conc]), where Vmax is the maximal rate of elimination, Km is the concentration at which the process runs at half maximal speed, and [cone] is the concentration of the drug. However, once the concentration falls below saturating levels first-order kinetics prevail. Once the saturating levels of drugs fall to ones eliminated via first-order kinetics, the half time can be measured from the linear portion of the In pt versus time relationship. Most elimination processes can be estimated by a one compartment model. This compartment can... 

The logarithmic plot is not linear, of course, since this is not a first-order reaction. Note, however, that even In [A], is linear in time to about 50 percent reaction. One cannot use these procedures to establish the kinetic order without data taken to at least two half-times, and preferably longer. 

Second-order kinetics, (a) Derive expressions for the half-time and lifetime of A if the rate law for its disappearance is v = fc[A]2 (b) calculate t]/i and t for the data presented in Section 2.2 (c) calculate the second half-life, t /i(2), i.e., the time elapsed between 50 percent and 75 percent completion, for the same reaction (d) compare fj/2(l) and fi/>(2), and contrast this result with that from first-order kinetics. 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

The half-life, f1/2, of a substance is the time needed for its concentration to fall to one-half its initial value. Knowing the half-lives of pollutants such as chlorofluoro-carbons allows us to assess their environmental impact. If their half-lives are short, they may not survive long enough to reach the stratosphere, where they can destroy ozone. Half-lives are also important in planning storage systems for radioactive materials, because the decay of radioactive nuclei is a first-order process. 

The concentration of the reactant does not appear in Eq. 7 for a first-order reaction, the half-life is independent of the initial concentration of the reactant. That is, it is constant regardless of the initial concentration of reactant, half the reactant will have been consumed in the time given by Eq. 7. It follows that we can take the initial concentration of A to be its concentration at any stage of the reaction if at some stage the concentration of A happens to be A], then after a further time tv2, the concentration of A will have fallen to 2[AJ, after a further tU2 it will have fallen to [A], and so on (Fig. 13.13). In general, the concentration remaining after n half-lives is equal to (t)" A 0. For example, in Example 13.6, because 30 days corresponds to 5 half-lives, after that interval [A ( = (j)5 A]0, or [A]0/32, which evaluates to 3%, the same as the result obtained in the example. 

Self-Tfst 13.9A Calculate (a) the number of half-lives and (b) the time required for the concentration of N20 to fall to one-eighth of its initial value in a first-order decomposition at 1000. K. Consult Table 13.1 for the rate constant. 

For a first-order process, calculate the rate constant, elapsed time, and amount remaining from the half-life (Example 13.6 and Self-Test 13.9). 

Dinitrogen pentoxide, N2Os, decomposes by first-order kinetics with a rate constant of 0.15 s 1 at 353 K. (a) What is the half-life (in seconds) for the decomposition of N2Os at 353 K (b) If [N2O5]0 = 0.0567 mol-L, what will be the concentration of N2Os after 2.0 s (c) How much time (in minutes) will elapse before the N205 concentration decreases from 0.0567 mol-L 1 to 0.0135 mol-L ... 

The half-life for the first-order decomposition of A is 355 s. How much time must elapse for the concentration of A to decrease to (a) one-fourth (b) 15% of its original value (c) one-ninth of its initial concentration ... 

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. 

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