First-order reaction half-time

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... 

Recall also from Chapter 15 that for first-order reactions, the time required for exactly half of the substance to react is independent of how much material is present. This constant time interval is the half-life, Equation... 

Note that for a first-order reaction, the time required for the reactant concentration to drop by some constant factor is a simple function of the rate constant. Thus, for instance, the half-life ri/2, which is the time required such that = [A]o for any starting concentration,... 

In contrast with a first-order reaction, the time required for the concentration of A to drop to one-half of its initial value in a second-order reaction depends on both the rate constant and the initial concentration. Thus, the value of h/2 increases as the reaction proceeds because the value of [A]0 at the beginning of each successive half-life is smaller by a factor of 2. Consequently, each half-life for a second-order reaction is twice as long as the preceding one (Figure 12.8). 

Model concentration of a chemical species constant (CG) at one boundary (z = h). Transport through the water column by eddy diffusion (K), and removal by first order reaction (half-life t). The other boundary of the water column (z = 0) is impermeable to the chemical species. Time to steady-state at the impermeable boundary shown for different values of dimensionless quotient h/ y/ (Kr). Equations 39y 43. 

The half-life tm) of a reaction is the time required for 50% of the starting material to be consumed. A rule of thumb is to follow the reaction to five or more half-lives to obtain an accurate first order rate constant. For a first order reaction the half-life is t /2 = ln(2)/l = 0.693 Ik. A related term isthelifetimeofaspecies, defined as l/Ii, where Iris the rate constant for the first order disappearance of the species. For a first order reaction, the time required for reaction does not depend upon how much reactant one starts with. For higher order reactions the half-life and lifetime do depend upon the concentrations of the reactants. 

Because no concentration term appears, for a first-order reaction, the time it takes to reach one-half the starting concentration is a constant and so it does not depend on the reactant concentration. 

Finally, there is a very popular concept connected to first-order reactions half-life. The half-life of a first-order reaction is the amount of time necessary for half of the original amount to react. We can use equation 20.14 to derive a simple expression for the half-life, ti/2. 

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... 

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... 

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. 

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. 

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. 

Therefore, if a plot of In [A] against t is linear, the reaction is first order and k can be obtained from the slope. For first-order reactions, it is customary to express the rate not only by the rate constant k but also by the half-life, which is the time required for half of any given quantity of a reactant to be used up. Since the half-life ti/2 is the time required for [A] to reach Aq/2, we may say that... 

The half-life of a first-order reaction is independent of the initial concentration. Thus, the time required for the reactant concentration to decrease from Uq to Oo/2 is the same as the time required to decrease from Uo/2 to a jA. This is not true for reactions other than first order. 

Another characteristic of first-order reactions is that the time it takes for half the reactant to disappear is the same, no matter what the concentration. This time is called the half-life ( 1/2). Applying Equation to a time interval equal to the half-life results in an equation for / i 2 When half the original concentration has been consumed,... 

One of the typical features of a (pseudo)-first order reaction is that a plot of the logarithm of the advancement of the reaction versus time (Fig. 2B) should give straight lines. However we observed deviation from linearity before the first half-life, in spite of the fact that another characteristic features of (pseudo)-first order reactions, namely that plots of the extent of reaction versus time were independant of the initial concentration (Fig. 3), was verified. We therefore investigated whether variation occured in the reaction conditions as a function of time. 

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 ... 

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. 

All that arithmetic is just to show you that the half-time for a first-order reaction depends only on k, not on how much A you have to start with. The whole point is that the bigger the k, the shorter the half-time, the faster the reaction. 

The half-life of a reactant is the time required for half of that reactant to be converted into products. For a first order reaction, the half-life is independent of concentration so that the same time is required to consume half of any starting amount or concentration of the reactant. On the other hand, the half-life of a second-order reaction does depend on the starting amount of the reactant. 

Statement (d) is incorrect it implies a constant rate during the first half-life. The rate of a first-order reaction actually decreases as time passes and reactant is consumed. 

Set I is the data for a first-order reaction we can analyze those items of data to determine the half-life. In the first 75 s, the concentration decreases by a bit more than half. This implies a half-life slightly less than 75 s, perhaps 70 s. This is consistent with the other time periods noted in the answer to Review Question 18 (b) and also to the fact that in the 150 s from 50 s to 200 s, the concentration decreases from 0.61 M to 0.14 M, a bit more than a factor-of-four decrease. The factor-of-four decrease, to one-fourth of the initial value, is what we would expect of two successive half-lives. We can determine the half-life... 

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 ... 

But then we notice that the time needed to decrease from 60 to 30 ixmoldm-3, or from 2 to 1 ixmoldm 3 will also be 10 min each. In fact, we deduce the important conclusion that the half-life of a first-order reaction is independent of the initial concentration of material. 

Is the reaction a pseudo first-order process The question says that the half-life t /2 of reaction is independent of initial concentration of ester, so the reaction must behave as though it was a first-order reaction in terms of [ester]. In other words, NaOH is in excess and its concentration does not vary with time. 

The reaction half-life, tV2, is the amount of time that it takes for a reactant concentration to decrease to one-half its initial concentration. For a first-order reaction, the half-life is a constant, independent of reactant concentration and has the following relationship ... 

A radioactive isotope may be unstable, but it is impossible to predict when a certain atom will decay. However, if we have a statistically large enough sample, some trends become obvious. The radioactive decay follows first-order kinetics (see Chapter 13 for a more in-depth discussion of first-order reactions). If we monitor the number of radioactive atoms in a sample, we observe that it takes a certain amount of time for half the sample to decay it takes the same amount of time for half the remaining sample to decay, and so on. The amount of time it takes for half the sample to decay is the half-life of the isotope and has the symbol t1/2. The table below shows the percentage of the radioactive isotope remaining versus half-life. 

Thus, the half-life period of a first-order reaction is independent of initial concentration of reactant. Irrespective of how many times the initial concentration of reactant changes, the half-life period will remain same. 

Problem 1.11 In a first order reaction the log (concentration of reactant) versus time plot was a straight line with a negative slope 0.50 x 104 sec-1. Find the rate constant and half-life period of 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. ... 

The latter is invariably used in the relaxation or photochemical approach to rate measurement (Sec. 1.8), rmd is the time taken for A to fall to 1/e (1/2.718) of its initial value. Half-lives or relaxation times are eonstants over the complete reaction for first-order or pseudo first-order reactions. The loss of reactant A with time may be described by a single exponential but yet may hide two or more concurrent first-order and/or pseudo first-order reactions. 

This equation resembles (1.26) but includes [A], the concentration of A at equilibrium, which is not now equal to zero. The ratio of rate constants, Atj/A , = K, the so-called equilibrium constant, can be determined independently from equilibrium constant measurements. The value of k, or the relaxation time or half-life for (1.47), will all be independent of the direction from which the equilibrium is approached, that is, of whether one starts with pure A or X or even a nonequilibrium mixture of the two. A first-order reaction that hides concurrent first-order reactions (Sec. 1.4.2) can apply to reversible reactions also. 

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