Reactant half-life

The formal definition for the half-life (or reactant half-life) of a reaction, symbolised t]/2, is that it is the time taken for the concentration of a reactant to fall by half... 

Reagents i, Ti -boiling dioxan, reactant half-life ca. 6 days ii, Ti°-boiling THF, 16 h... 

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. 

The functional dependence of the half-life on reactant concentration varies with the reactant order. From the integrated rate equations we obtain these results ... 

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... 

We already know that the higher the value of k, the more rapid the consumption of a reactant. Therefore, we should be able to deduce a relation for a first-order reaction that shows that, the greater the rate constant, the shorter the half-life. 

FIGURE 13.12 Thu ohange in concentration of the reactant in two first-order reactions plotted on the same graph When the first-order rate constant is large, the half-life of the reactant is short, because the exponential decay of the concentration of the reactant is then fast. 

EXAMPLE 13.6 Using a half-life to calculate the amount of reactant remaining... 

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. 

FIGURE 13.13 For first-order reactions, the half-life is the same whatever the concentration at the start of the chosen period. Therefore, it takes one half-life to fall to half the initial concentration, two half-lives to fall to one-tourth the initial concentration, three half-lives to fall to one-eighth, and so on. The boxes portray the composition ot the reaction mixture at the end of each half-life the red squares represent Inc reactant A and the yellow squares represent the product. 

A second-order reaction has a long tail of low concentration at long reaction times. The half-life of a second-order reaction is inversely proportional to the concentration of the reactant. 

Cl 13.39 Derive an expression for the half-life of the reactant A that decays by a third-order reaction with rate constant k. 

Half-life of a reactant in a first-order reaction ... 

For thermal reactions a variable temperature probe is necessary since optimum polarized spectra are usually obtained in reactions having a half-life for radical formation in the range 1-5 minutes. Reactant concentrations are usually in the range normally used in n.m.r. spectroscopy, although the enhancement of intensity in the polarized spectrum means that CIDNP can be detected at much lower concentrations. Accumulation of spectra from rapid repetitive scans can sometimes be valuable in detecting weak signals. 

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. 

Equation 6 would hold for a family of free radical initiators of similiar structure (for example, the frarw-symmetric bisalkyl diazenes) reacting at the same rate (at a half-life of one hour, for example) at different temperatures T. Slope M would measure the sensitivity for that particular family of reactants to changes in the pi-delocalization energies of the radicals being formed (transition state effect) at the particular constant rate of decomposition. Slope N would measure the sensitivity of that family to changes in the steric environment around the central carbon atom (reactant state effect) at the same constant rate of decomposition. 

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,... 

If it is certain that the reaction is indeed an irreversible first-order reaction, one can also determine how long it takes before 50 % of the reactant has been converted into products, as for any exponential decay the half-life, ty, is related to the rate constant k as... 

The reduction of Co(lll) by Fe(II) in perchloric acid solution proceeds at a rate which is just accessible to conventional spectrophotometric measurements. At 2 °C in 1 M acid with [Co(IlI)] = [Fe(II)] 5 x 10 M the half-life is of the order of 4 sec. Kinetic data were obtained by sampling the reactant solution for unreacted Fe(Il) at various times. To achieve this, aliquots of the reaction mixture were run into a quenching solution made up of ammoniacal 2,2 -bipyridine, and the absorbance of the Fe(bipy)3 complex measured at 522 m/i. Absorbancies of Fe(III) and Co(lll) hydroxides and Co(bipy)3 are negligible at this wavelength. With the reactant concentrations equal, plots of l/[Fe(Il)] versus time are accurately linear (over a sixty-fold range of concentrations), showing the reaction to be second order, viz. 

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 ... 

In the case of radio-active decay the rate is often expressed by the half-life, namely, die time required for half of the reactant to disappear. From Gq. (22) the half-life is given by t /2 = (hr 2)1 k. 

The concept of half-life also applies to chemical reactions. The half-life of a chemical reaction is the time it takes for the amount of one of the reactants to be reduced by half. In some reactions the reaction rate is determined by the concentration of one particular reactant as the reaction proceeds and the concentration of this reactant decreases, so does the rate of the reaction. This is the case for example, with amino acids, the components of proteins. Amino acids may occur in one of two different forms, the / and d forms (see Textbox 24). In living organisms, however, the amino acids occur only in the / form. After organisms die, the amino acids in the dead remains racemize and are gradually converted into the d form. Ultimately, the remaining amino acid, which is then known as a racemic mixture, consists of a mixture of 50% of the / form and 50% of the d form. 

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 (b) is correct. After each half-life—that is, after each 75 s—the amount of reactant remaining is half of the amount that was present at the beginning of that half-life. Statement (a) is incorrect the quantity of A remaining after 150 s is half of what was present after 75 s. Statement (c) is incorrect because different quantities of A are consumed in each 75 s of the reaction 1/2 of the original amount in the first 75 s, 1/4 of the original amount in the second 75 s, 1/8 of the original amount in the third 75 s, and so on. 

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. 

A zero-order reaction has a half life that varies proportionally to [A]0, therefore, increasing [A]0 increases the half-life for the reaction. A second-order reaction s half-life varies inversely proportional to [A]0, that is, as [A]0 increases, the half-life decreases. The reason for the difference is that a zero-order reaction has a constant rate of reaction (independent of [A]0). The larger the value of [A]0, the longer it will take to react. In a second-order reaction, the rate of reaction increases as the square of the [A]0, hence, for high [A]0, the rate of reaction is large and for very low [A]0, the rate of reaction is very slow. If we consider a bimolecular elementary reaction, we can easily see that a reaction will not take place unless two molecules of reactants collide. This is more likely when the [A]0 is large than when it is small. 

The half-life (t 1/2) of a reactant is the time required for its concentration to decrease to one-half its initial value. The rate of hydration of ethylene oxide (A) to ethylene glycol (C2H4O + H2O - C2H6O2) in dilute aqueous solution is proportional to the concentration of A with a proportionality constant kA = 4.11 X 10-5 s-1 at 20°C for a certain catalyst (HCIO4) concentration (constant). Determine the half-life (ti/2), or equivalent space-time (T1/2), in s, of the oxide (A) at 20°C, if the reaction is carried out... 

Fractional lifetime method. The half-life, tm, of a reactant is the time required for its concentration to decrease to one-half its initial value. Measurement of tll2 can be used to determine kinetics parameters, although, in general, any fractional life, r, can be... 

A second-order reaction may typically involve one reactant (A -> products, ( -rA) = kAc ) or two reactants ( pa A + vb B - products, ( rA) = kAcAcB). For one reactant, the integrated form for constant density, applicable to a BR or a PFR, is contained in equation 3.4-9, with n = 2. In contrast to a first-order reaction, the half-life of a reactant, f1/2 from equation 3.4-16, is proportional to cA (if there are two reactants, both ty2 and fractional conversion refer to the limiting reactant). For two reactants, the integrated form for constant density, applicable to a BR and a PFR, is given by equation 3.4-13 (see Example 3-5). In this case, the reaction stoichiometry must be taken into account in relating concentrations, or in switching rate or rate constant from one reactant to the other. 

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 ... 

For second-order reactions, the half-life does depend on the reactant concentration. We calculate it using the following formula ... 

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