First-order reaction reactant half-life

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. 

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. 

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 ... 

FIGURE 12.7 Concentration of a reactant A as a function of time for a first-order reaction. The concentration falls from its initial value, [Alo, to [A]q/2 after one half-life, to [A]0/4 after a second half-life, to [A]0/8 after a third half-life, and so on. For a first-order reaction, each half-life represents an equal amount of time. 

Equation 3-39 shows that in the first order reactions, the half-life is independent of the concentration of the reactant. This basis can be used to test whether a reaction obeys first order kinetics by measuring half-lives of the reaction at various initial concentrations of the reactant. 

Unlike t for the first-order reactions, the half-life of the second-order reaction is dependent on the initial concentration of reactants. It is not possible to derive a simple expression for the half-life of a second-order reaction with unequal initial concentrations. 

Thus for first-order reactions, the half-life is constant and independent of the initial reactant concentration and can be used directly to calculate the rate constant k. For non-first-order reactions, Eq. (7-171) can be linearized as follows ... 

This is a first-order reaction. The half-life of benzoyl peroxide at 100°C is 19.8 min. (a) Calculate the rate constant (in min ) of the reaction, (b) If the half-life of benzoyl peroxide is 7.30 h, or 438 min, at 70°C, what is the activation energy (in kJ/mol) for the decomposition of benzoyl peroxide (c) Write the rate laws for the elementary steps in the above polymerization process, and identify the reactant, product, and intermediates, (d) What condition would favor the growth of long, high-molar-mass polyethylenes ... 

Determining the rate constant and order of a reaction is tedious and time-consuming. For many studies, this detail is unwarranted and the half-life is measured instead. The half-life is the time required for half of the original concentration of reactant to disappear. For the particular case of a first-order reaction, the half-life ( i /2) is directly related to the reaction rate constant k by... 

In contrast to a first-order reaction, the half-life depends upon the initial concentration of the reactant and is not characteristic for the reaction. 

The half-life of a reaction, ti/2> i the time required for the concentration of a reactant to drop to one-half of its original value. For a first-order reaction, the half-life depends only on the rate constant and not on the initial concentration ti/2 = 0.693/fc. The half-life of a second-order reaction depends on both the rate constant and the initial... 

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. 

The half-life is the time needed for the reactant concentration to reach half its initial value for first-order reactions, the half-life is constant—that is, it is independent of concentration. 

The main point to note about eqn 6.13 is that/or a first-order reaction, the half-life of a reactant is independent of its concentration. It follows that if the concentration of A at some arbitrary stage of the reaction is [A], then the concentration will fall to [A] after an interval of (In 2)/k whatever the actual value of [A] (Fig. 6.10). 

In a first-order reaction the half-life is independent of the concentration(s) of the reactant(s). 

Rate Laws Experimental measurement of the rate leads to the rate law for the reaction, which expresses the rate in terms of the rate constant and the concentrations of the reactants. The dependence of rate on concentrations gives the order of a reaction. A reaction can be described as zero order if the rate does not depend on the concentration of the reactant, or first order if it depends on the reactant raised to the first power. Higher orders and fractional orders are also known. An important characteristic of reaction rates is the time required for the concentration of a reactant to decrease to half of its initial concentration, called the half-life. For first-order reactions, the half-hfe is independent of the initial concentration. 

A second order reaction does not follow exponential decay and one cannot talk about time constants, except when first order conditions are imitated (see below for solution under condition (b)). We have to revert to the term half life, which differs by a factor of ln2 from the time constant of a first order reaction. For a first order decay the half life (note its relation to the time constant, see section 2.1), which is defined as the time taken for half completion of a reaction, is independent of the starting point. For a second order decay the half life is inversely proportional to the reactant concentration under the condition Ca(0) = Cb(0). 

We can see from the above equation that for a first-order process the half-life does not depend upon the initial concentration. This is seen with the data on the solvolysis of 2-chloro-2-methylpropane in water, where from Table 3.1, the first half-life is about 210 sec, giving a rate constant = 3.3X10 sec , while for the second half-life, corresponding to the time necessary for the concentration of the reactant to be reduced from [A]q/2 to [A](/4, the value is virtually identical. Similarly, the half-lives for aU steps of this reaction are the same within experimental error. This provides an excellent technique for confirming the reaction order, and, as a working definition, it is normally accepted that if the... 

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... 

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... 

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. 

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

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