Half-life of a reactant

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

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 half-life of a reactant and the mean reaction time of a reaction are two measures of the time to reach equilibrium. The half-life ty2 is the time for the reactant to decrease to half of its initial concentration, or more generally, the time for it to decrease to halfway between the initial and the final equilibrium concentration. The mean reaction time t is roughly the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. The rigorous definition of the mean reaction time t is through the following equation (Equation 1-60) ... 

FIGURE 13.11 The half-life of a reactant is short if the first-order rate constant is large, because the exponential decay of the concentration of the reactant is then faster. 

If you recall, back in Chapter 5 we discussed half-life in the context of the decay of radioactive nuclei. In that chapter, we defined the half-life as the amount of time it took for one half of the original sample of radioactive nuclei to decay. Because the rate of decay only depends on the amount of the radioactive sample, it is considered a first-order process. Using the same logic, we can apply the concept of half-life to first-order chemical reactions as well. In this new context, the half-life is the amount of time required for the concentration of a reactant to decrease by one-half. The half-life equation from Chapter 5 can be used to determine the half-life of a reactant ... 

This relates the half-life of a reactant in a first-order reaction and its rate constant, k. In such reactions, the half-life does not depend on the initial concentration of A. This is not true for reactions having overall orders other than first order. 

The time requiredfor a reactant to reach half its original concentration is called the half-life of a reactant and is designated by the symbol fi/2. For example, we can calculate the half-life of the decomposition reaction discussed in Example 12.2. The data plotted in Fig. 12.4 show that the half-life for this reaction is 100 seconds. We can see this by considering the following numbers ... 

Half-life (of a reactant) the time required for a reactant to reach half of its original concentration. (12.4)... 

The half-life of a reactant is the time it takes for its concentration to fall to one-half its original value. Although this quantity can be defined for any reaction, it is particularly meaningful for first-order reactions. To see why, let s remrn to the system we considered in Example Problem 11.5, the photodissociation of ozone by UV light in the upper atmosphere. 

The half-life (q/2) of a reaction is the time it takes for half the reactant to react (or for half the product to form). To derive the half-life of a reactant in a first-order reaction, we begin with the equation shown at the bottom of column 1. 

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). 

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