The Half-Life of a Reaction

The half-life ty of a reaction is the time required for the concentration of a reactant to fall to one-half of its initial value. For example, if a reaction has a half-life of 100 seconds, and if the initial concentration of the reactant is 1.0 M, the concentration will fall to 0.50 M in 100 s. The half-life expression— which defines the dependence of half-life on the rate constant and the initial concentration—is different for different reaction orders. 

First-Order Reaction Half-Life From the deiinilion of half-life, and from the integrated rate law, we can derive an expression for the half-life. For a first-order reaction, the integrated rate law is  

At a time equal to the half-life (t = ti/2), the concentration is exactly half of the initial concentration ([A], = j[A]q). Therefore, when t = ti/2 we can write the following expression  

Solving for q/2, and substituting -0.693 for In j, we arrive at the expression for the half-life of a first-order reaction  

For a first-order reaction, the half-life is constant and independent of concentration. 

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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 ... 

Models may also be tested by utilizing the time required for a given fraction of a reactant to disappear, since this varies with the initial concentration in a fashion characteristic of the reaction order. For example, if the half-life of a reaction is defined as the time required for one-half of the initial amount of reactant to be consumed, then Eq. (4) may be written... 

For example, the half-life of a very fast reaction may he measured in microseconds. The half-life of a slow reaction may he measured in days. Knowing the half-life of a reaction is an easy way to tell how fast or how slow a reaction is. 

Briefly state why it is useful to know the half-life of a reaction. 

It is the half-life of a reaction that will govern the choice of the initiation method (Fig. 3.1) and it is the character of the reaction that will dictate the monitoring procedure (Sec. 3.8 on). 

The half-life of a reaction with a kinetic order higher than one is lengthened as the concentrations of the reactants are decreased (Sec. 1.3). Provided that there is still a sufficient change of concentration during the reaction to be accurately monitored, quite large rate constants may be measured if low concentration of reactants are used, even without recourse to the specialized techniques described in the previous section. 

An alternative method to determine the reaction order is the half-life method. The half life of a reaction (t /2) is the time it takes for 50% of the reactant(s) to be consumed. At time t /2 the concentration of A must then be [A]o/2. For a first-order reaction, Eq. 13.15 yields... 

One particular point of interest is the expression for the half-life of a reaction f,/2 this is the time required for one half of the reactant in question to disappear. A first order reaction is unique in that the half-life is independent of the initial concentration of the reactant. This characteristic is sometimes used as a test of whether a... 

The half-life of a reaction, symbolized by is the time required for the reactant concentration to drop to one-half of its initial value. Consider the first-order reaction... 

Integrated rate expressions can also be used to demonstrate that the half-life of a reaction varies systematically with [reactant], and so is diagnostic of the reaction order. 

As previously discussed, the half-life of a reaction is defined as the time it takes for the concentration of the reactant to fall to half of its initial value. Determining the half-life of a reaction as a function of the initial concentration makes it possible to calculate the reaction order and its specific reaction rate. 

The half-life of a reaction is the time required to consume half the reactant. 

The time scale in Table 2.3 is strictly the half-life of a reaction, which can be obtained by graphical analysis of rate laws cast into the mathematical form ... 

PROBLEM 6.4.2. The half-life of a reaction, t-i/2, is the time required for half the concentration of the relevant component to have disappeared. Obtain a relationship between x, and /C. ... 

The time required for a reactant to reach half of its original concentration is called the half-life of a reaction and is designated by ti/2- To illustrate this... 

It was suggested earlier that if the half life of a reaction was to be less than one millisecond, the product of the Boltzmann factor and the partial pressure of the species involved had to be greater than 10 . The consideration of time scales indicated that a reaction could only be considered to be balanced, that is, to depart negligibly from the equilibrium distribution of concentrations, if the half life was less than... 

The time required for a reactant to reach half of its original concentration is called the half-life of a reaction and is designated by t-yx- To illustrate this idea, we can calculate the half-life of the decomposition reaction discussed in Example 15.2. The data plotted in Fig. 15.4 show that the half-life for this reaction is 100 seconds. We can see this by considering the following numbers ... 

Analyze We are asked to estimate the half-life of a reaction from a graph of concentration versus time and then to use the half-life to calculate the rate constant 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... 

Equation (13.6) tells us that the half-life of a first-order reaction is independent of the initial concentration of the reactant. Thus, it takes the same time for the concentration of the reactant to decrease from 1.0 Af to 0.50 M, say, as it does for a decrease in concentration from 0.10 M to 0.050 M (Figure 13.12). Measuring the half-life of a reaction is one way to determine the rate constant of a first-order reaction. 

The half-life of a reaction is the amount of time required for half of the starting material to react. This is a useful device for determining how long it takes for a reaction to proceed to completion. Assume that it takes 100 minutes for 50% of a starting material to be converted to product. If the initial concentration of a reactant X is 1.0 M, then the concentration of X after 100 minutes will be 0.5 M. This is the concentration of starting material that remains after one half-life. After another 100 minutes, 50% of the remaining X will react, and the concentration of X is 0.25 M (after the second half-life). Another 100 minutes are required to bring the concentration to 0.125 M, and after a total reaction time of 500 minutes (five half-lives) the concentration of X is 0.0313 M. In other words, 0.9787 mol of X have reacted (97.87%) and, for a reaction to be complete, it must be allowed to proceed for more than five half-lives. 

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