The half-life for a first order reaction

The half-life is independent of the concentration, and takes the same value no matter what the extent of reaction. 

Question. Show that the decomposition of N20 follows first order kinetics, and find the rate constant and half-life. Why cannot the methods of Section 3.7 be used here  

Answer. We need to find loge[N20] to plot logL.[N20]f versus t. 

This value agrees with the value read directly from the experimental graph. 

The data is concentration/time data and is immediately adaptable to integrated rate methods. To use the methods of Section 3.7 the data has to be rate/ concentration data, which is not the manner in which it is presented in this problem. 

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Half-Life Method For a zero-order reaction the half-life (tll2) is proportional to the initial concentration. The half-life for a first-order reaction is independent of the initial concentration while a second-order reaction is proportional to 1/initial concentration. 

The constancy of the half-life for a first-order reaction is illustrated in Figure 12.7. Each successive half-life is an equal period of time in which the reactant concentration decreases by a factor of 2. We ll see in Chapter 22 that half-lives are widely used in describing radioactive decay rates. 

The half-life for a first-order reaction, therefore, is independent of the initial concentration of the chemical of interest. 

The half-life for second-order and other reactions depends on reactant concentrations and therefore changes as the reaction progresses. We obtained Equation 14.15 for the half-life for a first-order reaction by substituting [ A]o for [A], and fi/2 for... 

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 ... 

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. 

The above equation implies that the half-life of a first-order reaction is independent of concentration. This result in only true for a first-order reaction. 

As an example, consider the effect of temperature on a first-order rate constant for a process with an activation energy, Ea, that is considered to be independent of temperature. Table 16.1 summarizes the calculated effect of temperature on the half-life of a first-order reaction, based on Equation 16.9 and the assumption that the preexponential factor, A, is independent of temperature. 

The significance of the half-life of a first-order reaction becomes clear once we look at Fig. 13.12. We see that the concentration falls to half its initial value in the time tV2. The concentration falls to half its new value in the same time, and then to half that new value in another half-life. For each time interval equal to the half-life, the concentration falls to half its value at the start of the interval. 

The half-life of a first-order reaction is a constant because it depends only on the rate constant and not on the reactant concentration. This point is worth noting because reactions that are not first order have half-lives that do depend on concentration that is, the amount of time in one half-life changes as the reactant concentration changes for a non-first-order reaction. 

As shown in Equation (5.14a) and Equation (5.14b), the half-life and the shelf-life are constant and independent of the drug concentration, [A]0. For example, if the half-life of a first-order reaction is 124 days, it takes 124 days for a drug to decompose to 0.5 [A]0. Also it takes another 124 days for 50% of the remaining 50% of the drug to decompose. In Equation (5.13), the time required for 100% degradation cannot be calculated because In ([A] / [A]o) is an indefinite number. 

The period of half-life 6/2, or the time taken for half of the material to react, is the easiest way of visualizing a velocity constant. The half-life in a first order reaction is related to the velocity constant by the following relation which is obvious from equation (4),... 

The half-life of a second-order reaction is also significantly different from the half-life of a first-order reaction. As Equation 16.8 shows, the half-life of a second order reaction depends on the initial concentration of the reactant (for first-order reactions it does not) ... 

A general formula for the half-life of a first-order reaction can be derived from the integrated rate law for the general reaction,... 

An integrated rate law includes concentration and time as variables. In addition to reaction order, it allows determination of half-life, the time required for half of a reactant to be used up. The half-life of a first-order reaction does not depend on reactant concentration. 

At fixed conditions, the half-life of a first-order reaction is a constant, independent of reactant concentration. For example, the half-life for the first-order decomposition of N2O5 at 45 C is 24.0 min. The meaning of this value is that if we start with, say, 0.0600 mol/L of N2O5 at 45 C, after 24 min (one half-life). 

This is the general equation for the half-life of a first-order reaction. Equation (12.3) can be used to caicuiate t n if k is known or k if fi/2 is known. Note that for a first-order re action, the half-life does not depend on concentration. 

Note that the concentration of A molecules is halved at / = 20 s and is halved again at t = 40 s. We notice that the half-life is independent of the concentration of the reactant. A, and hence the reaction is first-order in A. The rate constant, k, can now be calculated using the equation for the half-life of a first-order reaction. 

We can obtain a mathematical expression for the half-life of a first-order reaction by substituting in the integrated rate law (Equation 11.5). By definition, when the reaction has been proceeding for one half-life (ti/2), the concentration of the reactant must be [X] = j[X]q. Thus we have... 

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