The Method of Half-Lives

In Section 2.2, the equation giving the half-hfe as a function of reaction order and initial concentration was derived. That equation can be written in the general form 

Because both k and n are constants, the last term will be a constant if the experimental conditions are not changed except for [AJ. Therefore, if several reactions are carried out with initial concentrations of A being [A] , [A]o/2, [A]o/4, etc., and the half-life for the reaction is determined in each case, a plot of In [A]q versus 1/2 will yield a straight hne that has a slope of — ( — 1), which allows n, the order of the reaction, to be determined. 

If the reaction is carried out using two different [AJ values, a ratio of two equations having the form of Eq. (3.1) gives 

Taking the logarithm of both sides of this equation, we obtain 

Factoring out ( — 1) on the right-hand side of this equation gives 

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The method of half-lives offers a simple, direct approach to the estimation of reaction order and one which involves the least amount of computation when only one species is involved in the rate equation. It becomes, however, fairly complicated when applied to systems in which there is more than one reactant (see Table II.2). 

The method of The half-life of a reaction, tj/j, is defined as the time it takes for the concen-tration of the reactant to fall to half of its initial value. By determining the half-life of a reaction as a function of the initial concentration, the reaction order and specific reaction rate can be determined. If two reactants are involved in the chemical reaction, the experimenter will use the method of excess in conjunction with the method of half-lives to arrange the rate law in the form... 

For the method of half-lives, taking the natural log of both sides of Equation (5-18),... 

The method of half-lives requires many experiments... 

P5-9. Straight forward problem involving the method of half lives. 

Decay of the nuclide itself. The conceptually simplest approach is to take a known quantity of the nuclide of interest, P, and repeatedly measure it over a sufficiently long period. The observed decrease in activity with time provides the half-life to an acceptable precision and it was this technique that was originally used to establish the concept of half-lives (Rutherford 1900). Most early attempts to assess half lives, such as that for " Th depicted on the front cover of this volume, followed this method (Rutherford and Soddy 1903). This approach may use measurement of either the activity of P, or the number of atoms of P, although the former is more commonly used. Care must be taken that the nuclide is sufficiently pure so that, for instance, no parent of P is admixed allowing continued production of P during the experiment. The technique is obviously limited to those nuclides with sufficiently short half-lives that decay can readily be measured in a realistic timeframe. In practice, the longest-lived isotopes which can be assessed in this way have half-lives of a few decades (e.g., °Pb Merritt et al. 1957). 

Different types of methods that are used in the study of the kinetics of reactions with an indication of the range of half lives to which each method applies mathced ... 

This method was suggested by Ostwald. The determination of half lives of a reaction at two different initial concentration may lead to the determination of n. 

B. Method of Half-lives. The time required to consume a given fraction, say one-half, of one of the starting materials will depend on the initial concentration of the reactants in a way which is fixed by the order of the reaction. If a number of experiments have been performed under conditions of very different initial concentrations of reactants, then it is possible from a comparison of the half-lives (or other fraction, whichever is more conveniently obtained from the data) in the different experiments bo decide the proper order of the reaction. 

The activation method requires the use of high-puiity targets, in order to exclude the influence of other nuclear reactions. Chemical procedures may be applied to separate the reaction products and to identify their atomic number Z. If short-lived radionuclides are to be measured, fast separation methods are required, for instance on-line separation in a gas stream that passes a temperature gradient (thermochromatography). In the case of half-lives of the order of milliseconds or less, however, only physical methods are applicable, in particular separation by a sequence of electric and magnetic fields. Stable or long-lived products may be determined by use of mass spectrometry, provided sufficient masses are available. 

The form of this equation immediately suggests that the approach to measure cross sections is similar to the one used for the determination of half-lives. However, cross section measurements are complicated by the fact that (jc is a function of the neutron energy and by the difficulty in obtaining a reliable value of the neutron flux. The product isotope of an (n, y) reaction can be either radioactive or stable. In the former case, either counting methods or mass spectrometric techniques can be utilized in order to obtain information concerning the capture cross section. When the product isotope is stable, only mass spectrometric methods are applicable. 

Heath and coworkers also developed a fast method (9, 10) for determining half-lives. Heath and Tumlinson (9) tested the correlation of the logarithm of gas chromatographic retention times on a liquid crystal liquid phase versus the logarithm of half-lives... 

A combined stopped-flow, continuous-flow method is described which allows the measurement of half-lives of first and second order reaction between s and lO s if the continuous flow... 

Measurement of specific activity. The half-life of a nuclide can be readily calculated if both the number of atoms and their rate of decay can be measured, i.e., if the activity A and the number of atoms of P can be measured, then X is known from A = XP. As instrumentation for both atom counting and decay counting has improved in recent decades, this approach has become the dominant method of assessing half-lives. Potential problems with this technique include the accurate and precise calibration of decay-counter efficiency and ensuring sufficient purity of the nuclide of interest. This technique provides the presently used half-lives for many nuclides, including those for the parents of the three decay chains, U, U (Jaffey et al. 1971), and Th. 

Conventional radiochemical methods for the determination of long-lived radionuclides at low concentration levels require a careful chemical separation of the analyte, e.g., by liquid-liquid, solid phase extraction or ion chromatography. The chemical separation of the interferents from the long-lived radionuclide at the ultratrace level and its enrichment in order to achieve low detection limits is often very time consuming. Inorganic mass spectrometry is especially advantageous in comparison to radioanalytical techniques for the characterization of radionuclides with long half-lives (> 104 a) at the ultratrace level and very low radioactive environmental or waste samples. 

The dating methods discussed up to now have been based on the use of long-lived radionuclides that are present in nature. Dating is also possible using extinct radionuclides, that is, nuclei whose half-lives are so short that if they existed at the time of formation of our solar system, they would have decayed away essentially completely by now. The nuclides 129I t /2 = 1.57 x 107 y) and 244Pu t /2 = 8.08 x 107 y) are noteworthy examples of this type of nuclide. 

Figure 11.5 Spontaneous fission half-lives, corrected according to the method of Swiatecki, vs. flssionability parameter x. (From R. Vandenbosch and J. R. Huizenga, Nuclear Fission. Copyright 1973 Academic Press. Reprinted by permission of Elsevier.)...

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