Reaction order, definition half-lives

Since all of the above-mentioned interconversion reactions are reversible, any kinetic analysis is difficult. In particular, this holds for the reaction Sg - Sy since the backward reaction Sy -+ Sg is much faster and, therefore, cannot be neglected even in the early stages of the forward reaction. The observation that the equilibrium is reached by first order kinetics (the half-life is independent of the initial Sg concentration) does not necessarily indicate that the single steps Sg Sy and Sg Sg are first order reactions. In fact, no definite conclusions about the reaction order of these elementary steps are possible at the present time. The reaction order of 1.5 of the Sy decomposition supports this view. Furthermore, the measured overall activation energy of 95 kJ/mol, obtained with the assumption of first order kinetics, must be a function of the true activation energies of the forward and backward reactions. The value found should therefore be interpreted with caution. 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

Fractional Change Method From the equations of half life tor reactions of various orders except first order reaction, time required to complete a definite fraction of the reaction is inversely proportional to af- where n is the order of reaction and a is initial concentration. 

Equation 11.2-25 is the formal definition of half-life, the time the second-order reaction takes to reduce the initial concentration to half the initial cq. Following the methodology of... 

When one half-life of a second-order reaction has elapsed (t = ty2), by definition, [A] = [A]o/2. Equation (15.5) then becomes... 

The expression for the half-life of a zero-order reaction can be obtained from the integrated rate law. By definition, [A] = [A]0/2 when t = tyj, so... 

Understand the definition of the term reaction half-life and use the method of successive half-lives as a preliminary check to determine whether a reaction is first-order overall or not. (Exercises 5.1, 5.2)... 

When one half-life of the second-order reaction has elapsed t = by definition,... 

In order to obtain flows of charged particles of a definite energy, accelerators (for instance, a cyclotron) are used. The irradiation with neutrons is carried out in reactors, where powerful flows of neutrons are formed due to the reaction ( ,/). At laboratory scale, small sources of neutrons are used, e.g., Ra-Be (reaction Be(a,n) C, 10 neutrons/s, but under a considerable y- background ) or Po-Be (not only less danger of y-radiation, but also less neutron yield and low half-life of Pb [140 days]). Neutron energies for these sources are in the range of 1-8 MeV. To get thermal neutrons, these sources are put inside a moderator, for instance, water or paraffin hydrocarbon. 

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... 

This analysis can, for example, be applied to multistep radioactive decay reactions and to isomerization reactions. In such multistep processes, every step is by definition a first-order process. An example of multistep radioactive decay is the Actinium series (see Lederer et ah, 1968), in which Bi alpha-decays to ° T1, which beta-decays to ° Pb with respective half-lives of 2.14 and 4.77 min. Therefore, in this two-step consecutive process, k J ki =/9 = 2.14/4.77 = 0.449, very close to the Acme point. Similarly, in the Radium series, Pb beta-decays to which beta-decays to Po, which then alpha-decays very rapidly (with a half-life of only 0.16 ms) to ° Pb. This multistep decay can be closely approximated by two steps, the first with a half-life of 27 min, the second with a half-life... 

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... 

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