Radioactive decay equations half-life

Suppose the initial number of nuclei of a radioactive nuclide is N0, and that the half-life is T. Then the amount of parent nuclei remaining at a time t can be written as Nx = NQ( /2)(tlT>. This relationship is called the radioactive decay equation. What is the number of daughter nuclei present at time t, expressed in terms of N0 and Nx ... 

The experimentally measured rates of decay of radioactive atoms show that the decay is first order, where the number of atoms decomposing in a unit of time is proportional to the number present—this can be expressed in the following equation dN/dt = —X N. Another term used for characterizing rate of decay is half-life (t 1/2), the time required for half of the initial number of atoms to decay. Isotopes are considered to be in secular equilibrium, when the rate of decay of the parent is equal to that of its daughter. 

The relationship between the decay constant X and the half-life tll2 can be derived from the general radioactive decay equation... 

The half life tin of a radioactive nuclide is defined as the lime required for one half the number of radioactive atoms in a sample to undergo decay. The half-life is thus the time required for N to decrease to NqI2. Substitution of Nq/2 for N in Equation 32-3 leads to... 

Irene Joliot-Curie (daughter of Pierre and Marie Curie) and her husband Frederic Joliot-Curie observed that when aluminum-27 is bombarded with alpha particles, neutrons and positrons (positive electrons) are emitted as part of the products. When the source of alpha particles is removed, neutrons cease to be produced, but positrons continue to be emitted. This observation suggested that the neutrons and positrons come from two separate reactions. It also indicated that a product of the first reaction is radioactive. After further investigation, they discovered that, when aluminum-27 is bombarded with alpha particles, phosphorus-30 and neutrons are produced. Phosphorus-30 is radioactive, has a half-life of 2.5 minutes, and decays to silicon-30 with the emission of a positron. The equations for these reactions are... 

As shown in Example, Equation is used to find a nuclear half-life from measurements of nuclear decays. Equation is used to find how much of a radioactive substance will remain after a certain time, or how long it will take for the amount of substance to fall by a given amount. Example provides an illustration of this t q)e of calculation. In Section 22-1. we show that Equation also provides a way to determine the age of a material that contains radioactive nuclides. 

The half-life of the process in Equation (8.32) is 5570 years. Following death, flora and fauna alike cease to breathe and eat, so the only 14C in a dead body will be the 14C it died with. And because the amounts of 14C decrease owing to radioactive decay, the amount of the 14C in a dead plant or person decreases whereas the amounts of the 12C and 13C isotopes do not. We see why the proportion of 14C decreases steadily as a function of time following the instant of death. 

Equation (8.33) suggests the half-life is independent of the amount of material initially present, so radioactive decay follows the mathematics of first-order kinetics. 

We start by inserting the known half-life h/2 into Equation (8.33) to obtain a rate constant of radioactive decay . 

The basic concepts of nuclear structure and isotopes are explained Appendix 2. This section derives the mathematical equation for the rate of radioactive decay of any unstable nucleus, in terms of its half life. 

Equation (9.6) is the basic equation describing the decay of all radioactive particles, and, when plotted out, gives the familiar exponential decay curve. The parameter X is characteristic of the parent nucleus, but is not the most readily visualized measure of the rate of radioactive decay. This is normally expressed as the half life (7/ 2). which is defined as the time taken for half the original amount of the radioactive parent to decay. Substituting N = Na/2 into the Equation (9.6) gives ... 

In these equations, In is the natural logarithm, Nt is the amount of isotope radioactive at some time t, N0 was the amount of isotope radioactive initially, and k is the rate constant for the decay. If you know initial and final amounts and if you are looking for the half-life, you would use equation (1) to solve for the rate constant and then use equation (2) to solve for t1/2. 

A radioactive isotope may be unstable, but it is impossible to predict when a certain atom will decay. However, if a statistically large enough sample is examined, some trends become obvious. The radioactive decay follows first-order kinetics (see Chapter 14 for a more in-depth discussion of first-order reactions and equations). If the number of radioactive atoms in a sample is monitored, it can be determined that it takes a certain amount of time for half the sample to decay it takes the same amount of time for half the remaining sample to decay and so on. The amount of time it takes for half the sample to decay is called the half-life of the isotope and is given the symbol r1/2. The table below shows the percentage of radioactive isotope remaining versus half-life. 

Know that the half-life, ty2, of a radioactive isotope is the amount of time it takes for one-half of the sample to decay. Know how to use the appropriate equations to calculate amounts of an isotope remaining at any given time, or use similar data to calculate the half-life of an isotope. 

Because a diffusion profile does not end abruptly (except for some special cases), it is necessary to quantify the meaning of diffusion distance. To do so, examine Equation 3-40a. Define the distance at which the concentration is halfway between Co and to be the mid-distance of diffusion, Xmid- The concept of Xmid is similar to that of half-life ti/2 for radioactive decay. From the definition, Xmid can be solved from the following ... 

Radioactive decay follows the same rate equation as first-order chemical kinetics (Section 2.5) the half-life t j2, the time required for one-half of the sample to decay, is given by (In 2)/(rate constant). Decay of a sample is considered arbitrarily to be practically complete after 10ti/2, which is 240,000 years for 239Pu. 

In the rubidium-strontium age dating method, radioactive 87Rb isotope with a natural isotope abundance of 27.85 % and a half-life of 4.8 x 1010 a is fundamental to the 3 decay to the isobar 87 Sr. The equation for the Rb-Sr method can be derived from Equation (8.9) ... 

Any radionuclide is characterised by its half-life r whose value is independent of the type of decay products that are created. Half-life is defined as the time required (from initial time t = 0) for the decomposition of half the atoms in the sample. The law of radioactive decay allows calculation of the number of atoms N left at time t in a population with N0 atoms initially. The integrated form of this law is given by the following equation ... 

Before going on to these alternative procedures, however, we should consider a special way by which true (not pseudo) first-order reactions are often considered. In these cases,/ = k. This consideration is especially applicable to radioactive decay processes. It is common practice to describe these true first-order reactions in terms of the time required for one-half of the material to decompose (this time is called the half-life, t ). In this special circumstance [A] = i[A]0 when t = t , and Equation 15-9 becomes... 

You know that radioactive decay is first-order, so it is necessary only to find the rate constant for decay, from which the half-life may be calculated by means of Equation 15-11. To make the first-order plot, first convert cpm to log cpm to get... 

A problem not mentioned in Chapter 15 is one that is very special for radioactive decay when the elapsed time given in the problem is insignificant in comparison with the half-life. Under such circumstances, Equation 26-2 is totally inappropriate, and the proper equation to use is Equation 26-1. In this case, consider —dN to be the number of atoms that disintegrate in a finite period of time df, which is negligible compared to q consider also that A remains constant during this same period of time. The following problem shows this application of Equation 26-1. 

Combining Equations 6.5 and 6.6 leads to a relationship that defines the half-life. Half-life, ty2, is a term used to describe the time necessary for one-half of the radioactive atoms initially present in a sample to decay. At the end of the first half-life, N in Equation 6.6 becomes A/(), and the result is shown in Equations 6.7 and 6.8. 

These equations say that if we know the value of either the decay constant k or the half-life t1/2r we can calculate the value of the other. Furthermore, if we know the value of fi/2, we can calculate the ratio of remaining and initial amounts of radioactive sample N/N0 at any time t by substituting the expression for k into the integrated rate law ... 

The half-life of a radioactive decay process is given by finding fj /2 in the equation... 

From day 6 onwards, the slope of the curve corresponds to the effective half-life of 131I on herbage, namely 5 d. Assuming that this continues indefinitely, the area under curve A in Fig. 3.6 is 1.4 m2 d l-1. This is equivalent to the transfer factor km, defined by equation (2.12). Values of Fm for 131I and 137Cs are about the same, but the radioactive decay of 131I reduces km compared with that for137Cs (Table 2.19). Also shown in Fig. 3.6 are values of C/ as deduced from measurements near... 

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 ... 

Tritium is hydrogen of mass number 3, having two neutrons and a proton in its nucleus. It is radioactive (half-life 12.4 years) in common with many isotopes having a large neutron-to-proton ratio, tritium decays with emission of an electron (called a beta ray). Such a decay can be represented by the nuclear equation (see also Chap. 27) ... 

The number of radioactive nuclei with a time function is smaller. The half-life is characteristic for each radionuclide and is defined as the period during which half of the number of radioactive nuclei decay. Every radionuclide decays according to equation (15.1) ... 

In that equation as applied to radioactive decay, JV stands for the number of atoms that have not yet decayed after a time (f), JV stands for the number of atoms you started with, e stands for a special number associated with natural logarithms that is approximately equal to 2.718, and X stands for a decay constant that is different for different atoms. From this, you can determine the half-life of any radioactive element. It s a fairly simple derivation, but since we re not going... 

Equation (2.3) indicates that the number of radioactive atoms present as well as the disintegration rate (activity) decrease exponentially with time. The time taken for half the radioactive atoms originally present to decay is called the half-life of the radionuclide. 

Whereas radioactive decay is never a reversible reaction, many first-order chemical reactions are reversible. In this case the characteristic life time is determined by the sum of the forward and reverse reaction rate constants (Table 9.5). The reason for this maybe understood by a simple thought experiment. Consider two reactions that have the same rate constant driving them to the right, but one is irreversible and one is reversible (e.g. k in first-order equation (a) of Table 9.5 and ki in first-order reversible equation (b) of the same table). The characteristic time to steady state tvill be shorter for the reversible reaction because the difference between the initial and final concentrations of the reactant has to be less if the reaction goes both ways. In the irreversible case all reactant will be consumed in the irreversible case the system tvill come to an equilibrium in which the reactant will be of some greater value. The difference in the characteristic life time between the two examples is determined by the magnitude of the reverse reaction rate constant, k. If k were zero the characteristic life times for the reversible and irreversible reactions would be the same. If k = k+ then the characteristic time for the reversible reaction is half that of the irreversible rate. 

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