Natural Radioactivity and the Stable Isotopes of Lead

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. 

The rate of decay of an unstable parent nucleus at any time t is proportional to the number (N) of atoms left (Faure, 1986 38). In other words, the rate at which the number of radioactive nuclei decline is proportional to the number left at that time. Expressed mathematically, this becomes  

The symbol X is introduced as the constant of proportionality, which is termed the decay constant of the parent nucleus, and is characteristic of that nucleus (with units of inverse time)  

The constant of integration C can be defined by setting the starting condition so that N=N0 at time t = 0, giving C= — In N0  

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  

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