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Radioactive decay is not always a one step phenomenon.Often times the parent nuclei changes into a radioactive daughter nuclei which also decays.The rate of decay remains constant throughout the decay process.There are three ways to show the exponential nature of half-life.If "A" represents the disintegration rate and "N" is number of radioactive atoms, then the direct relationship between them can be shown as below: \[ A \propto N \tag\] or mathematically speaking \[ A= \lambda N \tag \] where Since the decay rate is dependent upon the number of radioactive atoms, in terms of chemical kinetics, one can say that radioactive decay is a first order reaction process.Even though radioactive decay is a first order reaction, where the rate of the reaction depends upon the concentration of one reactant (r = k [A][B] = k [A}) , it is not affected by factors that alter a typical chemical reactions.The unstable nucleus then releases radiation in order to gain stability.For example, the stable element Beryllium usually contains 4 protons and 5 neutrons in its nucleus (this is not considered a very large difference).
Half-life and the radioactive decay rate constant λ are inversely proportional which means the shorter the half-life, the larger \(\lambda\) and the faster the decay. If the half-life were shorter, then the exponential decay graph would be steeper and the line would be decreasing at a faster rate; therefore, the amount of the radioactive nuclei would decrease as well.Also, radioactive decay is an exponential decay function which means the larger the quantity of atoms, the more rapidly the element will decay.Mathematically speaking, the relationship between quantity and time for radioactive decay can be expressed in following way: \[\dfrac = - \lambda N \tag\] or more specifically \[\dfrac = - \lambda N \tag\] or via rearranging the separable differential equation \[\dfrac = - \lambda dt \tag\] by Integrating the equation \[\ln N(t) = - \lambda t C \tag\] with There are two ways to characterize the decay constant: mean-life and half-life. As indicated by the name, mean-life is the average of an element's lifetime and can be shown in terms of following expression \[ N_t=N_o e^ \tag \] \[1 = \int^_ 0 c \cdot N_0 e^ dt = c \cdot \dfrac \tag\] Rearranging the equation: \[ c= \dfrac\] Half-life is the time period that is characterized by the time it takes for half of the substance to decay (both radioactive and non-radioactive elements).Carbon 14 (C-14) is produced in the upper atmosphere through the collision of cosmic rays with atmospheric 14N.
This radioactive carbon is incorporated in plants and respiration and eventually with animals that feed upon plants.Each type of decay emits a specific particle which changes the type of product produced.