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).In such cases, it is possible that the half-life of the parent nuclei is longer or shorter than the half-life of the daughter nuclei.Depending upon the substance, it is possible that both parent and daughter nuclei have similar half lives.
For a given element, the decay or disintegration rate is proportional to the number of atoms and the activity measured in terms of atoms per unit time.
In other words, the reaction rate does not depend upon the temperature, pressure, and other physical determinants.
However, like a typical rate law equation, radioactive decay rate can be integrated to link the concentration of a reactant with time.
\[ N_t=N_o\left( \dfrac \right)^ \tag \] \[ N_t=N_o e^ \tag \] By comparing Equations 1, 3 and 4, one will get following expressions \[ \ln = \ln(e^) = \ln (e^ ) \tag\] or with \(\ln(e) = 1\), then \[ \dfrac \ln \left( \frac \right) = \dfrac = -\lambda t \tag\] By canceling \(t\) on both sides, one will get following equation (for half-life) \[t_= \dfrac \approx \dfrac \tag \] or combining equations 1B and 11 \[ A = \dfrac N \tag\] Equation 11 is a constant, meaning the half-life of radioactive decay is constant.
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.