Activity & Half-Life
When a nucleus under goes decay it looses energy or mass. In the case of mass loss, changing from one isotope to another (and quiet often of a different element). It is impossible to predict which nucleus will decay in a sample. Every nucleus has the same probability of decaying, that is, the process of decay is completely random.
However, if there are enough nuclei in our sample then a fixed proportion will decay. The fixed proportion is given by where is the probability of an individual nucleus decaying in a given time frame (usually per second) and is the number of original isotope nuclei and is known as the decay constant.
The decay rate can be measured and is called the activity, measured in Becquerel, Bq: where is the time taken, in seconds, to record the activity.
The activity is also given by as the activity will be equal to the number of nuclei decaying.
Therefore, the negative sign indicating that this is a decay, i.e., the number of nuclei decreases.
Taking the limit as then this becomes
The solution to this equation is attained through integration: where is the number of the original nuclei remaining and is the initial number of nuclei of the particular isotope under consideration.
This is an exponential decay and when plotted gives a curve. Obtaining useful information about the decay from this curve can be more difficult than from a straight line, but a straight line form of the equation can be created by an algebraic process called linearisation. This is of the form where , , and . So plotting a graph of versus , the gradient give and the -intercept gives .
e.g. if there are 10000 nuclei present and 300 decay in 20 seconds, the decay constant is (300/10000)/20 = 0.0015s^-1. Half-Life Equation At time t=0, N=N0. At time T = t½ , N = 0.5N0. Substitute this into N=N0e(-λt): Picture This leads to the general form of the equation for half life:
A radioactive isotope may emit energy as photons of a specific energy, . The power (energy transferred per second), , where is the number of decays per second. Half life, is the time taken for the number of the original isotopes to decrease to half the initial mass, or, time taken for the activity to half.
the valley of nuclear stability - Radioactivity; Nuclear Radius https://chem.libretexts.org/Courses/Sacramento_City_College/SCC%3A_Chem_400_-_General_Chemistry_I/Text/20%3A_Radioactivity_and_Nuclear_Chemistry/20.04%3A_The_Valley_of_Stability%3A_Predicting_the_Type_of_Radioactivity https://www.radioactivity.eu.com/site/pages/Stability_Valley.htm