Derivation of the Law of Radioactive Decay:
The law of radioactive decay states that the rate of disintegration of radioactive nuclei in a sample is directly proportional to the number of undecayed nuclei present at that instant.
Let be the number of undecayed nuclei in a sample at time , and be the number of nuclei that decay in a small time interval . The rate of decay is .
According to the law:
where is the decay constant, a positive constant characteristic of the radioactive substance.
Rearranging the equation to separate the variables:
To find the number of nuclei remaining after a time , we integrate this equation. Let be the initial number of nuclei at time , and be the number of nuclei at time .
Taking the exponential of both sides:
This is the mathematical expression for the law of radioactive decay, which shows that the number of undecayed nuclei decreases exponentially with time.
Definition of Becquerel (Bq):
One becquerel is the unit of activity of a radioactive sample in the SI system. It is defined as one disintegration (or decay) per second.