Question

State the law of radioactive decay. Hence derive the expression $N = {N_0}{e^{ - \lambda t}}$ where symbols have their usual meanings.

Answer 1

Hint: We can solve this problem with the concept of radioactive decay. It is exponential in nature which has phenomena of spontaneous disintegration of the nucleus of an atom with emission of one or more radiation. Alpha, beta and gamma radiations are emitted from the radioactive decay. Law of radioactive decay is given by Henry Becquerel. In this phenomena, the nucleus of an unstable atom loses energy and disintegrates into one or more new nuclei which is also unstable.

Complete answer:
The elements or isotopes which emit radiation and undergo the phenomena of radioactivity known as radioactive elements. These elements undergo three types of radioactive decay such as: Alpha decay, Beta decay and Gamma decay. Radioactive elements have unstable nuclei. This process is a random process at the level of single atoms. The SI unit of the radioactivity is becquerel (Bq). One Bq is defined as one disintegration per second.

The law of radioactive decay states that for a particular time, the rate of radioactive disintegration is directly proportional to the number of nuclei of the elements present at that time. This law can be expressed as follows: $\dfrac{{dN}}{{dt}} \propto N$ or $\dfrac{{dN}}{{dt}} = - \lambda N$ , where $\lambda $is the proportionality constant known as radioactive decay constant.

If ${N_0}$ is the number of nuclei presents at the initial time and $N$ is the number of nuclei presents at the time $t$ then,
$\int\limits_{{N_0}}^N {\dfrac{{dN}}{N} = - \int\limits_0^t {\lambda dt = - \lambda \int\limits_0^t {dt} } } $
${\log _e}N - {\log _e}{N_0} = - \lambda t$
${\log _e}\left( {\dfrac{N}{{{N_0}}}} \right) = - \lambda t$
$\dfrac{N}{{{N_0}}} = {e^{ - \lambda t}}$
$N = {N_0}{e^{ - \lambda t}}$, hence the expression for the radioactive decay is derived.

Note:
We know that radioactivity is the physical phenomena of the radioactive elements such as uranium. That phenomena have a number of different applications in the medicine and industry. It is also used in the smoke alarms.