Question

How can I calculate the half-life of an isotope?

Answer 1

Hint: First we need to know the exact meaning of half life and an isotope. Half-life which is usually represented as \[{t_{\dfrac{1}{2}}}\] is the time which is required for a quantity to reduce to half of its initial value. Isotopes are variants of a particular element which has different neutron numbers, and consequently differ in nucleon number. All isotopes of a given element has the same proton number but different numbers of neutrons in each atom.

Complete step-by-step answer:
To measure the half life we first have to find out the number of radioactive decays per second and also the number of radioactive atoms.
The number of radioactive decays per second and the number of radioactive atoms are proportional to each other and this constant of proportionality is denoted as $\lambda $.
As we know the number of radioactive decays per second reduces the number of radioactive atoms, this decrease is denoted as $ - \dfrac{{dN}}{{dt}}$.
From this we can conclude that
 $
   - \dfrac{{dN}}{{dt}} = \lambda {\text{ times }}N \\
   - \dfrac{{dN}}{{dt}} = \lambda {\text{ }}N \\
  $
After further simplifications we get the equation for \[{t_{\dfrac{1}{2}}}\]
\[{t_{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda } = \dfrac{{0.693}}{\lambda }\]
A simpler equation is
\[{t_{\dfrac{1}{2}}} = 0.693\dfrac{{number{\text{ of atoms}}}}{{number{\text{ of decays}}}}\]
Add the number of atoms can be calculated by
$6.022 \times {10^{23}} \times \dfrac{{Mass{\text{ of substance}}}}{{Atomic{\text{ number}}}}$

Note: During natural radioactive decay, not all the atoms of an element are changed to atoms of another element instantaneously. This decay process will take time and there is a value in being able to express the rate at which a process occurs. One type of nuclear reaction is called the radioactive decay, where an unstable isotope of an element changes spontaneously and can emit radiation. One important measure of the rate at which a radioactive substance decays is called half-life.