Question

A sample of radioactive substance shows an intensity of 2.3 milli – curie at a time t and an intensity of 1.62 milli – curie after 600 second. What is the half-life period of radioactive material?

Answer 1

Hint The amount of time required for a radioactive element to reduce to half of its original amount through radioactive decay is called half-life and it is going to be denoted with a symbol $ {{t}_{1/2}} $ . every radioactive will have specific half-life.

Complete step by step answer:
- In the question it is given that a radioactive element shows an intensity of 2.3 milli-curie at a time ‘t’ and an intensity of 1.62 milli-curie after 600 sec. it is asked to calculate the half-life of that particular radioactive element.
- There is a formula to calculate the half-life of a radioactive element and it is as follows.
\[\lambda =\dfrac{2.303}{t}\log \left( \dfrac{{{N}_{0}}}{N} \right)\]
Here $ \lambda =\dfrac{0.693}{{{t}_{1/2}}} $ , $ {{t}_{1/2}} $ = half-life of radioactive element
t = intensity of radioactive element after some time = 600 sec
 $ ~{{N}_{0}} $ = intensity of radiation at initial time = 2.3 milli-curie
N = intensity of radiation after some time = 1.62 milli-curie
- Substitute all the known values in the above formula to get the formula to get half-life of the radioactive element and it is as follows.
\[\begin{align}
& \lambda =\dfrac{2.303}{t}\log \left( \dfrac{{{N}_{0}}}{N} \right) \\
& \dfrac{0.693}{{{t}_{1/2}}}=\dfrac{2.303}{t}\log \left( \dfrac{{{N}_{0}}}{N} \right) \\
& \dfrac{0.693}{{{t}_{1/2}}}=\dfrac{2.303}{600}\log \left( \dfrac{2.3}{1.62} \right) \\
& {{t}_{1/2}}=1187\sec \\
\end{align}\]
- Therefore the half-life of the radioactive element is 1187 seconds.

Note: Radioactive elements are not stable in nature. To get stability radioactive elements decays and releases various particles in the form of rays or elements. The elements which are released from radioactive elements are called daughter elements.