Question

After 1 Hour, ${\displaystyle \frac{1}{8}}$ of the initial mass of a certain radioactive isotope remains undecayed. What is the half-life of the isotope?

Answer 1

Hint: To describe the rate at which the isotope will decay and give off radiation, the half-life of an isotope is used. Using the half-life, the amount of radioactive material that will remain after a specified amount of time can be predicted. Calculate Half time by using the relation of mass.

Formula used:
$$HalfTime\propto {\displaystyle \frac{1}{mass}}$$

Complete Step-by-Step solution:
Half the original nuclei will disintegrate during the half-life of that substance. Although changing factors such as temperature, concentration, etc have accelerated or slowed chemical changes, these factors do not have an effect on half-life. Each radioactive isotope, independent of any of these factors, will have its own unique half-life.
$$HalfTime\propto {\displaystyle \frac{1}{mass}}$$
$$HalfTime\times mass=k$$

Let x be 1st half time and $$0.5\times m$$ be half time mass.
Given that after 1-hour mass = $${\displaystyle \frac{M}{8}}$$ or $$0.125M$$.
$$\Rightarrow x\times (0.5M)=1\times (0.125M)$$
$$x={\displaystyle \frac{0.125\times M}{0.5\times M}}$$
$$\therefore x=0.25hr$$
Or
$$\therefore x=15min$$

The half-life of the isotope is $$15min$$.

Note:
It is important to know about half-lives because it allows you to determine when it is safe to handle a sample of radioactive material. ... They need to be active long enough to treat the condition, but they should also have a half-life that is short enough so that healthy cells and organs are not injured. The half-lives of many radioactive isotopes have been determined and have been found to range from 10 billion years of extremely long half-lives to fractions of a second of extremely short half-lives. For selected elements, the table below illustrates half-lives.