Question

A radioactive substance decays from $100$ grams to $6.25$ grams in $100$ days. What is the half-life of this radioactive substance?

• A. $25days$
• B. $6.25days$
• C. $12.5days$
• D. $100days$
• E. $50days$

Answer 1

The correct option is A $25days$

The law of radioactive decay predicts how the number of the not decayed nuclei of a given radioactive substance decreases in the course of time.

Relation is $N=N_0{\left(1/2\right)}^{\frac{t}{\tau }}$

$N$ is undecayed part

$N_0$ is initial part

$\tau$ is half-life and t is time taken.

$N_0=100$

$N=6.25$

$N=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}$

$6.25=100{\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow \frac{6.25}{100}={\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow \frac{1}{16}={\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow {\left(\frac{1}{2}\right)}^4={\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow 4=\frac{t}{\tau }=\frac{100}{\tau }\Rightarrow \tau =100/4=25$

Half-life is $25$ days.