Question

Starting with a sample of pure ${}^{66}Cu,\frac{7}{8}$ of it decays into $Zn$ in $15$ minutes. The corresponding half-life is:

• A. $15$ minutes
• B. $10$ minutes
• C. $7\frac{1}{2}$ minutes
• D. $5$ minutes

Answer 1

The correct option is D $5$ minutes

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.

$7/8$ is decayed.so Remaining part is $1/8$

$N=1/8$

$N_0=1$

$N=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\frac{1}{8}=1\times {\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow {\left(\frac{1}{2}\right)}^3={\left(\frac{1}{2}\right)}^{\frac{t}{\tau }}\Rightarrow 3=\frac{t}{\tau }\Rightarrow 3=\frac{15}{\tau }\Rightarrow \tau =5$

Therefore, Half-life is 5 min.