Question

A radioactive isotope decays at such a rate that after $192$ minutes only $1/16$ of the original amount remains. Calculate $t_{\frac{1}{2}}$:

• A. $32min$
• B. $48min$
• C. $12min$
• D. $24min$

Answer 1

Answer: (B)

$48min$

For finding the final amount, we can use formula

$N_t=N_0\times {\frac{1}{2}}^{\frac{t}{t_{\frac{1}{2}}}}$

where,

$N_t=$ final amount of the isotope

$N_0=$ initial amount of the iostopes

$t=$ amount of time passed

and $t_{\frac{1}{2}}=$ half-life period

Here,

$t=192$ minutes

$\frac{N_t}{N_0}=\frac{1}{16}$

$\therefore \frac{1}{16}={\frac{1}{2}}^{\frac{192}{t_{\frac{1}{2}}}}$

$\therefore (\frac{1}{2}{)}^4={\frac{1}{2}}^{\frac{192}{t_{\frac{1}{2}}}}$

$\therefore 4=\frac{192}{t_{\frac{1}{2}}}$

$\therefore t_{\frac{1}{2}}=\frac{192}{4}=48min$