What is the half-life period of a radioactive material if its activity drops to 1 / 16 text th of its initial value of 30 years ?

Question

What is the half-life period of a radioactive material if its activity drops to 1 / 16 text th of its initial value of 30 years ? (A) 9.5 years (B) 8.5 years (C) 7.5 years (D) 10.5 years

Tardigrade Admin · Accepted Answer

A = A 0 e -λ t ⇒-λ t = ln (( A / A 0)) ⇒-( ln 2/ t 1 / 2) × 30= ln ((1/16)) ⇒-( ln 2/ t 1 / 2) × 30=-4 ln 2 ⇒ t 1 / 2=(30/4)=7.5 yrs

Q. What is the half-life period of a radioactive material if its activity drops to $1/1 6^{th}$ of its initial value of $30$ years ?

$9.5$ years

$8.5$ years

$7.5$ years

$10.5$ years

$A = A_{0} e^{- λ t}$
$\Rightarrow - λ t = ln (\frac{A}{A _{0}})$
$\Rightarrow - \frac{l n 2}{t _{1/2}} \times 30 = ln (\frac{1}{16})$
$\Rightarrow - \frac{l n 2}{t _{1/2}} \times 30 = - 4 ln 2$
$\Rightarrow t_{1/2} = \frac{30}{4} = 7.5 yrs$

Q. What is the half-life period of a radioactive material if its activity drops to 1/16th of its initial value of 30 years ?

9.5 years

8.5 years

7.5 years

10.5 years