Question

After 1 hr , 1/8 th of the initial mass of a certain radioactive isotopes remains undecayed. The half life of the isotopes is

Answer 1

Answer:

15 minutes

Explanation:

half time ∝ 1/mass.

∴ half time* mass = k(constant)

let x be 1st half time and 0.5*M be half time mass.

given that after 1 hour mass = M/8 or 0.125 M.

⇒ x*(0.5*M) = 1*(0.125*M)

    x = $\frac{0.125*M}{0.5*M}$

     x= 0.25 hours  

or

     x = 15 minutes

Answer 2

Answer:

The half life of the isotope is 15 minutes.  

Explanation:

Given the remaining undecayed radioactive isotope = 1/8th of initial mass.

Time taken to remain 1/8th of initial mass, $t = 1 \mbox{hour}=60\mbox{min}$

• Half-life time is defined as the time required for a radioactive isotope to reduce to half of its initial value.
• As the times increases the mass decreases. Thus, half time is inversely proportional to mass.

       $\therefore t_h\propto \dfrac{1}{m} \implies t_h\times m=\mbox{constant}$                            ...(1)

• Thus, for a particular isotope, the rate of change of mass  is a constant.

Let the initial mass of the isotope be $m$

Then after 60 min, the mass is

$\dfrac{1}{8}\times m= \dfrac{m}{8}$

Let the half time be $x$ and the respective half time mass $=\dfrac{1}{2} m$

Applying equation (1) for both cases,

$x\times \dfrac{1}{2} m= 60\times\dfrac{1}{8} m$

$\implies x= \dfrac{60\times2}{8} =15\mbox{min}$

Therefore, the half life of the isotope is 15 minutes.