The half life of a radioactive substance is 20 minutes. The approximate time interval \left ( t_{2}-t_{1} \right ) between the time t_{2} when \frac{2}{3} of it has decayed and time t_{1} when \frac{1}{3} of it had decayed is

  • Option 1)

    7 \: min

  • Option 2)

    14 \: min

  • Option 3)

    20 \: min

  • Option 4)

    28 \: min

 

Answers (1)

As we learnt in

Number of nuclei after disintegration -

N=N_{0}e^{-\lambda t} or A=A_{0}e^{-\lambda t}

- wherein

Number of nucleor activity at a time is exponentional function

 

 N=N_{0}e^{-\lambda t}

at t = t2,  N=\frac{N_{0}}{3}=N_{0}e^{-\lambda t_{2}}

or  e^{-\lambda t_{2}}=\frac{1}{3}                                            (1)

at  t = t1,  N=\frac{2}{3}N_{0}=N_{0}e^{-\lambda t_{1}}

or  e^{-\lambda t_{1}}=\frac{2}{3}                                            (2)

Divide (1) in (2)

\frac{e^{-\lambda t_{1}}}{e^{-\lambda t_{2}}}=2    or   {e^{+\lambda\left(t_{2}-t{_1} \right )}}=2

or   \lambda \left(t_{2}-t_{1})=ln2

or t_{2}-t_{1}=\frac{ln^{2}}{\lambda}=t_{1/2}=20\ minutes

Correct option is 3.


Option 1)

7 \: min

This is an incorrect option.

Option 2)

14 \: min

This is an incorrect option.

Option 3)

20 \: min

This is the correct option.

Option 4)

28 \: min

This is an incorrect option.

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