# The half life of a radioactive substance is 20 minutes. The approximate time interval $\dpi{100} \left ( t_{2}-t_{1} \right )$ between the time $\dpi{100} t_{2}$ when $\dpi{100} \frac{2}{3}$ of it has decayed and time $\dpi{100} t_{1}$ when $\dpi{100} \frac{1}{3}$ of it had decayed is Option 1) $7 \: min$ Option 2) $14 \: min$ Option 3) $20 \: min$ Option 4) $28 \: min$

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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