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(B) The rate of decay $R$ is proportional to the number of radioactive nuclei $N$,i.e.,$R = \frac{dN}{dt} \propto N$.Given the initial rate $R_1 = 200

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$60$

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(B) The rate of decay $R$ is proportional to the number of radioactive nuclei $N$,i.e.,$R = \frac{dN}{dt} \propto N$.Given the initial rate $R_1 = 200$ particles/s and final rate $R_2 = 25$ particles/s after time $t = 3$ hours.The ratio of rates is given by $\frac{R_2}{R_1} = \frac{N_2}{N_1} = \left( \frac{1}{2} ight)^n$,where $n = \frac{t}{T_{1/2}}$ is the number of half-lives.Substituting the values: $\frac{25}{200} = \frac{1}{8} = \left( \frac{1}{2} ight)^3$.Comparing the exponents,we get $n = 3$.Since $n = \frac{t}{T_{1/2}}$,we have $3 = \frac{3 	ext{ hours}}{T_{1/2}}$.Therefore,$T_{1/2} = 1$ hour.Since $1$ hour = $60$ minutes,the half-life period is $60$ minutes.

$A$ radioactive element emits $200$ particles per second. After $3$ hours,$25$ particles per second are emitted. The half-life period of the element will be .......... $minutes$.

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