JEE Main 2024 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

Radioactive nuclei of a certain species are produced in a sample at a constant rate of 1000 s^-1 and also decay with a half-life of 10 s. If the…

Numerical answer

Radioactive nuclei of a certain species are produced in a sample at a constant rate of

1000\ \mathrm{s^{-1}}

and also decay with a half-life of

10\ \mathrm{s}

. If the sample initially contains no such nuclei, the activity (in Bq) of the sample

30\ \mathrm{s}

after production starts is

Show answer and solution

Answer: 875

With production rate

R

and decay constant

\lambda

\dfrac{dN}{dt}=R-\lambda N

, so

N=\dfrac R\lambda(1-e^{-\lambda t})

and the activity is

A=\lambda N=R(1-e^{-\lambda t})

. Here

t=30\ \mathrm{s}=3T_{1/2}

, so

e^{-\lambda t}=2^{-3}=\dfrac18

. Therefore

A=1000\left(1-\dfrac18\right)=875\ \mathrm{Bq}

. The activity approaches the production rate (saturation) only for

t\gg T_{1/2}

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