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Dual Nature of Radiation question

2011 · Shift 1 · Q68
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Dual Nature of Radiation question

2011 · Shift 1 · Q68

JEE AdvancedPhysicsDual Nature of RadiationNumerical+3 / −1
The activity of a freshly prepared radioactive sample is 1010 disintegrations per second, whose mean life is 109 s. The mass of an atom of this radioisotope is 10 −-− 25 kg. The mass (in mg) of the radioactive sample is ‾\underline{\hspace{2cm}}​.
Numerical answer
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Correct answer: 1

  1. Use the relation between activity and number of undecayed nuclei

For a radioactive sample, A=λNA = \lambda NA=λN where:

  • A=1010 disintegrations s−1A = 10^{10}\ \text{disintegrations s}^{-1}A=1010 disintegrations s−1
  • mean life τ=109 s\tau = 10^{9}\ \text{s}τ=109 s
  • λ=1τ=10−9 s−1\lambda = \dfrac{1}{\tau} = 10^{-9}\ \text{s}^{-1}λ=τ1​=10−9 s−1

So, N=Aλ=Aτ=1010×109=1019N = \frac{A}{\lambda} = A\tau = 10^{10}\times 10^{9} = 10^{19}N=λA​=Aτ=1010×109=1019

  1. Find the total mass of the sample

Mass of one atom is given as m0=10−25 kgm_0 = 10^{-25}\ \text{kg}m0​=10−25 kg

Hence total mass, m=Nm0=1019×10−25=10−6 kgm = Nm_0 = 10^{19}\times 10^{-25} = 10^{-6}\ \text{kg}m=Nm0​=1019×10−25=10−6 kg

  1. Convert into mg

Since 1 kg=106 mg1\ \text{kg} = 10^{6}\ \text{mg}1 kg=106 mg we get 10−6 kg=10−6×106 mg=1 mg10^{-6}\ \text{kg} = 10^{-6}\times 10^{6}\ \text{mg} = 1\ \text{mg}10−6 kg=10−6×106 mg=1 mg

  1. Final answer

The mass of the radioactive sample is 1 mg\boxed{1\ \text{mg}}1 mg​

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