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Question

At time t=0, a container has N0 radioactive atoms with a decay constant λ. In addition, C numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at t=T ?
  1. CλeλTN0eλT
  2. CλeλT+N0eλT
  3. Cλ(1eλT)+N0eλT
  4. Cλ(1+eλT)+N0eλT


Solution

The correct option is C Cλ(1eλT)+N0eλT
At t=0 initial nucleon =N0
Additionally, at a constant rate (λNC)
=dNdt
t0dt=NN0(dNλNC)
t=1λln(λNC)NN0
t=1λln(λNCλN0C)
eλt=λNCλN0C
λNC=eλt(λN0C)
N=Cλ+eλtλ(λN0C)
=Cλ+N0eλtCλeλt
=Cλ[1e(λt)]+N0eλt
At t=TN=Cλ[1e(λT)]+N0eλT

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