Nuclei of a radioactive element A are being produced at a constant rate - The element has a decay constant - At time t-0- there are N0 nuclei of the element-Calculate the number N of nuclei of A at time t

For production, dN_Adt=α For decay, dN_Adt= λ N_A Net rate =dN_Adt=α λ N_A dN_Aα λ N_A=dt Integraing, ∫_N_0^NdN_Aα λ N_A=∫^t_0 dt ...

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Standard XII

Physics

Beta Decay

Nuclei of a r...

Question

Nuclei of a radioactive element A are being produced at a constant rate $α$ . The element has a decay constant $λ$ . At time $t = 0$ , there are $N_{0}$ nuclei of the element.
Calculate the number N of nuclei of A at time t

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Solution

For production, $\frac{d N_{A}}{d t} = α$

For decay, $\frac{d N_{A}}{d t} = - λ N_{A}$

Net rate $= \frac{d N_{A}}{d t} = α - λ N_{A}$

$\frac{d N_{A}}{α - λ N_{A}} = d t$

Integraing, $\int_{N_{0}}^{N} \frac{d N_{A}}{α - λ N_{A}} = \int_{0}^{t} d t$

$- \frac{1}{λ} l n (\frac{α - λ N}{α - λ N_{0}}) = t$

$l n (\frac{α - λ N}{α - λ N_{0}}) = - λ t$

$\frac{α - λ N}{α - λ N_{0}} = e^{- λ t}$

$α - λ N = (α - λ N_{0}) e^{- λ t}$

$N = \frac{1}{λ} [α - (α - λ N_{0}) e^{- λ t}]$

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Nuclei of a radioactive element A are being produced at a constant rate α. The element has a decay constant λ. At time t=0, there are N0 nuclei of the element.Calculate the number N of nuclei of A at time t

For production, dNAdt=αFor decay, dNAdt=−λNANet rate =dNAdt=α−λNAdNAα−λNA=dtIntegraing, ∫NN0dNAα−λNA=∫t0dt−1λln(α−λNα−λN0)=tln(α−λNα−λN0)=−λtα−λNα−λN0=e−λtα−λN=(α−λN0)e−λtN=1λ[α−(α−λN0)e−λt]

Nuclei of a radioactive element A are being produced at a constant rate $α$ . The element has a decay constant $λ$ . At time $t = 0$ , there are $N_{0}$ nuclei of the element.
Calculate the number N of nuclei of A at time t