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Radioactivity

Problem 6.232

A Bi210 radionuclide decays via the chain Bi210⟶λ1Po210λ2→ Pb206 (stable),  \mathrm{Bi}^{210} \underset{\lambda_{1}}{\longrightarrow} \mathrm{Po}^{210} \overrightarrow{\lambda_{2}} \mathrm{~Pb}^{206} \text { (stable), } where the decay constants are λ1=1.60⋅10−6 s−1,λ2=\lambda_{1}=1.60 \cdot 10^{-6} \mathrm{~s}^{-1}, \lambda_{2}= =5.80⋅10−8 s−1.=5.80 \cdot 10^{-8} \mathrm{~s}^{-1} . Calculate alpha- and beta-activities of the Bi210\mathrm{Bi}^{210} preparation of mass 1.00mg1.00 \mathrm{mg} a month after its manufacture.

Reveal Answer
 6.232. N˙β=N0λ1exp⁡(−λ1t)=0.72⋅1011 part./s, N˙α==N0(e−λ1t−e−λ2t)λˉ1λ2/(λ^2−λ1)=1.46⋅1011 part./s. Here N0 is the initial number of Bi210 nuclei. \begin{aligned} &\text { 6.232. } \dot{N}_{\beta}=N_{0} \lambda_{1} \quad \exp \quad\left(-\lambda_{1} t\right)=0.72 \cdot 10^{11} \quad \text { part./s, } \quad \dot{N}_{\alpha}=\\ &=N_{0}\left(\mathrm{e}^{-\lambda_{1} t}-\mathrm{e}^{-\lambda_{2} t}\right) \bar{\lambda}_{1} \lambda_{2} /\left(\hat{\lambda}_{2}-\lambda_{1}\right)=1.46 \cdot 10^{11} \text { part./s. Here } N_{0}\\ &\text { is the initial number of } \mathrm{Bi}^{210} \text { nuclei. } \end{aligned}
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