23P3Q09

9ai)

Decrease in mass after nuclear reaction

\displaystyle \begin{aligned}  \Delta m&=[400.19774-(393.54304+6.64466)]\times {{10}^{27}} \\ & =0.01004\times {{10}^{27}}\text{ kg} \end{aligned}

Energy released\displaystyle =\Delta m.{{c}^{2}}=(0.01004\times {{10}^{-27}}){{(3.00\times {{10}^{8}})}^{2}}=9.036\times {{10}^{-13}}\text{ J}

9aii)

The total momentum of a system is constant if no net external force acts on it.

9aiii)

By PCOM, the momentum of the daughter nucleus and the alpha particle are equal but opposite.            \displaystyle {{p}_{D}}={{p}_{\alpha }}

\displaystyle \begin{aligned}  \frac{K{{E}_{D}}}{K{{E}_{\alpha }}}&=\frac{{{p}_{D}}^{2}}{2{{m}_{D}}}\div \frac{{{p}_{\alpha }}^{2}}{2{{m}_{\alpha }}} \\ & =\frac{{{m}_{\alpha }}}{{{m}_{D}}} \\ K{{E}_{D}}&=\frac{4}{237}(8.784\times {{10}^{-13}}) \\ & =1.483\times {{10}^{-14}}\text{ J} \end{aligned}

9aiv)

Perhaps a gamma photon is also released. So some of the energy released is with the gamma photon.

9bi)

\displaystyle \begin{aligned}  A&=\lambda N \\ 8130&=(5.08\times {{10}^{-11}})N \\ N&=1.60\times {{10}^{14}} \end{aligned}

9bii)

\displaystyle {{t}_{1/2}}=\frac{\ln 2}{\lambda }=\frac{\ln 2}{5.08\times {{10}^{-11}}}=1.36\times {{10}^{10}}\text{ s}

9ci)

\displaystyle A=\lambda N

Activity is the product of the decay constant and the number of undecayed nuclei.

The beta emitter has a much higher initial activity since it has a higher decay constant, and the same initial number of undecayed nuclei.

9cii)

\displaystyle A=\lambda N

Initially, the activity of the beta emitter is higher than that of the americium-241.

However, as the number of beta emitters decrease more rapidly, the activity of the beta emitter will eventually be lower than that of the americium-241.

It takes americium-241 about 13 half-lives for its activity to decrease from 8130Bq to 1Bq. The beta emitter would have fallen to 1 Bq long before that.

COMMENT: The graph below depicts the situation if red sample has twice the decay constant as the blue. The two activities are equal within 1 half-life of the blue sample.

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