What is the alpha activity of a 10-kg sample of ${ }^{235} U$ that is used in a nuclear reactor?

Strategy We find the number of radioactive atoms by using Avogadro’s number and the gram-molecular weight. We find in Appendix 8 that ${ }^{235} U$ has a half-life for emitting α particles of $t_{1 / 2}=7.04 \times 10^{8}$ y. Then we use Equation (12.22) to find the activity.

$R=\lambda N(t)$ (12.22)

Question

What is the alpha activity of a 10-kg sample of [latex]{ }^{235} U[/latex] that is used in a nuclear reactor?

Strategy We find the number of radioactive atoms by using Avogadro’s number and the gram-molecular weight. We find in Appendix 8 that [latex]{ }^{235} U[/latex] has a half-life for emitting α particles of [latex]t_{1 / 2}=7.04 imes 10^{8}[/latex] y. Then we use Equation (12.22) to find the activity.

[latex]R=\lambda N(t)[/latex] (12.22)

Accepted Answer

The number of [latex]{ }^{235} U[/latex] atoms in a 10-kg sample is

[latex]N=M \frac{N_{ A }}{M\left({ }^{235} U ight)}[/latex]

[latex]\begin{aligned}N &=(10 kg )\left(\frac{10^{3} g }{1 kg } ight)\left(\frac{6.02 imes 10^{23} ext { atoms } / mol }{235 g / mol } ight) \&=2.56 imes 10^{25} ext { atoms }=2.56 imes 10^{25} ext { nuclei }\end{aligned}[/latex]

The activity is

[latex]\begin{aligned}R=\lambda N &=\frac{\ln (2) \cdot N}{t_{1 / 2}} \&=\frac{\ln (2) \cdot\left(2.56 imes 10^{25} ext { nuclei } ight)}{7.04 imes 10^{8} y } \&=2.52 imes 10^{16} ext { decays } / y =8.0 imes 10^{8} Bq\end{aligned}[/latex]

Q. 12.12

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