Radium-$223$ decays with a half-life of $11.4$ days. How long will it take for a $0.240-\mathrm{mol}$ sample of radium to decay to $1.50 \times 10^{-2}$ mol?

Write a partial decay series for $\mathrm{Rn}-220$ undergoing the sequential decays:

$$\alpha, \alpha, \beta, \alpha \text {. }$$

Suppose you a have a $100,000$-atom sample of a radioactive nuclide that decays with a half-life of $2.0$ days. How many radioactive atoms are left after $10$ days?

Write a partial decay series for Th-$232$ undergoing the sequential decays:

$$\alpha, \beta, \beta, \alpha \text {. }$$

Fill in the blanks in the partial decay series.

$$\begin{aligned} &{ }_{88}^{225} \mathrm{Ra} \longrightarrow{ }_{89}^{225} \mathrm{Ac}+ \underline{\phantom{\text{xxxxx}}}\\ &{ }_{89}^{225} \mathrm{Ac}\longrightarrow\underline{\phantom{\text{xxxxx}}}+{ }_2^4 \mathrm{He} \\ &\underline{\phantom{\text{xxxxx}}}\longrightarrow{ }_{85}^{217} \mathrm{At}+{ }_2^4 \mathrm{He} \\ &{ }_{85}^{217} \mathrm{At} \longrightarrow \underline{\phantom{\text{xxxxx}}}+{ }_2^4 \mathrm{He} \\ & \end{aligned}$$

Fill in the blanks in the partial decay series.

$$\begin{aligned} &{ }_{94}^{241} \mathrm{Pu} \longrightarrow{ }_{955}^{241} \mathrm{Am}+ \underline{\phantom{\text{xxxxx}}} \\ &{ }_{95}^{241} \mathrm{Am} \longrightarrow{ }_{93}^{237} \mathrm{~Np}+ \underline{\phantom{\text{xxxxx}}} \\ &{ }_{93}^{237} \mathrm{~Np} \longrightarrow\underline{\phantom{\text{xxxxx}}} + { }_2^4 \mathrm{He} \\ &\underline{\phantom{\text{xxxxx}}} \longrightarrow{ }_{92}^{233} \mathrm{U}+{ }_{-1}^0 \mathrm{e} \end{aligned}$$