Question

Identify the unknown isotope X in the following decays.

$$\begin{gathered} \text{a.}{}^{230}\mathrm{Th}\to \mathrm{X}+\alpha \\ \text{b.}{}^{35}\mathrm{S}\to \mathrm{X}+{\mathrm{e}}^{-}+\overline{\nu } \\ \text{c.}\mathrm{X}\to {}^{40}\mathrm{K}+{\mathrm{e}}^{+}+\nu \\ \text{d.}{}^{24}\mathrm{Na}\to {}^{24}\mathrm{Mg}+{\mathrm{e}}^{-}+\overline{\nu }\to \mathrm{X}+\gamma \end{gathered}$$

Short answer

Hence, the values of X are:

$$\begin{gathered} \text{a.)}X={}^{226}\mathrm{Ra} \\ \text{b.)}X={}^{35}\mathrm{Cl} \\ \text{c.)}X={}^{40}\mathrm{Ca} \\ \text{d.)}X={}^{24}\mathrm{Mg} \end{gathered}$$

Step-by-step solution

Given information

$$\begin{gathered} \text{a.}{}^{230}\mathrm{Th}\to \mathrm{X}+\alpha \\ \text{b.}{}^{35}\mathrm{S}\to \mathrm{X}+{\mathrm{e}}^{-}+\overline{\nu } \\ \text{c.}\mathrm{X}\to {}^{40}\mathrm{K}+{\mathrm{e}}^{+}+\nu \\ \text{d.}{}^{24}\mathrm{Na}\to {}^{24}\mathrm{Mg}+{\mathrm{e}}^{-}+\overline{\nu }\to \mathrm{X}+\gamma \end{gathered}$$

Explanation (a)

In the following decays, we need find unknown isotope X:

$\text{a.)}{}^{230}\mathrm{Th}\to X+\alpha$

The decay is:

$$\begin{gathered} {}^{230}\mathrm{Th}=X+{}^4\mathrm{He} \\ X={}^{226}\mathrm{Ra} \end{gathered}$$

Explanation (b)

The decay is:

Explanation (c)

The decay is:

Explanation (d)

The decay is: