Question

Identify the unknown isotope X in the following decays.

$$\begin{gathered} \text{a.}\mathrm{X}\to {}^{224}\mathrm{Ra}+\alpha \\ \text{b.}{}^{35}\mathrm{S}\to \mathrm{X}+{\mathrm{e}}^{-}+\overline{\nu } \\ \text{c.}\mathrm{X}\to {}^{40}\mathrm{K}+{\mathrm{e}}^{+}+\nu \\ \text{d.}{}^{24}\mathrm{Na}\to {}^{24}\mathrm{Mg}+{\mathrm{e}}^{-}+\overline{\nu }\to \mathrm{X}+\gamma \end{gathered}$$

Short answer

Hence, the values of X is:

$$\begin{gathered} \text{a.)}X={}^{228}\mathrm{Th} \\ \text{b.)}X={}^{207}\mathrm{Tl} \\ \text{c.)}X={}^7\mathrm{Li} \\ \text{d.)}X={}^{60}\mathrm{Ni} \end{gathered}$$

Step-by-step solution

Given information

$$\begin{gathered} \text{a.}\mathrm{X}\to {}^{224}\mathrm{Ra}+\alpha \\ \text{b.}{}^{35}\mathrm{S}\to \mathrm{X}+{\mathrm{e}}^{-}+\overline{\nu } \\ \text{c.}\mathrm{X}\to {}^{40}\mathrm{K}+{\mathrm{e}}^{+}+\nu \\ \text{d.}{}^{24}\mathrm{Na}\to {}^{24}\mathrm{Mg}+{\mathrm{e}}^{-}+\overline{\nu }\to \mathrm{X}+\gamma \end{gathered}$$

Explanation (a)

In the following decays, we will find for the unknown isotope X:

$\text{a.)}X\to {}^{224}\mathrm{Ra}+\alpha$

The decay is:

$$\begin{gathered} X={}^{224}\mathrm{Ra}+{}^4\mathrm{He} \\ X={}^{228}\mathrm{Th} \end{gathered}$$

Explanation (b)

The decay is:

Explanation (c)

The decay is:

Explanation (d)

The decay is: