Question

A radioactive nucleus undergoes a series of decays according to the sequence.

$\mathrm{X}\stackrel{\mathrm{\beta }}{\to }{\mathrm{X}}_1\stackrel{\mathrm{\alpha }}{\to }{\mathrm{X}}_2\stackrel{\mathrm{\alpha }}{\to }{\mathrm{X}}_3\to$

If the mass number and atomic number of ${\mathrm{X}}_3$ are $132$ and $69$ respectively, what is the mass number and atomic number of$X$?

Answer 1

Step1: Given data

Let the mass number of the nucleus $X$ is $A$ and atomic number of the nucleus is $Z$

The atomic number of the nucleus ${\mathrm{X}}_3$ is $69$

The mass number of the nucleus ${\mathrm{X}}_3$ is $132$

Step2: Write the radioactive reactions

We know that radioactive elements decay $\mathrm{\alpha }$-particles its mass number decreases $4units$ and its atomic number decreases by $2units$.

Also one ${}_{-1}{}^{°}\mathrm{\beta }$- particle decay does not effect on its mass number but atomic number increases by one unit.

${}_{\mathrm{Z}}{}^{\mathrm{A}}\mathrm{X}\stackrel{\mathrm{\beta }}{\to }{}_{\mathrm{Z}+1}{}^{\mathrm{A}}{\mathrm{X}}_1\stackrel{\mathrm{\alpha }}{\to }{}_{\mathrm{Z}+1-2}{}^{\mathrm{A}-4}\mathrm{X}_2\stackrel{\mathrm{\alpha }}{\to }{}_{\mathrm{Z}+1-2-2}{}^{\mathrm{A}-4-4}\mathrm{X}_3$

Step3: Finding the mass number

the mass number of ${\mathrm{X}}_3$ is $132$

$$\begin{gathered} A-4-4=132 \\ A=132+8 \\ A=140 \end{gathered}$$

Step4: Finding Atomic number

The atomic number of ${\mathrm{X}}_3$ is $69$

$$\begin{gathered} Z+1-2-2=69 \\ Z=69+3 \\ Z=72 \end{gathered}$$

Hence, mass number is $140$ and Atomic is $72$ of $X$.