Question

A radioactive element X disintegrates successively
$X\overset{{{\beta }^{-1}}}{\mathop{\to }}\,{{X}_{1}}\overset{\alpha }{\mathop{\to }}\,{{X}_{2}}\overset{{{\beta }^{-1}}}{\mathop{\to }}\,{{X}_{3}}\overset{\alpha }{\mathop{\to }}\,{{X}_{4}}$
If atomic number and atomic mass number of X are respectively 72 and 180, what are the corresponding values for ${{X}_{4}}$ ​?
A. 69, 172
B. 69, 176
C. 71, 176
D. 70, 172

Answer 1

Hint: We are given the atomic number and atomic mass of a radioactive element and also its disintegration. The element undergoes beta- decay and alpha-decay. We know the change in number of protons and neutrons during both processes. By using this we can find the solution.

Formula used:
Beta – decay,
${}_{-1}^{0}\beta $
Alpha – decay,
${}_{2}^{4}\alpha $

Complete step by step answer:
In the question it is said that a radioactive element disintegrates successively.
It also given that the element undergoes ${{\beta }^{-1}}$ decay and $\alpha $ decay alternatively
 We know that when an element undergoes ${{\beta }^{-1}}$ decay, its atomic number adds by 1 and mass number gets added by zero, i.e.
For beta decay, ${}_{-1}^{0}\beta $
When an element undergoes $\alpha $ decay its atomic number gets decreased by two and its mass number gets decreased by 4, i.e.
For $\alpha $-decay, ${}_{2}^{4}\alpha $
Here we are given the atomic number and mass number of the element X.
Atomic number = 72
Atomic mass = 180
We are asked to find the atomic number and atomic mass of ${{X}_{4}}$.
We know that ${{X}_{4}}$ is formed when X undergoes two $\beta $ - decays and two $\alpha $- decays.
Therefore we get,
Atomic number of ${{X}_{4}}$ = 72 + 1 – 2 +1 – 2 = 72 + 2 – 4 = 70
We can also find,
Atomic mass of ${{X}_{4}}$ = 180 + 0 – 4 + 0 – 4 = 180 – 8 = 172
Therefore the element ${{X}_{4}}$ has its atomic number and atomic mass as, ${}_{70}^{172}{{X}_{4}}$.

So, the correct answer is “Option D”.

Note: When a radioactive element undergoes beta – decay a beta particle is emitted from its atomic nuclei. During this process one of the neutrons in the nucleus gets converted into a proton and thus the atomic number increases by one. But there is no change in mass number because the mass of a beta – particle is very small than the mass of the atom.
During the process of alpha – decay the atomic nucleus of the radioactive element emits an alpha – particle. Since the alpha – particle consists of 2 protons and two neutrons, when the nucleus emits alpha – particle its mass number gets decreased by 4 and since there are 2 protons its atomic number gets reduced by 2.