Question

A radioactive nucleus undergoes a series of decays according to the sequence
$X\xrightarrow{\beta }{X_1}\xrightarrow{\alpha }{X_2}\xrightarrow{\alpha }{X_3}$, if the mass number and atomic number of ${X_3}$ are $172$ and $69$ respectively, what is the mass number and atomic number of \[X\]?

Answer 1

Hint: Radioactive decay also known as nuclear decay is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. The three common types of decay are alpha decay $(\alpha )$, beta decay $(\beta )$ and gamma decay $(\gamma )$.

Complete answer:
We are given the atomic number and mass of the \[X_3\], as we can see that the element \[X\] is undergoing beta decay to form the element \[{X_1}\]. Then the element \[{X_1}\] Is undergoing alpha decay to form \[{X_2}\] and then the element \[{X_2}\] undergoes alpha decay to form \[X_3\].
 In Alpha decay the mass is reduced by two and atomic number is reduced by four.
In Beta decay the mass is reduced by \[ - 1\] and the atomic number remains the same.
We will add these values from the mass and atomic number of \[X _3\]:
\[{X_3}\xleftarrow{\alpha }{X_2}\xleftarrow{\alpha }{X_1}\xleftarrow{\beta }X\]
\[{}_{69}^{172}{X_3}\] we will add the atomic number from \[4\] and mass by \[2\] because it is formed by the alpha decay, we get \[{}_{71}^{176}{X_2}\]. Again we will add atomic number by four and mass by two because it is again formed by alpha decay. Therefore, we get \[{}_{73}^{180}{X_1}\]. Now the element \[X\] is undergoing beta decay thus, the atomic mass will remain the same and mass will be added by \[ - 1\]. Therefore, we get \[{}_{72}^{180}X\].
So, the atomic number and mass of element \[X\] is \[{180_{}}an{d_{}}72\] respectively.

Note:
Alpha decay is a type of radioactive decay which emits an alpha particle that is identical to the nucleus of a helium \[ - 4\] atom \[{}_2^4He\]. Beta decay is a type of radioactive decay in which a beta particle is emitted from the atomic nucleus. The beta particle is a fast energetic electron \[{}_{ - 1}^0e\]. And gamma particles emit gamma waves, so they do not possess any mass or charge. Also in this question we need to find the atomic number and mass of \[X\] , so we will add the values of the mass and atomic number after every decay.