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

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.