Question

In order to form an element Q, an element P disintegrates by $\alpha$-emission and the new element suffers two further disintegrations, both by $\beta$-emission. Explain the fact that the elements P and Q are isotopes.

Answer 1

Given Data

1. Let element P is represented by ${}_Z{}^{A}P$ where A is the mass number and z is the atomic number.
2. After the emission of an $\alpha$-particle and two consecutive $\beta$-emissions, a new element Q is formed.

Emission of $\alpha$and $\beta$-particle

1. When the element P emits an $\alpha$-particle its mass number decreases by 4 and its atomic number decreases by 2. Let the element formed is M.
2. Represent this in the form of an equation we get${}_Z{}^{A}P\stackrel{-{}_2{}^{4}He}{\to }{}_{Z-2}{}^{A-4}M$.
3. Now for the $\beta$-emission of element M, the mass number will remain the same but the atomic number increases by 1 for each $\beta$-particle.
4. Representing this in the form of an equation we get ${}_{Z-2}{}^{A-4}M\stackrel{-{}_{-1}{}^{0}e}{\to }{}_{Z-2+1}{}^{A-4}M\stackrel{-{}_{-1}{}^{0}e}{\to }{}_{Z-2+1+1}{}^{A-4}Q$ that is ${}_Z{}^{A-4}Q$.

Final answer

1. Comparing elements P and Q we can say that in both the atoms atomic number is the same that is Z but the mass number is different.
2. Isotopes are the elements having the same atomic number but a different mass number.
3. So, as per the definition of isotope, elements P and Q are isotopes.