A \(\displaystyle {}^{232}_{\ \ 92}\text{U}\) nucleus emits an \(\displaystyle \alpha\) particle with kinetic energy \(\displaystyle = 5.32 \ \text{MeV}\). What is the daughter nucleus and what is the approximate atomic mass (in u) of the daughter atom? Ignore recoil of the daughter nucleus.
An \(\displaystyle \alpha\) particle is just a helium nucleus. In other words \(\displaystyle \alpha = {}^{4}_{2}\text{He}^{2+}\). For shortcut, they just write it like a helium atom, ie, \(\displaystyle {}^{4}_{2}\text{He}\).
Let \(\displaystyle {}^{A}_{Z}\text{X}\) be the daughter nucleus, then, we have this reaction:
\(\displaystyle {}^{232}_{\ \ 92}\text{U} \rightarrow {}^{A}_{Z}\text{X} + \alpha\)
Or
\(\displaystyle {}^{232}_{\ \ 92}\text{U} \rightarrow {}^{A}_{Z}\text{X} + {}^{4}_{2}\text{He}\)
This gives:
\(\displaystyle {}^{232}_{\ \ 92}\text{U} \rightarrow {}^{228}_{\ \ 90}\text{X} + {}^{4}_{2}\text{He}\)
So the daughter nucleus \(\displaystyle {}^{228}_{\ \ 90}\text{X}\) has an atomic number \(\displaystyle Z = 90\). Let us look at the periodic table and find out what is this element.
It is
thorium \(\displaystyle (\text{Th})\). Then the daughter nucleus is:
\(\displaystyle {}^{228}_{\ \ 90}\text{Th}\)
