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(A) Let the nucleus $P$ be represented as $_{Z}^{A}P$, where $A$ is the mass number and $Z$ is the atomic number.An alpha particle is represented as $

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Isotopes

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(A) Let the nucleus $P$ be represented as $_{Z}^{A}P$, where $A$ is the mass number and $Z$ is the atomic number.An alpha particle is represented as $_{2}^{4}\alpha$ and a $\beta^{-}$ particle (electron) is represented as $_{-1}^{0}e$.The decay process is: $_{Z}^{A}P \rightarrow _{Z'}^{A'}Q + 1(_{2}^{4}\alpha) + 2(_{-1}^{0}e)$.Conservation of mass number: $A = A' + 4 + 2(0) \Rightarrow A' = A - 4$.Conservation of atomic number: $Z = Z' + 2 + 2(-1) \Rightarrow Z = Z' + 2 - 2 \Rightarrow Z = Z'$.Since the atomic number $Z$ remains the same $(Z = Z')$, the nuclei $P$ and $Q$ have the same number of protons.Nuclei with the same atomic number but different mass numbers are called isotopes.

If a nucleus $P$ converts into a nucleus $Q$ by the decay of one alpha particle and two $\beta^{-}$ particles, then the nuclei $P$ and $Q$ are

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