The number of beta particles emitted by a radioactive substance is twice the number of alpha particles emitted

Question

The number of beta particles emitted by a radioactive substance is twice the number of alpha particles emitted by it. The resulting daughter is an:

Accepted Answer

1. **Decay Rules:** 
   - In **Alpha ($\alpha$) decay**, the nucleus emits a helium nucleus (${}_{2}^{4}\mathrm{He}$). The atomic number ($Z$) decreases by 2 and the mass number ($A$) decreases by 4 .
   - In **Beta ($\beta^-$) decay**, the nucleus emits an electron (${}_{-1}^{0}e$). The atomic number ($Z$) increases by 1 and the mass number ($A$) remains unchanged .
2. **Calculation:** 
   - Let the number of $\alpha$ particles emitted be $n$. The change in $Z$ due to $\alpha$ emission is $n 	imes (-2) = -2n$.
   - The number of $\beta$ particles emitted is given as twice the number of $\alpha$ particles, i.e., $2n$. The change in $Z$ due to $\beta$ emission is $2n 	imes (+1) = +2n$.
   - **Net change in atomic number ($Z$):** $\Delta Z = -2n + 2n = 0$.
3. **Conclusion:** The final daughter nucleus has the same atomic number ($Z$) as the parent nucleus but a different mass number (since $A$ decreases by $4n$). Atoms with the same atomic number but different mass numbers are defined as **isotopes** .

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