In a radioactive decay chain reaction, $$^{230}_{90}$$Th nucleus decays into $$^{214}_{84}$$Po nucleus. The ratio of the number of $$\alpha$$ to number of $$\beta^-$$ particles emitted in this process is _______.

The parent nucleus is $$^{230}_{90}\text{Th}$$ and the daughter nucleus is $$^{214}_{84}\text{Po}$$.

Let the number of $$\alpha$$ particles emitted be $$n_\alpha$$ and the number of $$\beta^-$$ particles emitted be $$n_\beta$$.

Mass-number (A) balance
Each $$\alpha$$ decay reduces the mass number by $$4$$, while $$\beta^-$$ decay does not change it.
Initial mass number $$= 230$$, final mass number $$= 214$$.
Therefore,
$$230 - 4n_\alpha = 214$$
$$\Rightarrow 4n_\alpha = 16$$
$$\Rightarrow n_\alpha = 4$$.

Atomic-number (Z) balance
Each $$\alpha$$ decay decreases the atomic number by $$2$$, while each $$\beta^-$$ decay increases it by $$1$$.
Starting atomic number $$= 90$$, final atomic number $$= 84$$.
Hence,
$$90 - 2n_\alpha + n_\beta = 84$$.
Substituting $$n_\alpha = 4$$:
$$90 - 2(4) + n_\beta = 84$$
$$90 - 8 + n_\beta = 84$$
$$82 + n_\beta = 84$$
$$\Rightarrow n_\beta = 2$$.

Required ratio
$$\frac{n_\alpha}{n_\beta} = \frac{4}{2} = 2$$.

Therefore, the ratio of the number of $$\alpha$$ particles to the number of $$\beta^-$$ particles emitted is 2.