Ratio of Alpha to Beta Particles in Thorium Decay

To find the ratio of the number of $\alpha$ particles to the number of $\beta^-$ particles emitted in the decay chain, we consider the nuclear reaction equation:

$^{230}_{90}\text{Th} \longrightarrow ^{214}_{84}\text{Po} + n_\alpha \cdot ^{4}_{2}\text{He} + n_\beta \cdot ^{0}_{-1}\text{e}$

where:

• $n_\alpha$ is the number of $\alpha$ particles emitted,
• $n_\beta$ is the number of $\beta^-$ particles emitted.

Step 1: Conservation of Mass Number ($A$)

By applying the law of conservation of mass number: $230 = 214 + 4 n_\alpha + 0 \cdot n_\beta$ $230 - 214 = 4 n_\alpha$ $16 = 4 n_\alpha \implies n_\alpha = 4$

Step 2: Conservation of Atomic Number ($Z$)

By applying the law of conservation of atomic number: $90 = 84 + 2 n_\alpha - 1 \cdot n_\beta$

Substitute $n_\alpha = 4$ into the equation: $90 = 84 + 2(4) - n_\beta$ $90 = 84 + 8 - n_\beta$ $90 = 92 - n_\beta \implies n_\beta = 2$

Step 3: Calculate the Ratio

The ratio of the number of $\alpha$ particles to the number of $\beta^-$ particles emitted is: $\text{Ratio} = \frac{n_\alpha}{n_\beta} = \frac{4}{2} = 2$