Animated Solution for Physics - Atoms and Nuclei: In a radioactive decay chain reaction, ${}_{90}^{230}\text{Th}$ nucleus decays into ${}_{84}^{214}\text{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$ particles emitted in this process is_________.

Imagine a heavy, unstable nucleus like Thorium-230. It's like a wobbly tower of blocks, desperate to find a stable configuration. To achieve this, it undergoes a series of transformations, spitting out pieces of itself.

These pieces are primarily alpha particles (helium nuclei) and beta particles (high-speed electrons). Our mission is to decode this exact sequence and find out how many of each particle were ejected when Thorium-230 finally settled down as Polonium-214.

We can represent this entire complex chain of events as a single, elegant nuclear equation.

Let the number of alpha particles emitted be $n$, and the number of beta particles emitted be $m$.

The reaction looks like this:

This equation is our roadmap. To solve it, we rely on two fundamental laws of the universe: the conservation of mass number and the conservation of atomic number.

Mass Number

First, let's look at the mass number, which is the total number of protons and neutrons. In any nuclear reaction, the total mass number before the decay must equal the total mass number after the decay.

Notice a crucial detail: beta particles have a mass number of zero. This means they don't affect the mass balance at all! This makes our job much easier. We can find the number of alpha particles just by looking at the change in mass.

Setting up the equation for mass numbers:

Now, we solve for $n$:

So, exactly 4 alpha particles were emitted during this decay chain.

Atomic Number

Next, we apply the conservation of atomic number, which represents the total electrical charge of the nucleus. The total charge before the decay must equal the total charge after.

Here, both alpha and beta particles play a role. Each alpha particle carries a charge of $+2$, and each beta particle carries a charge of $-1$.

Setting up the equation for atomic numbers:

We already know that $n=4$. Let's substitute this value into our equation:

Solving for $m$, we get:

This tells us that exactly 2 beta particles were emitted.

The question asks for the ratio of the number of alpha particles to the number of beta particles.

We have our values: $n=4$ and $m=2$.

The final answer is 2. This elegant integer is the result of the strict conservation laws governing the quantum world!