Animated Solution for Physics - Atoms and Nuclei: A uranium nucleus (atomic number $92$, mass number $238$) emits an alpha particle and the resulting nucleus emits $\beta$-particle. What are the atomic number and mass number of the final nucleus?

Imagine a nucleus as a bustling city of protons and neutrons. Sometimes, this city gets too crowded, or the balance of its inhabitants is just not right. To find peace and stability, the nucleus undergoes a transformation. This is the fascinating world of radioactivity!

In our problem, we start with a massive Uranium nucleus. It has an atomic number of $92$, meaning it has $92$ protons, and a mass number of $238$, which is the total count of protons and neutrons.

The Uranium nucleus decides it's too heavy. It undergoes alpha decay. But what exactly is an alpha particle?

An alpha particle is a tightly bound package of $2$ protons and $2$ neutrons. It's essentially a Helium nucleus! When the Uranium nucleus ejects this alpha particle, it loses these $4$ nucleons.

Mathematically, the mass number decreases by $4$:

And the atomic number decreases by $2$:

Our Uranium has transformed into a new element, Thorium, which we can call nucleus $\text{X}$ for now. It has an atomic number of $90$ and a mass number of $234$.

But the journey isn't over. This new nucleus $\text{X}$ is still not entirely stable. It has too many neutrons compared to its protons. To fix this, it undergoes beta decay.

During beta decay, the weak nuclear force works its magic. A neutron inside the nucleus transforms into a proton, and in the process, it spits out a high-speed electron, which we call a beta particle.

Because a neutron turned into a proton, the total number of nucleons hasn't changed. The mass number remains exactly the same at $234$.

However, we now have one extra proton! The atomic number increases by $1$:

And there we have it! After shedding an alpha particle and undergoing a beta transformation, our nucleus has finally reached its new identity.

The final nucleus has an atomic number of $91$ and a mass number of $234$. This beautiful sequence of decays is how nature constantly seeks balance at the most fundamental level.