Animated Solution for Physics - Atoms and Nuclei: If $M_o$ is the mass of an oxygen isotope ${}_8{\text{O}}^{17}$, $M_p$ and $M_n$ are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is

Imagine you have a perfectly built Lego castle. Now, imagine you take it apart and weigh all the individual Lego bricks. You would expect the total weight of the individual bricks to be exactly the same as the weight of the fully assembled castle, right? In our everyday macroscopic world, this is absolutely true. But when we dive into the quantum realm of the atomic nucleus, this intuitive logic completely shatters.

Let's explore this mind-bending phenomenon through the lens of an Oxygen-17 isotope, denoted as ${}_8{\text{O}}^{17}$.

First, we need to understand what exactly is inside this specific nucleus. The atomic number ($Z$) of Oxygen is $8$. This tells us that there are exactly $8$ protons residing in the nucleus. The mass number ($A$) is $17$, which represents the total number of nucleons (protons plus neutrons).

To find the number of neutrons ($N$), we simply subtract the number of protons from the total mass number:

So, our Oxygen-17 nucleus is a tightly packed cluster of $8$ protons and $9$ neutrons.

Now, let's do a thought experiment. Suppose we could place our intact Oxygen-17 nucleus on a highly precise quantum scale. Let's call its mass $M_o$.

Next, we take the nucleus apart and weigh the $8$ protons and $9$ neutrons individually. The mass of a single proton is $M_p$, so $8$ protons weigh $8M_p$. The mass of a single neutron is $M_n$, so $9$ neutrons weigh $9M_n$. The total mass of these separated, free nucleons is:

Here is where the universe plays a trick on us: the mass of the intact nucleus ($M_o$) is actually less than the sum of the masses of its individual parts ($8M_p+9M_n$). This difference is known as the mass defect ($\Delta m$).

Binding Energy

Where did this missing mass go? It didn't just vanish into thin air. According to Albert Einstein's legendary mass-energy equivalence principle, mass and energy are two sides of the same coin, related by the equation $E=m{c}^2$.

When the $8$ protons and $9$ neutrons come together to form the nucleus, the strong nuclear force binds them tightly. In doing so, a small amount of their mass is converted into a massive amount of energy, which is released into the universe. This is the Binding Energy.

To calculate this energy, we multiply the mass defect by the speed of light squared ($c^2$). In physics, binding energy is often expressed as the energy of the bound state minus the energy of the free state, which gives a negative value indicating a stable, lower-energy system.

Therefore, the expression for the nuclear binding energy is:

Expanding the negative sign, we get our final, elegant expression:

This formula beautifully encapsulates the profound truth that the whole is sometimes literally less than the sum of its parts, bound together by the fundamental forces of the cosmos.