Animated Solution for Physics - Atoms and Nuclei: Let $m_p$ be the mass of proton, $m_n$ the mass of neutron. $M_1$ the mass of ${}_{10}^{20}\text{Ne}$ nucleus and $M_2$ the mass of ${}_{20}^{40}\text{Ca}$ nucleus. Then

Mass Defect

To unravel this problem, we must first journey into the very heart of the nucleus. When protons and neutrons (collectively called nucleons) bind together to form a nucleus, a tremendous amount of energy is released. According to Einstein's legendary equation $E=m{c}^2$, this released energy must come from somewhere—it comes at the expense of mass!

This phenomenon is known as the mass defect. It dictates that the mass of any stable nucleus is strictly less than the sum of the masses of its individual, separated nucleons.

Let's apply this to the Neon nucleus (${}_{10}^{20}\text{Ne}$). It consists of $10$ protons and $10$ neutrons. Therefore, its nuclear mass $M_1$ must be less than the combined mass of these $20$ free particles:

This beautiful realization immediately confirms that option (d) is correct.

The energy equivalent of the mass defect is the Binding Energy (BE). It is the energy required to completely disassemble a nucleus into its constituent protons and neutrons. We can express the binding energy for both Neon and Calcium as follows:

For Neon (${}_{10}^{20}\text{Ne}$):

For Calcium (${}_{20}^{40}\text{Ca}$):

Binding Energy Per Nucleon

Total binding energy isn't a fair metric for comparing the stability of different nuclei, because heavier nuclei naturally have more binding energy simply due to having more nucleons. To level the playing field, physicists use the Binding Energy per Nucleon ($BE/A$).

For Neon ($A=20$):

For Calcium ($A=40$):

Nature's Blueprint

Now, we invoke one of the most famous graphs in all of physics: the Binding Energy Curve. Empirical data shows that for lighter nuclei (up to Iron-56), the binding energy per nucleon steadily increases as the mass number $A$ increases.

Since Calcium ($A=40$) is heavier than Neon ($A=20$), but both are lighter than Iron, Calcium is more tightly bound per nucleon than Neon. Mathematically, this translates to a strict inequality:

Let's substitute our expressions into this inequality and watch the physics transform into pure algebra:

First, we can cancel out $c^2$ from both sides. Next, multiply the entire inequality by $20$ to clear the denominators:

Now, let's split the fraction on the left side:

Notice the symmetry? The term $10(m_p+m_n)$ appears on both sides! We can subtract it away, leaving us with a much simpler relation:

To clean this up, we multiply both sides by $-1$. But beware the classic algebraic trap: multiplying an inequality by a negative number flips the inequality sign!

Finally, multiply by $2$ to isolate $M_2$:

This elegant derivation perfectly matches option (c). Thus, by combining the physical principles of the mass defect and the empirical stability curve, we have rigorously proven that both options (c) and (d) are the correct answers.