Q.Nuclei with magic no. of proton Z=2,8,20,28,50,82 and magic no. of neutrons N=2,8,20,28,50,82 and 126 are found to be very stable.
(i) Verify this by calculating the proton separation energy Sp for 120Sn (Z=50) and 121Sb (Z=51). The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by Sp=(MZ−1,N+MH−MZ,N)c2. Given 119In=118.9058u, 120Sn=119.902199u, 121Sb=120.903824u, 1H=1.0078252u.
(ii) What does the existence of magic number indicate?
Mass Energy Equivalence: From Intuition to the Formula
Imagine you have a lump of coal. You know you can burn it to get heat, and that heat can run a steam engine. The energy you get out seems to come from the chemical bonds in the coal. But what if I told you that the coal itself — just sitting there, not burning — already contains a staggering amount of energy locked inside its very mass? That is the core idea of mass-energy equivalence.
The Intuition: Mass is Frozen Energy
Think of mass as a kind of "frozen" or "stored" energy. When you burn coal, you are only releasing a tiny fraction of this stored energy — the energy in the chemical bonds. The rest of the mass remains as matter. But if you could somehow completely convert that lump of coal into pure energy, you would get an unimaginable amount — enough to power a city for years.
This is not a metaphor. Mass and energy are not two separate things that can be converted into each other like dollars and rupees. They are the same fundamental thing, just in different forms. Mass is a highly concentrated form of energy. Energy, when concentrated enough, behaves like mass.
The Precise Statement
The relationship is given by the most famous equation in physics:
E=mc2
Where:
E is the energy equivalent of the mass (in joules, J)
m is the mass (in kilograms, kg)
c is the speed of light in vacuum (3×108 m/s)
The speed of light is a huge number. Squaring it makes it enormous. This is why a tiny amount of mass corresponds to a colossal amount of energy.
What This Equation Actually Means
The equation tells you exactly how much energy is "stored" inside any object with mass m. If you could annihilate that mass completely, you would get E joules of energy.
Example: A 1 kg mass (like a litre of water) contains:
E=1×(3×108)2=9×1016 J
That is 90 quadrillion joules — roughly the energy released by a 20-megaton nuclear bomb. This is not energy you can normally access; it is locked inside the nucleus of atoms.
Where Does This Show Up in Real Life?
Nuclear Reactions: In nuclear fission (splitting atoms) or fusion (joining atoms), a tiny fraction of the mass of the nucleus is converted into energy. The mass of the products is slightly less than the mass of the reactants. The "missing" mass has become energy — exactly as E=mc2 predicts. This is how the Sun works and how nuclear power plants generate electricity.
Particle Physics: When a particle and its antiparticle meet, they annihilate completely into pure energy (usually gamma rays). The energy produced equals mc2 for the two particles. …
Why this formula?
Why E=mc2 — The Reasoning Behind Mass-Energy Equivalence
The formula E=mc2 is not a random guess. Einstein arrived at it by thinking deeply about what happens to energy when you move an object. The core insight: if an object gains energy, it must behave as if it has gained mass.
The Starting Point: Relativistic Momentum
In special relativity, the momentum of a particle is not simply p=mv. Instead, it is:
p=1−v2/c2m0v
where m0 is the rest mass (mass measured when the object is at rest). This formula already tells us something strange: as speed approaches c, momentum shoots toward infinity — no object with mass can reach the speed of light.
The Energy-Momentum Relation
Einstein then asked: what is the correct expression for kinetic energy that matches this new momentum? In classical physics, kinetic energy is K=21mv2. But that formula fails at high speeds.
The relativistic kinetic energy turns out to be:
K=1−v2/c2m0c2−m0c2
This looks odd — why subtract m0c2? Because when v=0, the first term becomes m0c2, and we want K=0 at rest. So the subtraction gives zero kinetic energy when the object is stationary.
Note
The term m0c2 appears naturally as a rest energy — energy that an object has simply because it has mass, even when completely at rest.
The Crucial Step: What Happens When You Add Energy?
Now consider a box that emits light (photons) in opposite directions. The light carries away energy. Classical physics says the box loses energy but its mass stays the same. Einstein showed this cannot be true.
The argument (simplified): if the box emits a pulse of light with energy E, the light carries momentum p=E/c. By conservation of momentum, the box recoils. But after the light is absorbed by the opposite wall, the box stops. The net effect: the box has moved slightly. Its center of mass has shifted — unless the energy carried by the light also carried mass.
Important
For the center of mass of the entire system (box + light) to remain stationary, the light must behave as if it has an effective mass m=E/c2. Therefore, energy itself has inertia.
The Full Formula
The total energy of any object — moving or at rest — is:
E=1−v2/c2m0c2
For an object at rest (v=0), this reduces to:
E=m0c2
For a moving object, the total energy is the sum of rest energy and kinetic energy:
E=m0c2+K
E=mc2
where m is the relativistic massm=1−v2/c2m0, or equivalently:
Sp(120Sn)≈10.64MeV versus Sp(121Sb)≈5.78MeV: it takes nearly twice the energy to pull a proton out of the magic-Z tin nucleus, confirming that magic numbers mark closed, unusually stable nuclear shells.
What Sp measures
The proton separation energy is the least energy needed to knock the most loosely bound proton out of a nucleus. A larger Sp means a more tightly bound, more stable nucleus. The given formula is just conservation of mass-energy:
Sp=(MZ−1,N+MH−MZ,N)c2,
where MZ−1,N is the daughter (one fewer proton, same N), MH is the hydrogen-atom mass supplying the removed proton, and MZ,N is the parent. Using atomic masses, the MH term automatically balances the electrons. Convert with 1u=931.5MeV/c2.
Method: Calculating Nucleon Separation Energy from Atomic Masses (Mass-Energy Equivalence)
This method applies to any question asking for the energy needed to remove a single nucleon (proton or neutron) from a nucleus, given atomic mass data — and for interpreting what a large or small separation energy tells you about nuclear stability.
Steps
Step 1: Identify the parent nucleus and the daughter left behind after removing one nucleon
Write down which nucleus loses the nucleon (the parent) and which nucleus remains (the daughter, with either Z or N reduced by one, depending on whether a proton or neutron is removed).
Step 2: Apply the given/general mass-difference formula, converting mass to energy
Separation energy is a direct application of E=mc2: it is the energy equivalent of the mass "left over" if you add up the pieces after separation and compare to the original parent. For a proton removal (using atomic masses, so the removed proton is represented by a neutral 1H atom to keep the electron count balanced):
Sp=(Mdaughter+M1H−Mparent)c2
(the analogous formula for a neutron uses the free neutron mass in place of M1H, with no daughter change in Z).
Step 3: Convert the mass difference to energy using 1u=931.5MeV/c2
Compute the mass difference in atomic mass units first (this keeps the arithmetic manageable and avoids rounding error), then multiply by 931.5 to convert directly to MeV. …