A full lesson: a cartoon movie to start, then E = mc² and 1 u = 931.5 MeV, the mass defect on a balance, binding energy calculated step by step, the binding energy per nucleon curve with its iron peak, why the curve is flat and what it says about fusion and fission, NEET shortcuts, problems to solve yourself, and 12 MCQs.
Two hydrogen atoms and two neutrons weigh 4.03298 u. Joined as helium-4 they weigh only 4.00260 u. The missing 0.03038 u left as 28.3 MeV of energy: that is the binding energy.
Ramesh uncle's kirana shop has an imaginary, super-precise scale that can weigh atoms in u. Kabir puts on two hydrogen atoms and two neutrons: 4.03298 u. Then he "sticks" them together into one helium-4 atom and weighs again: 4.00260 u. "Uncle, your scale is cheating!" The scale is fine. The whole is lighter than its parts, and the missing mass has gone out as energy.
An everyday hint: when you join two magnets, they snap together and you hear a click. Energy came out (sound, a little heat). To pull them apart again, you must put that energy back. A nucleus is the same, except the energy is so large that the mass change can actually be measured.
"Missing mass is not lost. It became energy, E = mc². The energy that leaves when a nucleus forms is exactly the energy you must give to break it: the binding energy."
"A chemical reaction, like burning a match, also releases energy. Why don't we see the mass change there?"
Chemical energies are a few eV per atom, about a million times smaller than nuclear energies (MeV). The mass change is real but less than a billionth of the mass, far too small to weigh.
Einstein's special relativity says mass is a form of energy: E = mc², with c ≈ 3 × 10⁸ m/s. Since c² = 9 × 10¹⁶ m²/s², a tiny mass is a huge energy. In a reaction, the total of mass energy and all other energy is conserved. Slide the mass, or pick a preset.
= 1.4924 × 10⁻¹⁰ J.
1.4924 × 10⁻¹⁰ / 1.602 × 10⁻¹³ = 931.5 MeV.
Mass in u × 931.5 = energy in MeV. Electron: 0.00055 u ≈ 0.511 MeV.
"Two conversions, two units. In kilograms, multiply by c² = 9 × 10¹⁶ to get joules. In u, multiply by 931.5 to get MeV. Never mix them."
Put Z protons and N neutrons (free and separate) on one pan, and the nucleus they form on the other. The free nucleons are always heavier. The difference is the mass defect ΔM. Choose a nuclide and watch the balance.
Always positive: the nucleus is lighter than its free nucleons.
Tables give neutral-atom masses. m_H = m_p + m_e, so the Z electron masses cancel on both sides.
8(1.00727) + 8(1.00866) − 15.99053 = 0.13691 u, using the nuclear mass 15.99053 u.
"Defect means parts minus whole, never whole minus parts. If your mass defect comes out negative, you have subtracted the wrong way round."
The binding energy E_b = ΔM c² is the energy needed to separate a nucleus into its free nucleons, and the energy released when they come together. Pick a nuclide and step through the calculation.
m_H = 1.007825 u, m_n = 1.008665 u, M(⁴He) = 4.002603 u.
Use atomic masses throughout.
Keep all six decimals until the end.
u × 931.5 → MeV.
This is the number that tells you how tightly the nucleus is bound.
"Five lines, always in the same order: count, add, subtract, multiply by 931.5, divide by A. In NEET, most marks are lost by rounding the masses too early."
A big nucleus naturally has a big E_b, so to compare nuclei we divide by A: E_bn = E_b / A. The larger E_bn, the more tightly bound the nucleus. Drag the marker along the curve (or use the slider), and tap the labelled nuclides.
Nearly constant over a huge range of nuclei.
The most tightly bound nuclei. Iron-56 and nickel-62 sit at the top.
Both ends are less tightly bound than the middle. ⁴He is a spike (7.07 MeV), much higher than its neighbours.
"Stability is decided by E_bn, not E_b. Uranium has a huge total binding energy, about 1800 MeV, but each nucleon is held less tightly than in iron."
"If you could move a nucleus towards the peak of this curve, would energy come out or go in?"
Energy comes out. Moving up the curve means each nucleon becomes more tightly bound, and the extra binding energy is released. Light nuclei move up by joining (fusion); heavy nuclei move up by splitting (fission).
NCERT draws three conclusions from the curve. Explore each one.
A nucleon is bound only to the few neighbours within the range of the nuclear force. Adding more nucleons far away does not bind it any more tightly.
2 ²H → ⁴He: 4(7.07) − 4(1.11) ≈ 23.8 MeV released. This powers the sun (Lesson 5).
E_bn rises from about 7.6 to 8.5 MeV: 240 × 0.9 ≈ 216 MeV per split (Lesson 4).
"One curve, three exam answers: flat because the force is short ranged, fusion for light nuclei, fission for heavy ones. Both processes climb towards iron."
1 MeV = 1.6 × 10⁻¹³ J. 1 g of mass = 9 × 10¹³ J.
Atomic masses, so electrons cancel. Divide by A for E_bn.
With E_bn: Q = Σ A·E_bn (after) − Σ A·E_bn (before). 2 ²H → ⁴He: 4 × 7.1 − 4 × 1.1 = 24 MeV.
Compare E_bn, not E_b. ⁴He (7.07) is more tightly bound than ⁷Li (5.61) although ⁷Li has more total E_b.
Peak ≈ 8.75 MeV near A = 56; flat ≈ 8 MeV for 30 < A < 170.
Energy must be supplied to break a nucleus: binding energy is positive.
"For energy released questions, never subtract masses if E_bn values are given. Multiply each E_bn by its A, then subtract before from after. Thirty seconds, full marks."
Use m_H = 1.007825 u, m_n = 1.008665 u, 1 u = 931.5 MeV/c². Cover the solution, write the formula, and solve on paper.
How much energy is equivalent to 1 g of mass? Express it in joules and in kilowatt-hours.
E = mc² with m in kg; 1 kWh = 3.6 × 10⁶ J.
The atomic mass of ²H is 2.014102 u. Find the binding energy of the deuteron. What is the minimum energy needed to split it into a proton and a neutron?
ΔM = m_H + m_n − M(²H).
The atomic mass of ⁵⁶₂₆Fe is 55.934939 u. Find its binding energy and binding energy per nucleon.
Z = 26, N = 30.
The binding energies per nucleon of ²H and ⁴He are 1.1 MeV and 7.0 MeV. Find the energy released when two deuterons fuse to form one ⁴He nucleus.
Total E_b = A × E_bn for each nucleus. Q = after − before.
A nucleus with A = 240 and E_bn = 7.6 MeV splits into two nuclei of A = 120, each with E_bn = 8.5 MeV. Estimate the energy released.
Same 240 nucleons before and after; each gains 0.9 MeV of binding.
Take 1 u = 931.5 MeV/c², c = 3 × 10⁸ m/s. Write the formula first, then click.
Score: 0 / 12
| If the question says… | Write this first | The trap |
|---|---|---|
| "energy equivalent of 1 u" | 931.5 MeV | Using 1.6 × 10⁻¹⁹ or 9 × 10¹⁶ with u |
| "mass defect" | (Z m_p + N m_n) − M | Subtracting the other way and getting a negative value |
| "atomic masses are given" | use m_H, not m_p | Mixing m_p with atomic masses (electron error) |
| "binding energy per nucleon" | E_b / A | Dividing by Z or by N |
| "most stable nucleus" | largest E_bn (A ≈ 56) | Choosing the largest E_b (uranium) |
| "energy released, E_bn given" | Q = ΣA·E_bn(after) − ΣA·E_bn(before) | Subtracting E_bn values without multiplying by A |
| "why is E_bn nearly constant" | short range, saturation of nuclear force | Answering "Coulomb repulsion" (that lowers E_bn for heavy nuclei) |
| "energy from 1 g / 1 kg" | E = mc² with m in kg | Forgetting to convert grams to kilograms |
Lesson 4 explains the nuclear force behind the curve, radioactivity, and fission of uranium-235. Lesson 5 uses the climb of light nuclei up the curve to explain fusion and the energy of the sun.
Values used: m_H = 1.007825 u, m_n = 1.008665 u, m_p = 1.00727 u, m_e = 0.00055 u, 1 u = 1.6605 × 10⁻²⁷ kg = 931.5 MeV/c², c = 2.9979 × 10⁸ m/s, 1 MeV = 1.602 × 10⁻¹³ J. Atomic masses from standard tables (AME); E_bn curve from measured values.
Follow the four steps: understand it, remember it, try it, then prove it with past papers.