Mass–Energy 💡

Einstein showed that mass itself stores energy. His famous relation is
\[ E = mc^{2}\tag{13.6} \]:contentReference[oaicite:0]{index=0}
Here E is the energy tied up in a mass m, and c ≈ 3 × 108 m s–1 is the speed of light.

Quick check-in ⚡

  • Converting just 1 g of matter releases \[ E = 10^{-3}\,(3\times10^{8})^{2} = 9\times10^{13}\;\text{J}. \]:contentReference[oaicite:1]{index=1} That is enough to power a city for days!

Nuclear Density 🌌

Because the nucleus packs lots of mass into a tiny volume, its density is mind-boggling. For the 56Fe nucleus (A=56) we find \[ \rho = \frac{\text{mass}}{\text{volume}} \approx 2.29\times10^{17}\;\text{kg m}^{-3}. \]:contentReference[oaicite:2]{index=2} Neutron stars squeeze matter to a similar density – they are like gigantic nuclei in space!

Nuclear Binding Energy 🔗

Mass defect (ΔM)

When neutrons and protons join to form a nucleus, the total mass drops just a little. That “missing” mass is called the mass defect \[ \Delta M = \bigl[ Zm_{p} + (A-Z)m_{n} – M \bigr]\tag{13.7} \]:contentReference[oaicite:3]{index=3} with Z protons, A – Z neutrons, and nuclear mass M.

Binding energy (Eb)

The energy that glues the nucleons together is \[ E_{b}= \Delta M\,c^{2}\tag{13.8} \]:contentReference[oaicite:4]{index=4} For example, in 16O the mass defect is 0.13691 u, giving \(E_{b}\approx127.5\;\text{MeV}\).:contentReference[oaicite:5]{index=5}

Binding energy per nucleon (Ebn)

A handy “stickiness index” is \[ E_{bn}= \frac{E_{b}}{A}\tag{13.9} \]:contentReference[oaicite:6]{index=6} It tells us the average energy needed to pull one nucleon out of the nucleus.

How Ebn changes with mass number 📈

  • For mid-size nuclei (30 < A < 170), \(E_{bn}\) stays almost flat at about 8 MeV.
  • The peak lies at A ≈ 56 with \(E_{bn}\approx8.75\;\text{MeV}\). Iron is close to this sweet spot.
  • Very light (A < 30) and very heavy (A > 170) nuclei have lower \(E_{bn}\).:contentReference[oaicite:7]{index=7}

Why does the curve flatten? 🤔

Each nucleon feels the strong nuclear force only from its few closest neighbors. This saturation means adding extra nucleons deep inside barely changes the binding energy per nucleon.:contentReference[oaicite:8]{index=8}

Energy tricks we exploit 🚀

  1. Fission: Split a very heavy nucleus (e.g., A ≈ 240) into two mid-mass fragments (≈120). Nucleons become more tightly bound, and energy is released.:contentReference[oaicite:9]{index=9}
  2. Fusion: Fuse two very light nuclei (A ≤ 10) into a heavier one. Again, binding per nucleon jumps, releasing energy – this powers the Sun!:contentReference[oaicite:10]{index=10}

NEET Fast-Track 🌟

  • \(E = mc^{2}\) – know it, love it, use it.
  • Mass defect → Binding energy via \(E_{b}= \Delta M\,c^{2}\).
  • Binding energy per nucleon peaks around A ≈ 56; drives both fission and fusion energy.
  • 1 u corresponds to 931.5 MeV of energy – a favorite conversion!:contentReference[oaicite:11]{index=11}
  • Nuclear density ≈ 1017 kg m–3 – illustrates the incredible compactness of nuclei.

Keep exploring – the nucleus is small but packs a punch! 💥