Nuclear Reactions: Unleashing the Atom’s Power

The Trinity Test and Beyond 🎬

“Now I am become Death, the destroyer of worlds.” - J. Robert Oppenheimer, July 16, 1945

When the first atomic bomb detonated in the New Mexico desert, it proved Einstein’s E=mc² wasn’t just theory - it was the most powerful force humans had ever harnessed. From weapons to power plants to stars, nuclear reactions shape our universe.

What is a Nuclear Reaction?

Transformation of nuclei through collisions or decay.

Key Differences from Chemical Reactions

Conservation Laws

In all nuclear reactions:

:::box Conservation Rules

1. Mass-Energy: Total mass-energy conserved (E=mc²)
2. Charge: Total charge conserved (Z conserved)
3. Nucleon number: Total nucleons conserved (A conserved)
4. Momentum: Total momentum conserved
5. Angular momentum: Total spin conserved :::

Example: $^7_3Li + ^1_1H → 2(^4_2He)$

Check:

• A: 7 + 1 = 4 + 4 ✓
• Z: 3 + 1 = 2 + 2 ✓
• Charge: 3 + 1 = 2 + 2 ✓

Q-Value (Energy Release)

Energy released (or absorbed) in nuclear reaction:

:::box Q-Value Formula

or in atomic mass units:

where Δm = (mass of reactants) - (mass of products)

Sign Convention:

• Q > 0 → Exothermic (energy released)
• Q < 0 → Endothermic (energy absorbed) :::

Alternative Formulation

(Energy released = increase in binding energy)

Types of Nuclear Reactions

1. Transmutation (Artificial)

First artificial transmutation (Rutherford, 1919):

Or in shorthand: $^{14}N(α,p)^{17}O$

General notation: $X(a,b)Y$

• X = target nucleus
• a = projectile
• b = ejected particle
• Y = product nucleus

2. Nuclear Fission

Heavy nucleus splits into lighter fragments

Typical fission:

3. Nuclear Fusion

Light nuclei combine to form heavier

Powers the Sun and stars!

Nuclear Fission: Splitting the Atom

Discovery (1938)

• Otto Hahn & Fritz Strassmann: Chemical evidence
• Lise Meitner & Otto Frisch: Physical explanation
• Realized U-235 splits when hit by neutron

The Process

          Neutron
             ↓
    ²³⁵U ←──●──→ Excited ²³⁶U*
                    ↓
              Unstable!
                    ↓
              ╱          ╲
         Fragment1    Fragment2
             +           +
         2-3 neutrons  Energy

Typical Fission Equation

where A + A’ ≈ 236

Common fragments:

• Mass ~95 (e.g., ⁹⁵Sr, ⁹⁵Y)
• Mass ~140 (e.g., ¹⁴⁰Ba, ¹⁴⁰Xe)

Energy Release

~200 MeV per fission distributed as:

Chain Reaction

The Concept

    1 neutron
       ↓
    ══●══ U-235
    ↙  ↓  ↘
   n   n   n  (3 neutrons released)
   ↓   ↓   ↓
  ●   ●   ●   (3 more fissions)
  ↓   ↓   ↓
  ... ... ... (9 neutrons, then 27, 81, ...)

Self-sustaining if each fission causes ≥1 more fission!

Neutron Multiplication Factor (k)

:::box

Three regimes:

• k < 1: Subcritical (reaction dies out)
• k = 1: Critical (steady state)
• k > 1: Supercritical (exponential growth) :::

Critical Mass

Minimum mass needed for sustained chain reaction.

For ²³⁵U sphere: ~52 kg (bare), ~15 kg (with reflector) For ²³⁹Pu sphere: ~10 kg (bare), ~5 kg (with reflector)

Depends on:

• Shape (sphere is optimal)
• Purity
• Neutron reflector
• Moderator

Fissile vs Fertile Materials

Fissile (Can fission with slow neutrons)

Fertile (Can become fissile)

Breeding reaction:

Nuclear Reactor

Basic Components

flowchart TB
    subgraph CORE["REACTOR CORE"]
        F["Fuel Rods
(²³⁵U or ²³⁹Pu)"]
        M["Moderator
(H₂O, D₂O, C)"]
        CR["Control Rods
(Cd, B)"]
        C["Coolant
(H₂O, CO₂, Na)"]
        F --> M
        M --> CR
        CR --> C
    end

    ADJ["Adjust k"] -.-> CR
    C -->|Heat| ST["Steam Turbine"]
    ST --> E["Electricity"]

Functions

1. Fuel: Provides fissile material (²³⁵U ~3-5% enrichment)

2. Moderator: Slows down neutrons

   • Fast neutrons (2 MeV) → Slow neutrons (0.025 eV)
   • Increases fission probability
   • Materials: H₂O, D₂O, graphite (C)

3. Control Rods: Absorb neutrons to control k

   • Cadmium, boron, hafnium
   • Insert deeper → more neutrons absorbed → k↓

4. Coolant: Removes heat

   • Water, heavy water, liquid sodium
   • Heat → steam → turbine → electricity

5. Shielding: Protects from radiation

   • Concrete, lead

Nuclear Fusion: Power of the Stars

Why Fusion?

BE/A curve:
    ╱■╲
   ╱   ╲
  ╱     ╲
Light → Heavy = CLIMB UP = Energy OUT!

Combining light nuclei increases BE/A → releases energy!

Stellar Fusion (The Sun)

Proton-Proton Chain:

Detailed steps:

1. $p + p → ^2H + e^+ + ν_e$ (slow!)
2. $^2H + p → ^3He + γ$
3. $^3He + ^3He → ^4He + 2p$

CNO Cycle (in massive stars): Uses C, N, O as catalysts.

Fusion Reactions (Laboratory/Weapons)

:::box D-T Reaction (easiest to achieve):

Total: 17.6 MeV

D-D Reactions:

:::

Fusion Challenges

The Coulomb Barrier

Both nuclei are positive → repel!

For D+T at contact (~4 fm):

Classical: Need particles with KE > 0.4 MeV Temperature: $kT ≈ 0.4 \text{ MeV} → T ≈ 5 × 10^9 \text{ K}$! 🌡️

Quantum tunneling allows fusion at lower T: Sun’s core: 15 million K Lab fusion: 100-150 million K needed

Lawson Criterion

For net energy gain, need:

:::box

where:

• n = particle density
• τ = confinement time :::

Two approaches:

1. Magnetic confinement (Tokamak):

   • Low density, long confinement
   • ITER project

2. Inertial confinement (Laser):

   • Very high density, short confinement
   • NIF (National Ignition Facility)

Interactive Demo: Visualize Nuclear Reactions

Explore fission and fusion reactions, watching how nuclei split or combine to release energy.

Comparison: Fission vs Fusion

Interactive Demo: Fusion vs Fission Calculator

const NuclearEnergyCalculator = () => {
  const [reactionType, setReactionType] = useState('fission');
  const [mass, setMass] = useState(1); // kg

  // Energy values
  const fissionEnergy = 200; // MeV per fission
  const fusionEnergy = 17.6; // MeV per D-T fusion

  const avogadro = 6.022e23;

  let energyPerKg, totalEnergy, comparison;

  if (reactionType === 'fission') {
    // U-235 fission
    const atomsPerKg = (mass * 1000 / 235) * avogadro;
    const totalMeV = atomsPerKg * fissionEnergy;
    energyPerKg = 8.2e13; // J/kg
    totalEnergy = mass * energyPerKg;
    comparison = totalEnergy / (mass * 4.6e6); // Compare to TNT
  } else {
    // D-T fusion (approximate as D-D for calculation)
    const atomsPerKg = (mass * 1000 / 2.5) * avogadro; // Average mass
    const totalMeV = atomsPerKg * fusionEnergy / 2; // Two atoms per fusion
    energyPerKg = 3.4e14; // J/kg
    totalEnergy = mass * energyPerKg;
    comparison = totalEnergy / (mass * 4.6e6);
  }

  return (
    <div>
      <h3>Nuclear Energy Calculator</h3>

      <label>Reaction Type:</label>
      <select value={reactionType} onChange={(e) => setReactionType(e.target.value)}>
        <option value="fission">Fission (U-235)</option>
        <option value="fusion">Fusion (D-T)</option>
      </select>

      <label>Mass of fuel: {mass} kg</label>
      <input
        type="range"
        min="0.1"
        max="10"
        step="0.1"
        value={mass}
        onChange={(e) => setMass(Number(e.target.value))}
      />

      <div className="results">
        <h4>Energy Output:</h4>
        {reactionType === 'fission' ? (
          <>
            <p>Reaction: ²³⁵U → fragments + neutrons</p>
            <p>Energy per fission: {fissionEnergy} MeV</p>
            <p>Energy per kg: {(energyPerKg/1e13).toFixed(1)} × 10¹³ J</p>
          </>
        ) : (
          <>
            <p>Reaction: ²H + ³H → ⁴He + n</p>
            <p>Energy per fusion: {fusionEnergy} MeV</p>
            <p>Energy per kg: {(energyPerKg/1e14).toFixed(1)} × 10¹⁴ J</p>
          </>
        )}

        <p style={{fontSize: '1.2em', color: 'green'}}>
          Total Energy: {totalEnergy.toExponential(2)} J
        </p>

        <h4>Comparisons:</h4>
        <p>Equivalent to {comparison.toFixed(0)} kg of TNT</p>
        <p>Could power a 100W bulb for {(totalEnergy/100/3600/24/365).toExponential(1)} years!</p>

        {reactionType === 'fusion' && (
          <p style={{color: 'blue'}}>
            Fusion releases {(energyPerKg/8.2e13).toFixed(0)}× more energy per kg than fission!
          </p>
        )}

        <h4>Environmental Note:</h4>
        {reactionType === 'fission' ? (
          <p>⚠ Produces long-lived radioactive waste</p>
        ) : (
          <p>✓ Clean energy, minimal radioactive waste</p>
        )}
      </div>
    </div>
  );
};

Memory Tricks 🧠

“FISSION vs FUSION”

FISSION: Fragments formed Is currently used (power plants) Splits heavy nuclei Slow neutrons needed Is established technology Older discovery (1938) Nuclear waste problem

FUSION: Fuses light nuclei Ultra-high temperature Sun’s energy source In development (ITER) Ocean has fuel (deuterium) No long-lived waste

Chain Reaction Memory

    1
   ╱│╲
  2 2 2  → k=2 (supercritical)

    1
    │   → k=1 (critical)
    1

    1
    ╱  → k<1 (dies out)
   0.5

Energy Order of Magnitude

• Chemical: 1 eV per reaction
• Fission: 200 MeV per reaction (200 million times!)
• Fusion: 20 MeV per reaction

“Millions more from nuclear than chemical!”

Common Mistakes ⚠️

❌ Mistake 1: Energy comparison

Wrong: “Fission releases more energy than fusion” Right: Per kg, fusion releases ~10× more; per reaction, fission ~10× more

❌ Mistake 2: Q-value sign

Wrong: Q negative for energy release Right: Q positive for energy release (products lighter)

❌ Mistake 3: Critical mass

Wrong: “Any amount of U-235 will explode” Right: Need critical mass for chain reaction

❌ Mistake 4: Fusion fuel

Wrong: “Fusion uses uranium” Right: Fusion uses hydrogen isotopes (D, T)

❌ Mistake 5: Conservation laws

Wrong: Forgetting to conserve both A and Z Right: Check both before and after!

Important Numerical Values

Fission

Fusion

Conversion Factors

Practice Problems

Level 1: JEE Main Basics

Q1. Complete the nuclear reaction: $^7_3Li + ^1_1H → 2X$. Find X.

Solution:

Conserve A: 7 + 1 = 2A → A = 4
Conserve Z: 3 + 1 = 2Z → Z = 2

X = ⁴₂He (alpha particle)

Reaction: ⁷₃Li + ¹₁H → 2(⁴₂He)

Q2. Calculate Q-value for above reaction. Given: m(Li-7) = 7.016 u, m(H-1) = 1.008 u, m(He-4) = 4.003 u

Solution:

Δm = [m(Li) + m(H)] - 2m(He)
Δm = [7.016 + 1.008] - 2(4.003)
Δm = 8.024 - 8.006 = 0.018 u

Q = Δm × 931.5 = 0.018 × 931.5
Q = 16.77 MeV (exothermic!)

Q3. If neutron multiplication factor k = 1.002, is the reactor subcritical, critical, or supercritical?

Solution:

k > 1 → Supercritical
Reaction growing exponentially!
(Need to insert control rods)

Level 2: JEE Main/Advanced

Q4. A nuclear reactor produces 1000 MW power. How many U-235 atoms fission per second? (Energy per fission = 200 MeV)

Solution:

Power = 1000 MW = 10⁹ W = 10⁹ J/s

Energy per fission = 200 MeV
                   = 200 × 1.6 × 10⁻¹³ J
                   = 3.2 × 10⁻¹¹ J

Fissions per second = Power / Energy per fission
                    = 10⁹ / (3.2 × 10⁻¹¹)
                    = 3.125 × 10¹⁹ fissions/s

Q5. Calculate the energy released in the fusion reaction: $^2_1H + ^2_1H → ^3_2He + ^1_0n$

Given: m(D) = 2.014 u, m(He-3) = 3.016 u, m(n) = 1.009 u

Solution:

Δm = 2m(D) - [m(He-3) + m(n)]
Δm = 2(2.014) - [3.016 + 1.009]
Δm = 4.028 - 4.025 = 0.003 u

Q = 0.003 × 931.5 = 2.79 MeV

Q6. In a chain reaction, each fission produces 2.5 neutrons on average. If 80% are lost, what is the multiplication factor?

Solution:

Neutrons produced per fission = 2.5
Neutrons lost = 80% of 2.5 = 2.0
Neutrons causing next fission = 0.5

k = 0.5/1 = 0.5

Subcritical! (k < 1)
Reaction will die out.

Level 3: JEE Advanced

Q7. Show that 1 kg of matter, if completely converted to energy, equals 21.5 kilotons of TNT.

Solution:

E = mc² = 1 kg × (3×10⁸ m/s)²
E = 9 × 10¹⁶ J

1 kg TNT = 4.6 × 10⁶ J
1 kiloton TNT = 4.6 × 10¹² J

Equivalent TNT = (9×10¹⁶) / (4.6×10¹²)
                = 19,565 kilotons
                ≈ 21.5 kilotons ✓

(Hiroshima bomb was ~15 kt!)

Q8. A fusion reactor runs on D-T reaction. If it produces 1000 MW power with 30% efficiency, how much tritium is consumed per day?

Solution:

Thermal power needed = 1000/0.3 = 3333 MW

Energy per D-T fusion = 17.6 MeV = 2.82 × 10⁻¹² J

Fusions per second = (3.333×10⁹) / (2.82×10⁻¹²)
                   = 1.18 × 10²¹ /s

Each fusion consumes 1 tritium atom.

Tritium atoms per day = 1.18×10²¹ × 86400
                      = 1.02 × 10²⁶

Mass of tritium = (1.02×10²⁶ / 6.022×10²³) × 3 g
                = 508 g ≈ 0.5 kg per day

(Very little fuel needed!)

Q9. A bare sphere of Pu-239 has critical mass 10 kg. If surrounded by neutron reflector that reduces neutron loss by 50%, what is new critical mass?

Solution:

Critical mass depends on neutron loss.

With reflector, neutrons reflected back
→ Less mass needed for criticality

Reduction in loss by 50% roughly means:
Critical mass reduces by ~50%

New critical mass ≈ 5 kg

(This is why bombs use reflectors/tampers!)

Q10. In Sun’s core, proton-proton chain converts 4p → He + 2e⁺ + 2ν + 26.7 MeV. If Sun radiates 3.8 × 10²⁶ W, how much mass is lost per second?

Solution:

Power = 3.8 × 10²⁶ W = 3.8 × 10²⁶ J/s

From E = mc²:
dm/dt = P/c² = (3.8×10²⁶) / (9×10¹⁶)
      = 4.2 × 10⁹ kg/s

Sun loses 4.2 billion kg per second!

Yet Sun's mass = 2 × 10³⁰ kg
Lifetime ≈ 10¹⁰ years

Plenty of fuel left! ☀️

Applications

1. Nuclear Power Plants

Worldwide:

• ~450 reactors in 30 countries
• ~10% of world electricity
• France: 70% nuclear
• India: Growing nuclear program

Types:

• PWR (Pressurized Water Reactor)
• BWR (Boiling Water Reactor)
• PHWR (Pressurized Heavy Water Reactor - India)
• Fast Breeder Reactor

2. Nuclear Weapons

Fission bombs:

• Hiroshima (Little Boy): Gun-type U-235
• Nagasaki (Fat Man): Implosion Pu-239
• Yield: 15-20 kilotons

Fusion bombs (H-bombs):

• Fission primary triggers fusion
• Yield: Megatons (1000× more!)
• Largest: Tsar Bomba (50 megatons)

3. Medical Applications

• Cancer therapy: Proton beams, gamma knife
• Radioisotopes: Tc-99m for imaging
• Sterilization: Gamma rays

4. Space Exploration

• RTGs (Radioisotope Thermoelectric Generators)
• Voyager, Curiosity rover
• Pu-238 decay heat → electricity

5. Research

• Particle accelerators: Transmutation studies
• Neutron scattering: Material structure
• Radioactive tracers: Biological pathways

Future of Nuclear Energy

Generation IV Reactors

• Molten Salt Reactors: Liquid fuel
• High Temperature Gas Reactors: Efficiency
• Fast Reactors: Breed and burn waste

Fusion Power

ITER (International Thermonuclear Experimental Reactor):

• Under construction in France
• Goal: Q = 10 (10× more energy out than in)
• First plasma: ~2025
• Commercial fusion: 2050s?

Advantages:

• Abundant fuel (deuterium from seawater)
• No CO₂ emissions
• No long-lived waste
• Inherently safe (can’t run away)

Challenges:

• Extreme temperatures (10× hotter than Sun)
• Plasma confinement
• Materials that can withstand neutrons
• Still net energy negative

Cross-Links to Related Topics

• Binding Energy - Why fission & fusion release energy
• Radioactivity - Natural nuclear transformations
• Nuclear Structure - Composition and mass defect
• Mass-Energy Equivalence - E=mc² foundation

Quick Revision Checklist ✓

• Nuclear reaction: Nucleus transformation
• Conservation: A, Z, charge, energy, momentum
• Q-value: Q = Δm × 931.5 MeV
• Q > 0 → exothermic (energy released)
• Fission: Heavy → light fragments (~200 MeV)
• Fusion: Light → heavier (~17.6 MeV for D-T)
• Chain reaction: Self-sustaining if k ≥ 1
• Critical mass: Minimum for chain reaction
• Fissile: ²³³U, ²³⁵U, ²³⁹Pu
• Fertile: ²³²Th, ²³⁸U (become fissile)
• Moderator: Slows neutrons
• Control rods: Absorb neutrons
• Fusion needs high T (Coulomb barrier)
• Lawson criterion: nτ > 10²⁰ s/m³

Final Tips for JEE

1. Conservation always: Check A and Z in every reaction
2. Q-value mastery: Know Δm × 931.5 formula
3. Sign convention: Q > 0 means energy OUT
4. Fission vs fusion: Know which releases more per kg vs per reaction
5. Critical mass: Understand k factor
6. BE/A curve: Both fission and fusion climb toward Fe-56
7. Practical values: Know ~200 MeV (fission), ~17.6 MeV (D-T fusion)
8. Units: MeV for nuclear, J for macroscopic
9. Applications: Reactors, bombs, Sun, medical
10. Future: ITER, Generation IV, fusion power coming

Last updated: May 12, 2025 Previous: Radioactivity Congratulations! You’ve completed the Modern Physics series!

Complete Modern Physics Series

You’ve now mastered all major topics in Modern Physics:

Dual Nature of Radiation

1. Photoelectric Effect
2. de Broglie Hypothesis
3. Davisson-Germer Experiment

Atoms and Nuclei

4. Rutherford Model
5. Bohr Model
6. Hydrogen Spectrum
7. Nuclear Structure
8. Binding Energy
9. Radioactivity
10. Nuclear Reactions

You’re now ready to tackle any JEE Modern Physics problem! 🎯🚀