Radioactivity: Nature’s Nuclear Clock

The Marie Curie Legacy 🎬

Before Oppenheimer harnessed nuclear energy, Marie Curie discovered radioactivity - winning two Nobel Prizes and revolutionizing physics! Her work with polonium and radium revealed that atoms aren’t immutable - they can spontaneously transform, ticking away like cosmic clocks.

“Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less.” - Marie Curie

Discovery of Radioactivity

Historical Timeline

1896: Henri Becquerel

• Discovered uranium emits “rays”
• Accidentally left uranium salt on photographic plate
• Plate darkened even wrapped in black paper!
• Not phosphorescence - something new! ⚛️

1898: Marie & Pierre Curie

• Discovered polonium (Po) and radium (Ra)
• Coined term “radioactivity”
• Isolated radioactive elements

1899: Rutherford

• Identified α and β radiation
• Different penetrating powers

1900: Villard

• Discovered γ radiation
• Most penetrating type

Types of Radioactive Decay

α-Decay (Alpha Decay)

Emission of helium nucleus ($^4_2He$ or α)

:::box

Changes:

• Mass number: A → A - 4
• Atomic number: Z → Z - 2
• New element (2 places left in periodic table) :::

Example:

Characteristics:

• Heavy nuclei (A > 200) typically
• Low penetration (~few cm in air)
• Stopped by paper
• High ionization
• Energy: 4-9 MeV typically

β⁻-Decay (Beta-minus Decay)

Neutron converts to proton

:::box

Changes:

• Mass number: A unchanged
• Atomic number: Z → Z + 1
• New element (1 place right in periodic table)
• Electron and antineutrino emitted :::

Example:

Characteristics:

• Neutron-rich nuclei
• Moderate penetration (~few mm in Al)
• Continuous energy spectrum (due to neutrino)
• Medium ionization

β⁺-Decay (Beta-plus/Positron Decay)

Proton converts to neutron

:::box

Changes:

• Mass number: A unchanged
• Atomic number: Z → Z - 1
• New element (1 place left)
• Positron and neutrino emitted :::

Example:

Note: Requires M(parent) > M(daughter) + 2m_e (threshold!)

Electron Capture (EC)

Nucleus captures inner shell electron

:::box

Changes:

• Mass number: A unchanged
• Atomic number: Z → Z - 1
• X-rays emitted (electron vacancy filled)
• Competes with β⁺ decay :::

Example:

γ-Decay (Gamma Decay)

Excited nucleus releases energy

:::box

Changes:

• No change in A or Z!
• Same element, lower energy state
• Pure electromagnetic radiation
• Often follows α or β decay :::

Example:

Characteristics:

• Very high penetration (many cm of Pb)
• No mass or charge
• Electromagnetic wave (photon)
• Low ionization
• Energy: keV to MeV

Comparison of Radiation Types

Radioactive Decay Law

The Fundamental Law

:::box Decay Law

where:

• N(t) = number of nuclei at time t
• N₀ = initial number of nuclei
• λ = decay constant (characteristic of isotope)
• t = time :::

Meaning: Radioactive decay is:

• Random (can’t predict individual decay)
• Spontaneous (doesn’t need external trigger)
• Exponential (constant fraction decays per unit time)

Activity (Rate of Decay)

:::box

Units:

• SI: Becquerel (Bq) = 1 decay/second
• Old: Curie (Ci) = 3.7 × 10¹⁰ Bq :::

Half-Life (T₁/₂)

Time for half the nuclei to decay:

:::box Half-Life Formula

After n half-lives:

where $n = \frac{t}{T_{1/2}}$ :::

Interactive Demo: Visualize Nuclear Decay

Watch radioactive nuclei decay exponentially over time, demonstrating half-life concepts.

Mean Life (τ)

Average lifetime of a nucleus:

:::box

:::

Relation:

Interactive Demo: Decay Calculator

const RadioactiveDecaySimulator = () => {
  const [isotope, setIsotope] = useState('C14');
  const [time, setTime] = useState(5730);
  const [N0, setN0] = useState(1000000);

  const halfLives = {
    'C14': 5730, // years
    'U238': 4.47e9, // years
    'Ra226': 1600, // years
    'I131': 8.02, // days
    'Co60': 5.27 // years
  };

  const T_half = halfLives[isotope];
  const lambda = 0.693 / T_half;
  const N_t = N0 * Math.exp(-lambda * time);
  const A_t = lambda * N_t;
  const A_0 = lambda * N0;
  const n_halfLives = time / T_half;
  const percentRemaining = (N_t / N0) * 100;

  return (
    <div>
      <h3>Radioactive Decay Simulator</h3>

      <label>Isotope:</label>
      <select value={isotope} onChange={(e) => setIsotope(e.target.value)}>
        <option value="C14">Carbon-14 (T½=5730 y)</option>
        <option value="U238">Uranium-238 (T½=4.47 Gy)</option>
        <option value="Ra226">Radium-226 (T½=1600 y)</option>
        <option value="I131">Iodine-131 (T½=8 d)</option>
        <option value="Co60">Cobalt-60 (T½=5.27 y)</option>
      </select>

      <label>Initial nuclei: {N0.toExponential(2)}</label>
      <input
        type="range"
        min="100000"
        max="10000000"
        step="100000"
        value={N0}
        onChange={(e) => setN0(Number(e.target.value))}
      />

      <label>Time elapsed: {time} {isotope === 'I131' ? 'days' : 'years'}</label>
      <input
        type="range"
        min="0"
        max={T_half * 3}
        value={time}
        onChange={(e) => setTime(Number(e.target.value))}
      />

      <div className="results">
        <h4>Decay Parameters:</h4>
        <p>Half-life: {T_half} {isotope === 'I131' ? 'days' : 'years'}</p>
        <p>Decay constant: {lambda.toExponential(3)} per year</p>
        <p>Mean life: {(1/lambda).toFixed(0)} {isotope === 'I131' ? 'days' : 'years'}</p>

        <h4>At time t = {time}:</h4>
        <p>Number of half-lives: {n_halfLives.toFixed(2)}</p>
        <p>Remaining nuclei: {N_t.toExponential(2)}</p>
        <p>Percent remaining: {percentRemaining.toFixed(1)}%</p>
        <p>Initial activity: {A_0.toExponential(2)} Bq</p>
        <p>Current activity: {A_t.toExponential(2)} Bq</p>

        <div className="visual">
          <div style={{
            width: '100%',
            height: '30px',
            backgroundColor: '#ccc',
            position: 'relative'
          }}>
            <div style={{
              width: `${percentRemaining}%`,
              height: '100%',
              backgroundColor: 'blue',
              transition: 'width 0.3s'
            }} />
          </div>
          <p>Visual: Fraction remaining</p>
        </div>

        {percentRemaining < 1 && (
          <p style={{color: 'red'}}>
            ⚠ Less than 1% remaining - essentially all decayed!
          </p>
        )}
      </div>
    </div>
  );
};

Decay Series (Radioactive Chains)

Some isotopes decay through multiple steps:

Uranium-238 Series

²³⁸U (4.5 Gy)
  ↓ α
²³⁴Th (24 d)
  ↓ β⁻
²³⁴Pa (1.2 min)
  ↓ β⁻
²³⁴U (246,000 y)
  ↓ α
  ... (many steps)
  ↓
²⁰⁶Pb (stable)

Total: 14 steps, ending at stable lead-206!

Other Series

1. Thorium-232 series → ²⁰⁸Pb
2. Uranium-235 series → ²⁰⁷Pb
3. Neptunium-237 series → ²⁰⁹Bi (extinct in nature)

Memory Tricks 🧠

“ALPHA BETA GAMMA” Comparison

Alpha: All Atomic mass lost (4 units) Large and heavy (He nucleus) Paper stops it High ionization Always from heavy nuclei

Beta: Basically electron Energy spectrum continuous Transmutation (Z changes by 1) Aluminum stops it

Gamma: Go through everything And no charge Mass-less photon Maximum penetration After other decays usually

Decay Formula Memory

“Number equals N-zero times e to the minus lambda-t”

Half-life:

“Point-six-nine-three over lambda”

Half-Life Pattern

After 1 T½: ½ remains
After 2 T½: ¼ remains
After 3 T½: ⅛ remains
After n T½: (½)ⁿ remains

"Every half-life, half remains!"

Common Mistakes ⚠️

❌ Mistake 1: Half-life formula

Wrong: T₁/₂ = 1/λ Right: T₁/₂ = 0.693/λ (or ln2/λ)

❌ Mistake 2: Activity vs number

Wrong: A = N Right: A = λN (activity is rate, not number!)

❌ Mistake 3: Decay direction

Wrong: α decay increases Z Right: α decay decreases Z by 2, A by 4

❌ Mistake 4: Beta decay mass

Wrong: β⁻ decay decreases mass number Right: β⁻ keeps A same, only Z changes

❌ Mistake 5: Exponential vs linear

Wrong: N = N₀ - λt (linear decay) Right: N = N₀e^(-λt) (exponential decay)

Important Half-Lives

Common Isotopes

Carbon-14 Dating

The Principle

1. Cosmic rays create ¹⁴C in atmosphere
2. Living organisms maintain constant ¹⁴C/¹²C ratio
3. After death, ¹⁴C decays (T₁/₂ = 5730 y)
4. Measure remaining ¹⁴C → determine age!

:::box

or

where A₀ is activity in living material, A is current activity. :::

Useful range: Up to ~50,000 years

Example: Dead wood has ¼ the ¹⁴C activity of living wood.

Equilibrium in Decay Series

Secular Equilibrium

Parent half-life » daughter half-life:

Eventually:

Daughter decays as fast as it’s produced!

Example: ²³⁸U → ²³⁴Th system

Transient Equilibrium

Parent half-life > daughter half-life (but comparable):

Practice Problems

Level 1: JEE Main Basics

Q1. Complete the decay: $^{238}_{92}U → ? + ^4_2He$

Solution:

α decay: A decreases by 4, Z decreases by 2

A: 238 - 4 = 234
Z: 92 - 2 = 90

²³⁴₉₀Th (Thorium-234)

Q2. A radioactive sample has half-life 10 days. What fraction remains after 30 days?

Solution:

n = t/T₁/₂ = 30/10 = 3 half-lives

Fraction = (1/2)³ = 1/8

Q3. The decay constant of a sample is 0.01 per year. Find the half-life.

Solution:

T₁/₂ = 0.693/λ = 0.693/0.01 = 69.3 years

Level 2: JEE Main/Advanced

Q4. ²²⁶Ra (T₁/₂ = 1600 y) decays to ²²²Rn (T₁/₂ = 3.8 d). Initially pure Ra sample. After long time, what is ratio of Ra to Rn decay rates?

Solution:

At secular equilibrium:
λ_Ra × N_Ra = λ_Rn × N_Rn

Decay rates: A_Ra = A_Rn

Ratio = 1:1

Even though numbers are different (N_Rn << N_Ra),
decay rates are equal at equilibrium!

Q5. A sample’s activity decreases to 6.25% of original in 2 hours. Find: (a) Half-life (b) Decay constant

Solution:

A/A₀ = 6.25/100 = 1/16 = (1/2)⁴

4 half-lives in 2 hours
T₁/₂ = 2/4 = 0.5 hours = 30 minutes

λ = 0.693/T₁/₂ = 0.693/0.5 = 1.386 h⁻¹

Q6. A ¹⁴C sample has activity 15 disintegrations/minute/gram. Living carbon has 16 dpm/g. Find age of sample. (T₁/₂ = 5730 y)

Solution:

A/A₀ = 15/16

t = (T₁/₂/ln2) × ln(A₀/A)
t = (5730/0.693) × ln(16/15)
t = 8267 × ln(1.0667)
t = 8267 × 0.0645
t ≈ 533 years

Level 3: JEE Advanced

Q7. Two radioactive samples A and B have equal activities initially. After 4 days, activity of A is 1/16 and B is 1/4 of initial. Find: (a) Ratio of half-lives (b) Ratio of activities after 8 days

Solution:

(a) For A: A_A/A₀ = 1/16 = (1/2)⁴
    4 days = 4 half-lives of A
    T_A = 1 day

    For B: A_B/A₀ = 1/4 = (1/2)²
    4 days = 2 half-lives of B
    T_B = 2 days

    T_A : T_B = 1:2

(b) After 8 days:
    For A: n = 8/1 = 8 half-lives
    A_A = A₀(1/2)⁸ = A₀/256

    For B: n = 8/2 = 4 half-lives
    A_B = A₀(1/2)⁴ = A₀/16

    A_A/A_B = (1/256)/(1/16) = 16/256 = 1/16

Q8. ²³⁸U decays to ²⁰⁶Pb with T₁/₂ = 4.5 × 10⁹ years. A rock has U:Pb ratio of 1:1. Find age of rock.

Solution:

Initially: N₀ of U, 0 of Pb
After time t: N of U, (N₀-N) of Pb

Given: N/(N₀-N) = 1/1
∴ N = N₀ - N
∴ 2N = N₀
∴ N = N₀/2

One half-life has passed!
Age = T₁/₂ = 4.5 × 10⁹ years

Actually, should verify:
N/N₀ = e^(-λt) = 1/2
λt = ln2
t = (ln2)/λ = T₁/₂ ✓

Q9. A radioactive sample undergoes two types of decay with constants λ₁ and λ₂. Show that effective half-life is:

Solution:

Effective decay constant: λ_eff = λ₁ + λ₂

(Each decay mode contributes independently)

Total decay rate:
dN/dt = -(λ₁ + λ₂)N = -λ_eff N

Solution: N = N₀ e^(-λ_eff t)

Half-life: N₀/2 = N₀ e^(-λ_eff T_eff)
1/2 = e^(-λ_eff T_eff)
ln(1/2) = -λ_eff T_eff
-ln2 = -λ_eff T_eff

T_eff = ln2/λ_eff = ln2/(λ₁ + λ₂) ✓

Physical meaning: Multiple decay paths
→ faster overall decay → shorter half-life

Q10. Show that mean life τ = 1/λ equals the average lifetime of radioactive nuclei.

Solution:

Mean life = ∫₀^∞ t × P(t) dt

where P(t) = probability of decay at time t

P(t)dt = (fraction surviving to t) × (prob of decay in dt)
       = (N/N₀) × (λ dt)
       = e^(-λt) × λ dt

τ = ∫₀^∞ t × λ e^(-λt) dt

Integration by parts:
Let u = t, dv = λe^(-λt)dt
du = dt, v = -e^(-λt)

τ = [-te^(-λt)]₀^∞ + ∫₀^∞ e^(-λt) dt
  = 0 + [-e^(-λt)/λ]₀^∞
  = 0 - (-1/λ)
  = 1/λ ✓

Also: τ = T₁/₂/ln2 = T₁/₂/0.693 ≈ 1.44 T₁/₂

Applications of Radioactivity

1. Medical

• Diagnosis: PET scans (¹⁸F), X-rays
• Treatment: Cancer therapy (⁶⁰Co, ¹³¹I)
• Sterilization: γ-rays kill bacteria

2. Dating

• Carbon-14: Organic materials (50,000 y)
• Uranium-Lead: Rocks (millions of years)
• Potassium-Argon: Volcanic rocks

3. Industrial

• Thickness gauges: β-rays
• Smoke detectors: ²⁴¹Am (α-emitter)
• Tracers: Follow fluid flow

4. Research

• Rutherford scattering: α-particles
• Tracer studies: Biological pathways
• Neutron activation: Element analysis

Nuclear Stability

Stable vs Unstable

Stable nuclei:

• Balanced N/Z ratio
• “Valley of stability” on N-Z plot
• ~270 stable isotopes

Unstable nuclei (radioactive):

• Too many neutrons → β⁻ decay
• Too many protons → β⁺ decay or EC
• Too heavy → α decay
• Excited state → γ decay

N-Z Plot (Stability Curve)

N
│       Heavy
│      (α decay)
│    ╱
│   ╱  Neutron-rich
│  ╱   (β⁻ decay)
│ ╱
│╱______ Stable
│     ╱ (N≈Z for light,
│    ╱   N>Z for heavy)
│   ╱ Proton-rich
│  ╱  (β⁺ or EC)
│ ╱
└─────────────→ Z
    Light

Cross-Links to Related Topics

• Nuclear Structure - Composition of nucleus
• Binding Energy - Why some nuclei unstable
• Nuclear Reactions - Induced transmutations
• Rutherford Model - α-scattering experiments

Quick Revision Checklist ✓

• α decay: A-4, Z-2, ⁴He emitted
• β⁻ decay: A same, Z+1, e⁻ emitted
• β⁺ decay: A same, Z-1, e⁺ emitted
• γ decay: A and Z same, photon emitted
• Decay law: N = N₀e^(-λt)
• Activity: A = λN
• Half-life: T₁/₂ = 0.693/λ
• After n half-lives: N = N₀(1/2)^n
• Mean life: τ = 1/λ = 1.44T₁/₂
• Penetration: α < β < γ
• Ionization: α > β > γ
• ¹⁴C dating: T₁/₂ = 5730 years

Final Tips for JEE

1. Master decay equations: Know what changes in each type
2. Exponential decay: Not linear! N = N₀e^(-λt)
3. Half-life formula: T₁/₂ = 0.693/λ (not 1/λ!)
4. Activity = rate: A = λN (dN/dt)
5. Conservation laws: Check A and Z before/after
6. Units: Bq (SI) vs Ci (old)
7. Carbon dating: Know T₁/₂ = 5730 years
8. Penetration order: α < β < γ (remember!)
9. Multiple half-lives: Use (1/2)^n
10. Physical meaning: Random, spontaneous, exponential

Last updated: May 8, 2025 Previous: Binding Energy Next: Nuclear Reactions