Nuclear Chemistry

Radioactivity, Decay, and Nuclear Reactions

Types of Radioactive Decay

Alpha (α) Decay

₂₃₈U → ₂₃₄Th + ₄He²⁺ (α particle)

Particle: Helium nucleus (₂⁴He or ₂⁴α)

Mass number: decreases by 4

Atomic number: decreases by 2

Beta (β⁻) Decay

₁₄C → ₁₄N + ₀e⁻ (β⁻ particle)

Particle: Electron (₋₁⁰e or ₋₁⁰β)

Mass number: unchanged

Atomic number: increases by 1

n → p⁺ + e⁻ + ν̄e (antineutrino)

Beta (β⁺) Decay (Positron Emission)

₂₂Na → ₂₂Ne + ₀e⁺ (β⁺ particle)

Particle: Positron (₊₁⁰e or ₊₁⁰β)

Mass number: unchanged

Atomic number: decreases by 1

Gamma (γ) Radiation

₆₀Co* → ₆₀Co + γ

Particle: High-energy photon (₀⁰γ)

Mass number: unchanged

Atomic number: unchanged

Often accompanies other decay types

Radioactive Decay Kinetics

First-Order Decay Law

N(t) = N₀ e-λt

or

ln(N/N₀) = -λt

N(t) = number of nuclei at time t

Nâ‚€ = initial number of nuclei

λ = decay constant (s⁻¹)

t = time elapsed

Half-Life (t1/2)

t1/2 = ln(2) / λ = 0.693 / λ

Also:

N(t) = N₀ (1/2)t/t₁/₂

Definition: Time required for half of the radioactive nuclei to decay

Activity (A)

A = λN = A₀ e-λt

Activity: Rate of decay (decays per second)

Units:

• Becquerel (Bq) = 1 decay/s

• Curie (Ci) = 3.7 × 10¹⁰ Bq

Mass-Energy Equivalence

Einstein's Equation

E = mc²

E = energy (J)

m = mass (kg)

c = speed of light = 3.00 × 10⁸ m/s

Mass Defect (Δm)

Δm = (Σmreactants - Σmproducts)

For nucleus:

Δm = [Zmp + Nmn] - mnucleus

Z = number of protons

N = number of neutrons

mp = 1.00728 u (proton mass)

mn = 1.00867 u (neutron mass)

1 u = 1.66054 × 10⁻²⁷ kg = 931.5 MeV/c²

Binding Energy (BE)

BE = Δm × c²

In MeV:

BE (MeV) = Δm (u) × 931.5

Binding energy: Energy required to disassemble nucleus into separate nucleons

Higher BE/nucleon: More stable nucleus

Peak stability: Iron-56 (⁵⁶Fe) at ~8.8 MeV/nucleon

Binding Energy Per Nucleon

BE/nucleon = BE / A

A = mass number (total nucleons)

Worked Examples

Example 1: Half-Life Calculation

Problem: ¹⁴C has t1/2 = 5,730 years. How much of a 100 g sample remains after 17,190 years?

Solution:

Number of half-lives: n = t / t1/2

n = 17,190 / 5,730 = 3.00 half-lives

N(t) = N₀ × (1/2)n

N(t) = 100 g × (1/2)³

N(t) = 100 g × (1/8)

N(t) = 12.5 g remains

87.5 g has decayed to ¹⁴N

Example 2: Binding Energy Calculation

Problem: Calculate the binding energy of ⁴He (mass = 4.00260 u).

Solution:

⁴He has 2 protons, 2 neutrons

Mass of separated nucleons:

2 × 1.00728 u + 2 × 1.00867 u = 4.03190 u

Mass defect:

Δm = 4.03190 - 4.00260 = 0.02930 u

Binding energy:

BE = 0.02930 u × 931.5 MeV/u

BE = 27.3 MeV

BE/nucleon = 27.3 / 4 = 6.83 MeV/nucleon

Example 3: Decay Constant

Problem: ¹³¹I has t1/2 = 8.0 days. Calculate λ and the fraction remaining after 24 days.

Solution:

λ = 0.693 / t1/2

λ = 0.693 / 8.0 days = 0.0866 day⁻¹

Fraction remaining:

N/N₀ = e-λt = e-0.0866×24

N/Nâ‚€ = e-2.08 = 0.125

12.5% remains (1/8 of original)

This is 3 half-lives: (1/2)³ = 1/8 ✓

Common Half-Lives

IsotopeDecay ModeHalf-LifeUse/Note
¹⁴Cβ⁻5,730 yearsCarbon dating
²³⁸Uα4.5 × 10⁹ yearsGeological dating
¹³¹Iβ⁻8.0 daysMedical imaging
⁹⁹ᵐTcγ6.0 hoursMedical imaging
²²⁶Raα1,600 yearsHistorical radium
³H (tritium)β⁻12.3 yearsExit signs, tracers

Common Mistakes

⚠️

Confusing Mass Number and Atomic Number Changes

Alpha: A-4, Z-2; Beta⁻: A same, Z+1; Beta⁺: A same, Z-1

⚠️

Wrong Unit Conversions

1 u = 931.5 MeV/c², not just 931.5 MeV!

⚠️

Mass Defect Sign

For stable nuclei, mass defect is POSITIVE (products lighter than reactants)

Related Tools

Nuclear Binding Energy Calculator

Mass defect and E = Δmc²

Nuclear Binding Energy

Binding energy formula

Half-Life Formula

Radioactive decay

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