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
N₀ = 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/N₀ = 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
| Isotope | Decay Mode | Half-Life | Use/Note |
|---|---|---|---|
| ¹⁴C | β⁻ | 5,730 years | Carbon dating |
| ²³⁸U | α | 4.5 × 10⁹ years | Geological dating |
| ¹³¹I | β⁻ | 8.0 days | Medical imaging |
| ⁹⁹ᵐTc | γ | 6.0 hours | Medical imaging |
| ²²⁶Ra | α | 1,600 years | Historical radium |
| ³H (tritium) | β⁻ | 12.3 years | Exit 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)