2.3 Nuclear Chemistry: Radiation, Decay Kinetics & Nuclear Reactions
Key Takeaways
- Nuclear stability depends on the balance between electrostatic proton repulsion and the attractive strong nuclear force, establishing a stable neutron-to-proton (N/Z) ratio ranging from 1.0 in light nuclei to 1.5 in heavy elements.
- Radionuclides decay toward stability via alpha emission, beta-minus emission, positron emission, electron capture, or spontaneous fission while strictly conserving mass number and atomic number.
- Radioactive decay follows first-order kinetics defined by ln(N_t/N_0) = -kt and a constant half-life t_1/2 = 0.693/k, enabling precise radiometric dating and diagnostic medical dosimetry.
- Nuclear binding energy arises from the mass defect via E = (Δm)c², with peak stability near iron-56 explaining why nuclear fission of heavy nuclei and nuclear fusion of light nuclei both liberate immense energy.
2.3 Nuclear Chemistry: Radiation, Decay Kinetics & Nuclear Reactions
While chemical reactions involve valence electron shifts, nuclear chemistry examines transformations within atomic nuclei, involving large mass-energy conversions and first-order decay kinetics.
Nuclear Stability and the Band of Stability
Competing Nuclear Forces
Nuclear stability is determined by two opposing forces:
- Electrostatic Repulsion: Long-range repulsive force among positive protons.
- Strong Nuclear Force: Extremely powerful short-range (< 1.5 × 10⁻¹⁵ m) attraction among all nucleons.
Neutrons supply strong nuclear attraction without adding electrostatic repulsion.
Neutron-to-Proton Ratio (N/Z)
The band of stability tracks stable N/Z ratios:
- Light Nuclei (Z ≤ 20): Stable N/Z is approximately 1.0 (e.g., ¹²₆C, ¹⁶₈O).
- Heavy Nuclei (20 < Z ≤ 83): Repulsion demands more neutrons; stable N/Z rises to 1.5 (e.g., ²⁰⁸₈₂Pb: 126 n / 82 p = 1.54).
- Z > 83: Electrostatic repulsion overcomes the strong force. All nuclides with Z > 83 are radioactive.
Magic Numbers
Nuclei with completed shells (magic numbers: 2, 8, 20, 28, 50, 82, 126) exhibit high stability. "Doubly magic" nuclides like ⁴₂He (2p, 2n) and ²⁰⁸₈₂Pb (82p, 126n) are exceptionally stable.
Modes of Radioactive Decay & Nuclear Equations
| Decay Mode | Symbol | ΔA | ΔZ | Nuclear Scenario |
|---|---|---|---|---|
| Alpha (α) Decay | ⁴₂He or ⁴₂α | -4 | -2 | Heavy nuclides (Z > 83) reducing overall mass. |
| Beta-Minus (β⁻) Decay | ⁰₋₁e or ⁰₋₁β | 0 | +1 | Neutron-rich nuclei (above band of stability, high N/Z). |
| Positron (β⁺) Emission | ⁰₊₁e or ⁰₊₁β | 0 | -1 | Neutron-poor nuclei (below band of stability, low N/Z). |
| Electron Capture (EC) | ⁰₋₁e (reactant) | 0 | -1 | Neutron-poor nuclei capturing inner orbital electron. |
| Gamma (γ) Emission | ⁰₀γ | 0 | 0 | Metastable excited nucleus releasing energy. |
| Spontaneous Fission | Fragments + ¹₀n | Splits | Splits | Extremely heavy transuranic elements (Z ≥ 92). |
Decay Mechanisms
- Alpha Decay: ²³⁸₉₂U --> ²³⁴₉₀Th + ⁴₂He
- Beta-Minus Decay: ¹₀n --> ¹₁p + ⁰₋₁e + ν̄_e ==> ¹⁴₆C --> ¹⁴₇N + ⁰₋₁e
- Positron Emission: ¹₁p --> ¹₀n + ⁰₊₁e + ν_e ==> ¹¹₆C --> ¹¹₅B + ⁰₊₁e
- Electron Capture: ⁸¹₃₇Rb + ⁰₋₁e --> ⁸¹₃₆Kr (emits characteristic X-ray)
- Gamma Emission: ⁹⁹ᵐ₄₃Tc --> ⁹⁹₄₃Tc + ⁰₀γ
Balancing Nuclear Reactions
- Total mass numbers (A) must be equal on both sides.
- Total atomic numbers (Z) must be equal on both sides.
Kinetics of Radioactive Decay
Radioactive decay is a first-order kinetic process:
Rate = -dN/dt = kN [k = decay constant]
Integrated Rate Law and Half-Life
Integrating yields:
ln(N_t / N₀) = -kt or N_t = N₀ e^(-kt)
The half-life (t₁/₂) is the time for half the sample to decay:
t₁/₂ = ln(2) / k ≈ 0.693 / k
For n half-lives (n = t / t₁/₂):
N_t = N₀ · (1/2)ⁿ
Multi-Step Calculation Example
A sample contains 80.0 mg of Iodine-131 (t₁/₂ = 8.02 days).
- Decay Constant: k = 0.693 / 8.02 days = 0.0864 day⁻¹
- Remaining Mass after 24.06 Days: n = 24.06 / 8.02 = 3.0 half-lives; N_t = 80.0 mg × (1/2)³ = 10.0 mg
- Time to Reach 5.0% Initial Activity: ln(0.050) = -(0.0864)t ==> -2.996 = -0.0864 t ==> t = 34.7 days
Mass Defect, Binding Energy & Fission vs. Fusion
Mass Defect (Δm) and Binding Energy
A nucleus weighs less than its constituent free protons and neutrons:
Δm = [Z · m_p + (A - Z) · m_n] - m_nucleus
This mass loss represents binding energy (E_b):
ΔE = (Δm)c² [1 amu = 931.5 MeV]
Binding Energy Curve
Plotting binding energy per nucleon (E_b / A) peaks at Iron-56 (8.79 MeV/nucleon):
- Fission: Heavy nuclei (e.g., ²³⁵U) split into medium-mass fragments with higher E_b / A, releasing ~200 MeV per event. In fission reactors, control rods made of cadmium or boron regulate neutron flux, while moderators like graphite slow neutrons to thermal speeds to maintain a steady chain reaction with a critical mass.
- Fusion: Light nuclei (e.g., ²₁H and ³₁H) fuse into heavier nuclei (⁴₂He), releasing ~17.6 MeV with greater energy per gram than fission. Temperatures > 10⁷ K are required to overcome electrostatic repulsion.
Practical Applications
Radiometric Dating
- Carbon-14 Dating: Cosmic neutrons produce ¹⁴C in the atmosphere (¹⁴₇N + ¹₀n --> ¹⁴₆C + ¹₁p). Living organisms maintain constant ¹⁴C/¹²C ratios. After death, ¹⁴C decays (t₁/₂ = 5,730 years), dating organic artifacts up to ~50,000 years.
- Uranium-Lead Dating: ²³⁸U decays to ²⁰⁶Pb (t₁/₂ = 4.47 × 10⁹ years), dating minerals and rocks (~4.5 billion years).
Medical and Industrial Technologies
- Diagnostic Tracers: Technetium-99m (t₁/₂ = 6.0 hours) emits clean 140 keV gamma rays for organ imaging without particulate radiation damage.
- Positron Emission Tomography (PET): Fluorine-18 emits positrons that annihilate with electrons (e⁺ + e⁻ --> 2 γ at 511 keV, 180°), localizing active tumors.
- Oncology: Cobalt-60 gamma rays and Iodine-131 beta emissions destroy cancerous tissue.
- Smoke Detectors: Americium-241 alpha emissions ionize air; smoke entering interrupts current, triggering the alarm.
When a radioactive nuclide undergoes beta-minus (β-) decay, how do its mass number (A) and atomic number (Z) change?
A medical radioisotope has a half-life of 6.0 hours. If an initial dose has an activity of 320 mCi, what activity remains in the patient after exactly 24.0 hours?
Complete the following balanced nuclear equation by identifying the missing particle X: 235/92 U + 1/0 n -> 141/56 Ba + 92/36 Kr + 3 X
Why do both the nuclear fission of very heavy elements (such as Uranium-235) and the nuclear fusion of very light elements (such as Hydrogen-2 and Hydrogen-3) release vast amounts of energy?