Radioactivity as a Spontaneous Nuclear Transformation
Radioactivity is the spontaneous transformation of an unstable nucleus into a different nucleus (or a different nuclear energy state), accompanied by the emission of particles and/or electromagnetic radiation. “Spontaneous” means the decay occurs without needing an external trigger such as heating, pressure, or chemical reaction.
In practice, we describe a radioactive transformation with a nuclear decay equation. The equation is not just bookkeeping: it encodes the physical conservation laws that every decay must satisfy.
Conservation Rules Used in Decay Equations
1) Conservation of charge (atomic number)
The total electric charge before and after the decay must be the same. In nuclear notation this is tracked by the atomic number Z. When you balance a decay equation, the sum of Z values on the left must equal the sum of Z values on the right.
2) Conservation of nucleon number (mass number)
The total number of nucleons (protons + neutrons) is conserved in ordinary nuclear decays. This is tracked by the mass number A. Balance by ensuring the sum of A values is the same on both sides.
3) Conservation of energy and momentum
Energy and momentum are always conserved, but they are not always obvious from the written equation because some energy can appear as:
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- Kinetic energy of emitted particles and the recoiling daughter nucleus
- Gamma radiation (photons) emitted when the daughter nucleus de-excites
- Neutrinos/antineutrinos that carry away energy and momentum (crucial in beta decay)
A key practical implication: in many decays (especially beta decay), you cannot infer the emitted particle energies from A and Z alone; the energy is shared among multiple products to satisfy momentum conservation.
Common Emissions and How They Change A and Z
| Decay type | Emitted particle | Symbol | Change in A | Change in Z | Notes |
|---|---|---|---|---|---|
| Alpha (α) | Helium nucleus | ^4_2He | -4 | -2 | Often from heavy nuclei |
| Beta minus (β−) | Electron + antineutrino | ^0_-1e + \bar{\nu}_e | 0 | +1 | Neutron → proton in nucleus |
| Beta plus (β+) | Positron + neutrino | ^0_+1e + \nu_e | 0 | -1 | Proton → neutron; requires sufficient energy |
| Electron capture (EC) | Captured electron + neutrino | e^- + \nu_e | 0 | -1 | Alternative to β+; often followed by X-rays |
| Gamma (γ) | Photon | \gamma | 0 | 0 | De-excitation; changes energy state only |
When writing equations, it is common to omit neutrinos in introductory balancing exercises because they do not affect A or Z. However, remember they are physically important for energy and momentum conservation in beta decays.
How to Write and Balance Nuclear Decay Equations (Step-by-Step)
Step 1: Write the parent nuclide and the known emitted particle(s)
Example template:
^A_ZX → (daughter) + (emitted particle)Step 2: Use nucleon number conservation to determine the daughter’s A
Add up A on the right and set it equal to the left.
Step 3: Use charge conservation to determine the daughter’s Z
Add up Z on the right and set it equal to the left.
Step 4: Identify the element from Z
Once Z is known, the element symbol is determined by the periodic table.
Step 5: Check reasonableness and note possible accompanying radiation
For example, gamma emission may follow alpha or beta decay if the daughter is produced in an excited state.
Worked Examples
Example 1: Alpha decay
Suppose a nucleus undergoes alpha decay:
^238_92U → (daughter) + ^4_2HeBalance A: 238 = A_daughter + 4 → A_daughter = 234
Balance Z: 92 = Z_daughter + 2 → Z_daughter = 90
Identify element: Z=90 is thorium (Th).
^238_92U → ^234_90Th + ^4_2HeExample 2: Beta minus decay (β−)
In β− decay, the nucleus increases its atomic number by 1 while keeping the same mass number:
^14_6C → (daughter) + ^0_-1eBalance A: 14 = A_daughter + 0 → A_daughter = 14
Balance Z: 6 = Z_daughter + (-1) → Z_daughter = 7
Identify element: Z=7 is nitrogen (N).
^14_6C → ^14_7N + ^0_-1e ( + \bar{\nu}_e )The antineutrino is shown in parentheses to emphasize it is required physically even if not needed for balancing A and Z.
Example 3: Beta plus decay (β+)
^22_11Na → (daughter) + ^0_+1eBalance A: 22 = A_daughter
Balance Z: 11 = Z_daughter + (+1) → Z_daughter = 10 (neon, Ne)
^22_11Na → ^22_10Ne + ^0_+1e ( + \nu_e )Example 4: Electron capture (EC)
Electron capture is written with an electron on the left side:
^7_4Be + ^0_-1e → (daughter) ( + \nu_e )Balance A: 7 + 0 = A_daughter → A_daughter = 7
Balance Z: 4 + (-1) = Z_daughter → Z_daughter = 3 (lithium, Li)
^7_4Be + ^0_-1e → ^7_3Li ( + \nu_e )Example 5: Gamma emission (no change in A or Z)
If a nucleus is produced in an excited state, it can emit a gamma photon:
^99m_43Tc → ^99_43Tc + \gammaThe m indicates a metastable (excited) nuclear state. Only the energy state changes; A and Z remain the same.
Balancing Practice: A and Z as a “Two-Equation System”
Many problems reduce to solving two simple equations:
- Nucleon number:
A_parent = A_daughter + A_emitted - Charge:
Z_parent = Z_daughter + Z_emitted
For electron capture, include the captured electron on the left:
A_parent + 0 = A_daughterZ_parent + (-1) = Z_daughter
Exercises: Identify the Missing Particle or Nuclide
Instructions: For each transformation, fill in the missing item marked ?. Use conservation of A and Z. (Neutrinos are omitted unless explicitly requested.)
Set A: Identify the missing emitted particle
- 1)
^210_84Po → ^206_82Pb + ? - 2)
^131_53I → ^131_54Xe + ? - 3)
^18_9F → ^18_8O + ? - 4)
^57_27Co → ^57_26Fe + ? - 5)
^99m_43Tc → ^99_43Tc + ?
Set B: Identify the missing daughter nuclide
- 6)
^226_88Ra → ? + ^4_2He - 7)
^3_1H → ? + ^0_-1e - 8)
^40_19K → ? + ^0_+1e - 9)
^201_80Hg + ^0_-1e → ? - 10)
^60_27Co → ^60_28Ni + ?
Set C: Short decay chains (track A and Z across multiple steps)
For each chain, fill in the missing nuclide(s) or particle(s).
- 11)
^238_92U → ^234_90Th + ?then^234_90Th → ^234_91Pa + ? - 12)
^222_86Rn → ^218_84Po + ?then^218_84Po → ? + ^4_2He - 13)
^14_6C → ^14_7N + ?then^14_7N* → ^14_7N + ?
Answer key (hide until you try)
Show answers
- 1)
^4_2He(alpha) - 2)
^0_-1e(β−) - 3)
^0_+1e(β+) - 4)
^0_+1e(β+ or EC is also possible in some nuclides; here the balancing points to β+ if written as emission) - 5)
\gamma - 6)
^222_86Rn - 7)
^3_2He - 8)
^40_18Ar - 9)
^201_79Au(electron capture) - 10)
^0_-1e(β−) - 11) First missing:
^4_2He; second missing:^0_-1e - 12) First missing:
^4_2He; second missing nuclide:^214_82Pb - 13) First missing:
^0_-1e; second missing:\gamma
Unstable Nuclide vs Hazardous Material
A common confusion is to equate “unstable” with “dangerous.” They are related but not the same.
Unstable nuclide (a nuclear property)
A nuclide is unstable if it can lower its nuclear energy by transforming into another nuclide or energy state. This is a statement about the nucleus itself, independent of how much of it you have.
Hazardous material (depends on amount and emission type)
The hazard from a radioactive material depends strongly on:
- Activity: the number of decays per second (measured in becquerels, Bq). A tiny amount of a very active nuclide can produce many decays per second; a large amount of a weakly active nuclide may produce few.
- Radiation type and energy: alpha, beta, gamma (and sometimes neutrons) differ in penetration and biological impact.
- Exposure pathway: external exposure vs inhalation/ingestion (internal exposure). Alpha emitters, for example, are often low external hazards but can be serious internal hazards if taken into the body.
- Shielding and distance: gamma rays generally require denser shielding than beta particles; alpha particles are stopped by very thin layers but are dangerous if internal.
Practical comparison: a sealed source that emits mostly gamma radiation can be hazardous at a distance because gamma rays penetrate; a small speck of an alpha emitter might be hard to detect externally yet be hazardous if inhaled. Therefore, “unstable” tells you a decay can happen; “hazardous” requires considering how many decays occur and what comes out, plus the exposure scenario.
