Soddy and Fajans' radioactive displacement law
In alpha decay, mass number falls by 4 and atomic number by 2. In beta decay, mass number stays the same and atomic number rises by 1.
The law and the symbols
A nucleus is written with two numbers. In ₉₂U²³⁸, the number at the bottom left is the atomic number Z = 92 (the number of protons). The number at the top right is the mass number A = 238 (protons + neutrons). In 1913, Frederick Soddy and Kasimir Fajans gave the radioactive displacement law. It tells how Z and A change when a nucleus gives out radiation. The original nucleus is the parent nucleus; the new one formed is the daughter nucleus. The daughter is a new element, so it is displaced (moved) in the periodic table.
The three rules
- α★Alpha decay: parent (Z, A) → daughter (Z − 2, A − 4) + ₂He⁴Example: ₉₂U²³⁸ → ₉₀Th²³⁴ + ₂He⁴. The daughter moves two places back in the periodic table.
- β★Beta decay: parent (Z, A) → daughter (Z + 1, A) + ₋₁e⁰Example: ₉₀Th²³⁴ → ₉₁Pa²³⁴ + ₋₁e⁰. A neutron in the nucleus changes into a proton and an electron, and the electron is thrown out. The daughter moves one place forward.
- γGamma decay: excited nucleus (Z, A) → same nucleus (Z, A) + γOnly energy is given out, so A and Z do not change. It usually follows an alpha or beta decay.
★ key step
Quick check: what changes
| Decay | Mass number A | Atomic number Z | Position in the periodic table |
|---|---|---|---|
| Alpha (α) | Decreases by 4 | Decreases by 2 | 2 places to the left |
| Beta (β) | No change | Increases by 1 | 1 place to the right |
| Gamma (γ) | No change | No change | Same place |
Worked example: completing decay equations
(a) Radium-226 (₈₈Ra²²⁶) gives out an alpha particle. Write the daughter nucleus. (b) Carbon-14 (₆C¹⁴) gives out a beta particle. Write the daughter. (c) ₉₂U²³⁸ finally becomes stable lead, ₈₂Pb²⁰⁶. How many alpha and beta particles are given out?
- (a) Alpha: A − 4 = 226 − 4 = 222; Z − 2 = 88 − 2 = 86. Element 86 is radon.
- ₈₈Ra²²⁶ → ₈₆Rn²²² + ₂He⁴. Check: 226 = 222 + 4 and 88 = 86 + 2.
- (b) Beta: A stays 14; Z + 1 = 6 + 1 = 7. Element 7 is nitrogen.
- ₆C¹⁴ → ₇N¹⁴ + ₋₁e⁰. Check: 14 = 14 + 0 and 6 = 7 + (−1).
- (c) Only alpha decay changes A. The fall in A = 238 − 206 = 32, so the number of alphas = 32 ÷ 4 = 8.
- 8 alphas lower Z by 8 × 2 = 16, giving 92 − 16 = 76. But lead has Z = 82.
- Each beta raises Z by 1, so the number of betas = 82 − 76 = 6.
(a) ₈₆Rn²²² (b) ₇N¹⁴ (c) 8 alpha and 6 beta particles
Carbon-14 in every living thing slowly turns into nitrogen-14 by beta decay. Scientists use this to find the age of ancient objects, such as charcoal from the Keeladi excavation near Madurai.
2 marks: “State Soddy and Fajans' displacement law for alpha decay.” When a nucleus emits an alpha particle, its mass number decreases by 4 and its atomic number by 2: ₉₂U²³⁸ → ₉₀Th²³⁴ + ₂He⁴.
2 marks: “Why does beta decay raise the atomic number but not the mass number?” A neutron turns into a proton and an electron, and the electron leaves. Protons rise by 1, but the total number of protons and neutrons stays the same.
4 marks: “₉₀Th²³² emits one alpha particle and then two beta particles. Find the final nucleus.” After α: A = 228, Z = 88. After two β: A = 228, Z = 90. Final nucleus: ₉₀Th²²⁸, an isotope of the parent.
α: A minus 4, Z minus 2 (two steps back). β: same A, Z plus 1 (one step forward). γ: no change.