Chapter 21: Nuclear Physics (Numerical Problems)

Interactive Study Notes Preview

Chapter Overview & SLOs

How do we compute the constituent subatomic particles from a nuclear symbol? A specific atomic nucleus is structurally quantified using the standard symbol ${}_{Z}^{A}X$. The mass number A represents the total count of nucleons (protons + neutrons), while the atomic number Z represents the proton count. The total neutron count N is computed using N = A - Z.

Example 1 - Polonium-209: For ${}_{84}^{209}Po$: Z = 84 protons, A = 209 nucleons, N = 209 - 84 = 125 neutrons.

How are radioactive transmutation pathways balanced mathematically? Unstable isotopes undergo spontaneous decay. Both mass number (A) and atomic number (Z) must balance on both sides of the reaction arrow.

Alpha (α) Decay: Ejects a helium nucleus (${}_{2}^{4}He$). Equation: ${}_{Z}^{A}X → {}_{Z-2}^{A-4}Y + {}_{2}^{4}He$

Example 2 - Americium-241 alpha decay: ${}_{95}^{241}Am → {}_{93}^{237}Np + {}_{2}^{4}He$ (Americium-241 transmutes into Neptunium-237)

Beta (β⁻) Decay: Ejects a high-speed electron (${}_{-1}^{0}e$). Equation: ${}_{Z}^{A}X → {}_{Z+1}^{A}Y + {}_{-1}^{0}e$

Example 3 - Beta decay: ${}_{6}^{14}C → {}_{7}^{14}N + {}_{-1}^{0}e$ (Carbon-14 decays to Nitrogen-14)

What formulas govern the exponential decay of radioactive samples over time? The fraction of remaining radioactive nuclei after n half-lives is: N/N₀ = (1/2)ⁿ, where n = number of half-lives. Total elapsed time t = n × T₁/₂ (where T₁/₂ is the half-life).

Example 4 - Carbon dating: Carbon-14 has half-life T₁/₂ = 5730 years. If an archaeological specimen retains 25% of its initial C-14 concentration, then N/N₀ = 0.25 = (1/2)ⁿ. Since 0.25 = (1/2)², n = 2 half-lives. Age of specimen = 2 × 5730 = 11,460 years.

Example 5 - Radioactive decay step reduction: If a sample of 64,000 atoms decays to 1,000 atoms:

• N/N₀ = 1,000/64,000 = 1/64 = (1/2)⁶, so n = 6 half-lives
• Elapsed time t = 6 × T₁/₂

Summary of formulas:

| Formula | Description | |---------|-------------| | N = A - Z | Neutron count = mass number - atomic number | | ${}_{Z}^{A}X → {}_{Z-2}^{A-4}Y + {}_{2}^{4}He$ | Alpha decay | | ${}_{Z}^{A}X → {}_{Z+1}^{A}Y + {}_{-1}^{0}e$ | Beta decay | | N/N₀ = (1/2)ⁿ | Fraction remaining after n half-lives | | t = n × T₁/₂ | Elapsed time = number of half-lives × half-life |

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we calculate proton, neutron, and total nucleon totals from nuclear notation ${}_{Z}^{A}X$? Students will isolate structural configurations for heavy radioactive metals, proving that Polonium-209 (${}_{84}^{209}Po$) yields exactly 125 neutrons ($N = A - Z = 209 - 84 = 125$).
• How do we formulate balanced nuclear transmutations for alpha and beta decay mechanisms? Students will determine unknown daughter elements by conserving total mass numbers and atomic numbers, such as detailing the alpha decay of Americium-241 into Neptunium-237 (${}_{93}^{237}Np$).
• How do we solve exponential decay problems using the half-life ratio equations $t = n \times T_{1/2}$ and $N = N_0(1/2)^n$? Students will evaluate multi-step sample reductions, such as computing how a 64,000 atom trace degrades down to 1,000 atoms across 6 full half-life steps.
• How do we apply radioactive half-life variables to compute the age of ancient organic matter via carbon dating? Students will analyze remaining isotope percentages to deduce archaeological chronology, mapping a 25% remaining trace of Carbon-14 back to an age of 11,460 years.

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