Key Points: Atoms & Nuclei — Core Concepts

Rutherford's Experiment:

• Alpha particles fired at thin gold foil → most pass undeflected (empty space), few deflect at large angles, very rarely bounce back (~180°).
• Atom has a tiny dense nucleus (~10^{-15} m) with all positive charge; electrons orbit far away (~10^{-10} m).
• Distance of closest approach: d = 2k$Ze^{2}$/KE_α (upper limit on nuclear size).

Bohr Model:

• Three postulates: stationary orbits, quantized angular momentum L = nℏ, photon emission during transitions.
• Radius: r_n = 0.529 $n^{2}$/Z Å. Proportional to $n^{2}$, inversely proportional to Z.
• Velocity: v_n = $2.18 \times 10^{6}$ Z/n m/s. Decreases with n, increases with Z.
• Energy: E_n = −13.6 $Z^{2}$/$n^{2}$ eV. Negative = bound state. Ground state (n=1) = most negative = most tightly bound.
• KE = −E (always positive). PE = 2E (always negative). Check: KE + PE = E.

Hydrogen Spectral Series:

• Formula: 1/λ = R(1/n_{1}^{2} − 1/n_{2}^{2}), R = $1.097 \times 10^{7}$ $m^{-1}$.
• Lyman: UV (n_{1}=1). Balmer: Visible (n_{1}=2). Paschen: IR (n_{1}=3). Brackett/Pfund: Far IR (n_{1}=4,5).
• Longest wavelength in series = first line (lowest n_{2} = n_{1}+1). Shortest wavelength = series limit (n_{2}=∞).
• Spectral lines from level n: N = n(n−1)/2.

Nuclear Structure:

• Nuclear radius: R = 1.2 × A^(1/3) fm. Volume ∝ A → constant density ≈ $2.3 \times 10^{17}$ kg/$m^{3}$.
• Mass defect: $\Delta m$ = [Zm_p + (A−Z)m_n] − M (always positive for stable nuclei).
• Binding energy: BE = $\Delta m$ × 931.5 MeV (1 u = 931.5 MeV/$c^{2}$).
• BE/A peaks at Fe-56 (8.75 MeV/n) — the most stable nucleus.
• Fusion (light nuclei) and fission (heavy nuclei) both move toward Fe-56 peak → both release energy.

Radioactive Decay:

• Decay law: N = $N_{0}$e^(−λt). After n half-lives: N = $N_{0}$/2ⁿ.
• Half-life: t_{1}/_{2} = 0.693/λ. Activity: A = λN (SI unit: becquerel, Bq).
• Mean life: τ = 1/λ = 1.443 t_{1}/{2} (always > t{1}/_{2}). At t=τ: N = $N_{0}$/e = 36.8% remains.
• Alpha decay: $\Delta A$ = −4, $\Delta Z$ = −2 (He-4 emitted).
• Beta-minus: $\Delta A$ = 0, $\Delta Z$ = +1 (n→p, electron+antineutrino emitted).
• Gamma: $\Delta A$ = 0, $\Delta Z$ = 0 (photon, nuclear de-excitation).