Modern PhysicsTop Priority (3 Qs in JEE)

Atoms & Nuclei Formulas

Bohr atomic model, energy levels, Rydberg formula, radioactive decay law, half-life, and nuclear binding energy.

8 Formulas•822+ Questions in Bank

Core Formulas & Mathematical Identities

r_n = 0.529 \frac{n^2}{Z} \text{ \AA}, \quad v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} = c \alpha \frac{Z}{n}

Quantized orbit radius and electron speed for hydrogen-like species.

Examiner Tip: Radius scales as n²/Z; velocity scales as Z/n.

E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}, \quad K_n = -E_n, \quad U_n = 2E_n

Quantized energy levels in hydrogenic atoms.

Examiner Tip: Ionization energy from ground state is +13.6 Z² eV.

\frac{1}{\lambda} = R Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \quad (R \approx 1.097 \times 10^7 \text{ m}^{-1})

Wavelengths of emitted photons during transitions between orbits.

Examiner Tip: Lyman (n₁=1, UV), Balmer (n₁=2, Visible), Paschen (n₁=3, IR).

N(t) = N_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}, \quad \tau_{\text{mean}} = \frac{1}{\lambda} = 1.44 T_{1/2}

Statistical exponential decay of radioactive nuclei.

Examiner Tip: Fraction remaining after n half-lives is (1/2)^n.

R = R_0 A^{1/3} \; (R_0 \approx 1.2 \text{ fm}), \quad \Delta E = \Delta m \times 931.5 \text{ MeV}

Nuclear volume scales directly with mass number A (constant nuclear density ~ 2.3 × 10¹⁷ kg/m³).

Examiner Tip: Peak binding energy per nucleon occurs near Iron-56 (~8.8 MeV/nucleon).

\sqrt{\nu} = a(Z - b), \quad \lambda_{\text{min}} = \frac{hc}{eV} = \frac{12400}{V \text{ (volts)}} \text{ \AA}

Frequency of characteristic X-ray lines where b is screening constant (b = 1 for K-alpha lines). Duane-Hunt minimum cutoff wavelength.

Examiner Tip: Plot of √ν vs atomic number Z is a straight line, which originally ordered the periodic table by atomic number.

Q = [m_{\text{reactants}} - m_{\text{products}}] c^2 = \sum BE_{\text{products}} - \sum BE_{\text{reactants}}

Net energy liberated or absorbed in nuclear transmutation. Exoergic if Q > 0, endoergic if Q < 0.

Examiner Tip: Threshold kinetic energy for endoergic reaction: K_{th} = |Q| (1 + m_{projectile} / m_{target}).

A(t) = \lambda N(t) = A_0 e^{-\lambda t}, \quad 1 \text{ Curie (Ci)} = 3.7 \times 10^{10} \text{ Bq}

Decay rate per second. In successive decay A -> B -> C with secular equilibrium: λ_A N_A = λ_B N_B.

Examiner Tip: Parallel radioactive decay: effective decay constant λ_eff = λ₁ + λ₂; 1/T_{eff} = 1/T₁ + 1/T₂.

Frequently tested misconceptions in Atoms & Nuclei