type: formula_sheet | subtopic: Complete Formula Reference

Rutherford's Model

$d = \frac{2kZe^2}{KE_\alpha} \quad [L] = \text{m}$ where k = $9 \times 10^{9}$ N $m^{2}$ $C^{-2}$, Z = atomic number of target, e = $1.6 \times 10^{-19}$ C.

Bohr Model (Hydrogen-like atoms)

Orbital radius: $r_n = \frac{a_0 n^2}{Z} = \frac{0.529\, n^2}{Z}\ \text{Å} \quad [L] = \text{m (or Å)}$

Orbital velocity: v_n = \frac{2.18 \times 10^6\, Z}{n}\ \text{m/s} \quad [$LT^{-1}$] = \text{m/s}

Total energy: E_n = -\frac{13.6\, Z^2}{n^2}\ \text{eV} \quad [ML^2$T^{-2}$] = \text{eV or J}

Kinetic energy: KE_n = -E_n = +\frac{13.6\, Z^2}{n^2}\ \text{eV} \quad [ML^2$T^{-2}$] = \text{eV}

Potential energy: PE_n = 2E_n = -\frac{27.2\, Z^2}{n^2}\ \text{eV} \quad [ML^2$T^{-2}$] = \text{eV}

Angular momentum (quantized): L_n = n\hbar = \frac{nh}{2\pi} \quad [ML^2$T^{-1}$] = \text{J·s}

Time period: $T_n \propto \frac{n^3}{Z^2} \quad [T] = \text{s}$

Equivalent orbital current: $I_n = \frac{e}{T_n} \propto \frac{Z^2}{n^3} \quad [A] = \text{A}$

Hydrogen Spectral Series (Rydberg Formula)

\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad n_2 > n_1 \quad [$L^{-1}$] = \text{m}^{-1} $R = 1.097 \times 10^7\ \text{m}^{-1}\text{ (Rydberg constant)}$

Number of spectral lines from level n: $N = \frac{n(n-1)}{2}$

Nuclear Physics

Nuclear radius: R = R_0 $A^{1/3}$, \quad R_0 = 1.2\ \text{fm} = 1.2 \times 10^{-15}\ \text{m} \quad [L]

Nuclear volume: $V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3\, A \propto A$

Nuclear density: \rho = \frac{Am_u}{V} = \text{constant} \approx 2.3 \times 10^{17}\ \text{kg/m}^3 \quad [$ML^{-3}$]

Mass defect: $\Delta m = \left[Zm_p + (A-Z)m_n\right] - M \quad [M] = \text{u or kg}$

Binding energy: BE = \Delta m \times 931.5\ \text{MeV} \quad [ML^2$T^{-2}$] = \text{MeV or J} $\left(1\ \text{u} = 931.5\ \text{MeV}/c^2\right)$

Binding energy per nucleon: $\frac{BE}{A} = \frac{\Delta m \times 931.5}{A}\ \text{MeV/nucleon}$

Radioactive Decay

Decay law: N(t) = N_0\, $e^{-\lambda t}$ \quad \text{(dimensionless ratio)}

Activity: A(t) = \lambda N = A_0\, $e^{-\lambda t}$ \quad [$T^{-1}$] = \text{Bq (= s}^{-1}\text{)}

Half-life: $t_{1/2} = \frac{0.693}{\lambda} = \frac{\ln 2}{\lambda} \quad [T] = \text{s}$

After n half-lives: $N = \frac{N_0}{2^n}, \quad \text{where } n = \frac{t}{t_{1/2}}$

Mean life: $\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{0.693} = 1.443\, t_{1/2} \quad [T] = \text{s}$

Key relation: $\tau > t_{1/2} \text{ always. At } t=\tau\text{: } N = N_0/e \approx 0.368\, N_0$