Key Points: All Formulas with Dimensional Analysis

Rutherford: $d = \frac{2kZe^2}{KE_\alpha} \quad [d] = [L] = \text{m}$

Bohr Model: $r_n = \frac{0.529\, n^2}{Z}\ \text{Å} \quad [r] = [L] = \text{m}$

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

$E_n = -\frac{13.6 Z^2}{n^2}\ \text{eV} \quad [E] = [ML^2T^{-2}]$

$KE_n = -E_n \quad (\text{positive}) \qquad PE_n = 2E_n \quad (\text{negative})$

$L_n = n\hbar = \frac{nh}{2\pi} \quad [L] = [ML^2T^{-1}] = \text{J·s}$

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

$N_{\text{lines}} = \frac{n(n-1)}{2}$

Nuclear Physics: $R = R_0 A^{1/3},\ R_0 = 1.2\ \text{fm} \quad [R] = [L]$

$\rho = \frac{Am_u}{(4/3)\pi R^3} \approx 2.3 \times 10^{17}\ \text{kg/m}^3 \quad [ML^{-3}] = \text{constant}$

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

$BE = \Delta m \times 931.5\ \text{MeV} \quad [ML^2T^{-2}] \quad (1\text{ u} = 931.5\ \text{MeV}/c^2)$

Radioactive Decay: $N(t) = N_0 e^{-\lambda t} = \frac{N_0}{2^{t/t_{1/2}}} \quad [\lambda] = [T^{-1}] = \text{s}^{-1}$

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

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

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

Distance of Closest Approach: $d = \frac{2kZe^2}{KE_\alpha},\ k = 9 \times 10^9\ \text{Nm}^2\text{C}^{-2},\ e = 1.6 \times 10^{-19}\ \text{C}$