Electromagnetic Radiation

$c = \nu\lambda \quad (c = 3 \times 10^8 \text{ m/s})$ $E = h\nu = \frac{hc}{\lambda} \quad (h = 6.626 \times 10^{-34} \text{ J·s})$

Photoelectric Effect

$KE = h\nu - h\nu_0 = h(\nu - \nu_0)$ $\phi = h\nu_0 \quad \text{(work function)}$

Bohr Model (Hydrogen-Like Atoms, Z = atomic number)

$E_n = -\frac{13.6 \, Z^2}{n^2} \text{ eV} = -\frac{2.18 \times 10^{-18} \, Z^2}{n^2} \text{ J}$ $r_n = \frac{0.529 \, n^2}{Z} \text{ Å}$ $v_n = \frac{2.18 \times 10^6 \, Z}{n} \text{ m/s}$ $L = mvr = \frac{nh}{2\pi} = n\hbar$ $T_n = \frac{2\pi r_n}{v_n} \propto \frac{n^3}{Z^2}$

Energy of Transition

$\Delta E = E_{n_2} - E_{n_1} = 13.6 \, Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV} \quad (n_2 > n_1, \text{ emission})$

Hydrogen Spectrum (Rydberg Formula)

$\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \quad (R_H = 1.097 \times 10^7 \text{ m}^{-1})$ $\text{For hydrogen-like ions: } \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

Spectral Lines

$\text{Lines from level } n = \frac{n(n-1)}{2}$

de Broglie Wavelength

$\lambda = \frac{h}{mv} = \frac{h}{p}$ $\text{For accelerated particle: } \lambda = \frac{h}{\sqrt{2mKE}} = \frac{h}{\sqrt{2meV}}$

Heisenberg Uncertainty Principle

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$ $\Delta x \cdot \Delta v \geq \frac{h}{4\pi m}$ $\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$

Quantum Numbers & Orbital Capacity

$l = 0, 1, 2, \ldots, (n-1)$ $m_l = -l, -(l-1), \ldots, 0, \ldots, +(l-1), +l \quad \text{(total: } 2l+1 \text{ values)}$ $m_s = \pm\frac{1}{2}$ $\text{Electrons per subshell} = 2(2l+1)$ $\text{Electrons per shell} = 2n^2$

Node Formulas

$\text{Total nodes} = n - 1$ $\text{Angular nodes} = l$ $\text{Radial nodes} = n - l - 1$

Virial Theorem (Bohr Model)

$KE = -E_n = +|E_n|$ $PE = 2E_n \quad \text{(negative)}$ $E_{total} = E_n = KE + PE$