First Law and Work

$\Delta U = q + w \quad \text{(IUPAC)}$

$w_{\text{free}} = 0, \quad w_{\text{irrev}} = -P_{ext}\Delta V, \quad w_{\text{rev}} = -nRT\ln\frac{V_2}{V_1}$

$|w_{\text{free}}| < |w_{\text{irrev}}| < |w_{\text{rev}}|$

Enthalpy

$H = U + PV \implies \Delta H = \Delta U + \Delta n_g RT$

$\Delta n_g = n_g(\text{products}) - n_g(\text{reactants}) \quad \text{(gaseous species only)}$

$\Delta H_{rxn}^\circ = \sum\Delta H_f^\circ(\text{products}) - \sum\Delta H_f^\circ(\text{reactants}) \quad \text{(Hess's Law)}$

$\Delta H_{\text{bond}} = \sum BE_{\text{broken}} - \sum BE_{\text{formed}}$

Heat Capacities

$C_p - C_v = R \quad \text{(all ideal gases)}$

$q_v = nC_v\Delta T = \Delta U \quad \text{(isochoric)}$

$q_p = nC_p\Delta T = \Delta H \quad \text{(isobaric)}$

$\gamma = \frac{C_p}{C_v}: \quad \text{Monoatomic: }\frac{5}{3},\quad \text{Diatomic: }\frac{7}{5},\quad \text{Non-linear triatomic: }\frac{4}{3}$

Entropy

$\Delta S = \frac{q_{rev}}{T} \quad \text{(reversible process)}$

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0 \quad \text{(2nd Law, spontaneous)}$

Gibbs Free Energy and Equilibrium

$G = H - TS \implies \Delta G = \Delta H - T\Delta S$

$\Delta G < 0: \text{spontaneous},\quad \Delta G = 0: \text{equilibrium},\quad \Delta G > 0: \text{non-spontaneous}$

$\Delta G^\circ = -RT\ln K = -2.303\,RT\log K$

$T_{\text{crossover}} = \frac{\Delta H}{\Delta S} \quad \text{(when }\Delta H\text{ and }\Delta S\text{ have same sign)}$

Spontaneity Cases

Constants

$R = 8.314\ \frac{\text{J}}{\text{mol·K}},\quad \ln 2 = 0.693,\quad \ln 10 = 2.303,\quad 1\ \text{L·atm} = 101.3\ \text{J}$