The first law $\Delta U = q + w$ uses IUPAC convention: work ON system is positive, heat IN to system is positive. For isothermal ideal gas expansion: $w = -nRT\ln(V_2/V_1)$; for irreversible: $w = -P_{ext}\Delta V$; for free expansion: $w = 0$.

In $\Delta H = \Delta U + \Delta n_g RT$, only gaseous moles are counted in $\Delta n_g$. Solids and liquids are excluded entirely.

Hess's Law: $\Delta H_{rxn}^\circ = \Sigma\Delta H_f^\circ(\text{products}) - \Sigma\Delta H_f^\circ(\text{reactants})$. Combustion is always exothermic; atomization and ionization are always endothermic.

Gibbs equation $\Delta G = \Delta H - T\Delta S$: spontaneous when $\Delta G < 0$. At equilibrium $\Delta G = 0$ (not $\Delta G^\circ$). Crossover temperature $T = \Delta H/\Delta S$ when both have the same sign.

Four spontaneity cases: (1) $\Delta H(-), \Delta S(+)$: always spontaneous; (2) $\Delta H(+), \Delta S(-)$: never spontaneous; (3) $\Delta H(-), \Delta S(-)$: low T spontaneous; (4) $\Delta H(+), \Delta S(+)$: high T spontaneous.