Integrated Rate Laws and Half-Life

Each order has a signature linear plot: zero order $[A]$ vs. $t$; first order $\ln[A]$ vs. $t$; second order $1/[A]$ vs. $t$. First-order decay dominates the MCAT: $\ln[A] = \ln[A]_0 - kt$ $t_{1/2} = \frac{0.693}{k}$ Only first-order half-life is independent of starting concentration.

Activation Energy, Mechanisms, and Catalysis

The Arrhenius equation $k = Ae^{-E_a/RT}$ shows rate constants grow with temperature and shrink with activation energy. In multistep mechanisms, the slow (rate-determining) step sets the rate law; intermediates are formed then consumed, while catalysts are regenerated and lower $E_a$ without shifting equilibrium. Collision theory requires sufficient energy and correct orientation.

Key Takeaways

• Orders come from experiment, never from stoichiometric coefficients.
• Match the linear plot to the order: $\ln[A]$ linear means first order.
• Catalysts speed both directions equally; equilibrium position is unchanged.
• The rate-determining step's molecularity dictates the observed rate law.