Chemical kinetics Study Guide

📖 Core Concepts Chemical kinetics – study of how fast reactions proceed and why rates differ. Rate law – expresses rate as \( \text{rate}=k\prodi[\text{reactant}i]^{ni} \); \(k\) = rate constant, \(ni\) = experimental order. Activation energy (\(Ea\)) – minimum energy a molecule must have to react; lower \(Ea\) → faster reaction. Arrhenius equation – \(k = A\,e^{-Ea/(RT)}\); shows temperature dependence of \(k\). Catalyst – provides an alternative pathway with lower \(Ea\); not consumed, does not shift equilibrium. Kinetic vs. thermodynamic control – fastest‑forming product dominates early (kinetic); most stable product dominates at equilibrium (thermodynamic). Curtin–Hammett principle – when rapidly interconverting reactants give different products, product ratios depend on transition‑state energies, not on reactant equilibrium concentrations. Kinetic isotope effect – replacing an atom with a heavier isotope changes the rate, revealing which bonds break in the rate‑determining step. --- 📌 Must Remember Rate ∝ collision frequency × fraction with energy ≥ \(Ea\). Higher concentration → more collisions → higher rate (law of mass action). Temperature rise: each 10 °C ≈ 2–3× increase in rate (via Arrhenius). Catalysts accelerate both forward and reverse reactions equally; equilibrium constant unchanged. Pressure for gases acts like concentration: increase → faster rate. Photochemical reactions require light of a specific wavelength to create an excited reactant. Euler vs. Runge‑Kutta: Euler is simple but less accurate; Runge‑Kutta (3rd‑order) gives better precision for ODEs. --- 🔄 Key Processes Deriving a rate law Vary one reactant’s concentration, keep others constant → measure initial rate. Plot \(\log(\text{rate})\) vs. \(\log[\text{reactant}]\); slope = reaction order \(n\). Using the Arrhenius equation Take natural log: \(\ln k = \ln A - \frac{Ea}{R}\frac{1}{T}\). Plot \(\ln k\) vs. \(1/T\); slope = \(-Ea/R\), intercept = \(\ln A\). Applying the Curtin–Hammett principle Identify rapidly interconverting conformers/reactants (A ↔ B). Determine activation barriers to each product (ΔG‡\(A\), ΔG‡\(B\)). Product ratio ≈ \(e^{-(\Delta G^{\ddagger}A-\Delta G^{\ddagger}B)/RT}\). Numerical integration (Euler) \(C{t+Δt}=Ct + (\text{rate at }t)\times Δt\). Use small \(Δt\) for acceptable error. Runge‑Kutta (3rd‑order) step Compute three slopes \(k1, k2, k3\) using intermediate points; combine to update concentration with higher accuracy. --- 🔍 Key Comparisons Kinetics vs. Thermodynamics Kinetics: how fast → rate constant \(k\). Thermodynamics: whether it proceeds → ΔG, equilibrium constant \(K\). Catalyst vs. Reactant concentration Catalyst: lowers \(Ea\) without being consumed; effect independent of concentration of reactants. Concentration: raises collision frequency; no change in energy barrier. Euler vs. Runge‑Kutta Euler: single slope, simple, larger truncation error. Runge‑Kutta: multiple slopes, more accurate, slightly more work. Kinetic control vs. Thermodynamic control Kinetic: product formed fastest dominates (often less stable). Thermodynamic: most stable product dominates after equilibrium is reached. --- ⚠️ Common Misunderstandings “Catalyst changes equilibrium.” – False; it only speeds the approach to equilibrium. “Higher temperature always makes a reaction faster indefinitely.” – Very high \(T\) can lead to side reactions or decomposition, altering the pathway. “Rate law follows stoichiometry.” – Only true for elementary steps; overall reactions often have fractional or zero orders. “Pressure only matters for gases.” – True for gas‑phase collisions; liquids/solids are largely unaffected. “Photons always increase rate.” – Only if the photon energy matches an absorption band that leads to a reactive excited state. --- 🧠 Mental Models / Intuition Collision‑energy picture: imagine reactants as billiard balls; rate depends on how often they hit (concentration, surface area) and whether the hit is hard enough (temperature, \(Ea\)). Energy‑profile diagram: catalyst flattens the hill (lower \(Ea\)) but the valley (ΔG) stays the same → same equilibrium. “Fast vs. stable” analogy: a sprinter (kinetic product) finishes the race first but may be less fit than the marathon runner (thermodynamic product) who wins in the long run. --- 🚩 Exceptions & Edge Cases Diffusion‑controlled reactions: at very low concentrations, rate may be limited by how fast molecules find each other, giving fractional orders. Fall‑off region (high pressure): rate coefficients deviate from simple pressure‑proportionality; require Lindemann or Troe formulations. Enzyme saturation: Michaelis–Menten kinetics deviate from simple mass‑action when substrate concentration ≫ \(KM\). Isotope effects: large when the isotopic substitution occurs at a bond broken in the rate‑determining step; negligible otherwise. --- 📍 When to Use Which Arrhenius plot → when you have rate constants at several temperatures and need \(Ea\). Euler method → quick estimate for linear or mildly curved concentration vs. time curves; small step size required. Runge‑Kutta → stiff or highly non‑linear rate equations, or when high accuracy is needed for simulation. Spectrophotometry → ideal when one species absorbs uniquely at a wavelength and reaction is fast enough for real‑time monitoring. Stochastic simulation → when dealing with low‑number molecular events (e.g., enzyme turnover in single cells). --- 👀 Patterns to Recognize Rate ∝ [A]^m [B]^n → look for linearity in log‑log plots to extract \(m, n\). Straight line in \(\ln k\) vs. \(1/T\) → confirms Arrhenius behavior; curvature hints at a change in mechanism. Product distribution unchanged by concentration → likely under thermodynamic control. Rapid interconversion of reactants + different products → apply Curtin–Hammett. Large kinetic isotope effect (> 5) → bond to the isotopic atom is broken in the rate‑determining step. --- 🗂️ Exam Traps “Catalyst appears in the rate law.” – Remember it lowers \(Ea\) but is not part of the concentration term. “Higher pressure always increases rate constant \(k\).” – Only true for simple collisions; at very high pressures, fall‑off can reduce effective \(k\). “Reaction order equals stoichiometric coefficient.” – Only for elementary steps; overall order must be determined experimentally. “Arrhenius pre‑exponential factor \(A\) is a universal constant.” – It is specific to each reaction and reflects collision frequency and orientation. “Photochemical reactions are always faster than thermal ones.” – Speed depends on quantum yield and subsequent steps, not just light absorption. ---