Chemical Kinetics: Rate Laws & Arrhenius Equation

Chemical kinetics studies the speed of chemical reactions and the factors that influence them. JEE focuses heavily on rate law determination, integrated rate equations, half-life calculations, and the Arrhenius equation for temperature dependence.

Rate of Reaction: For aA + bB → cC + dD, rate = -(1/a)d[A]/dt = -(1/b)d[B]/dt = (1/c)d[C]/dt = (1/d)d[D]/dt. Instantaneous rate is the slope of the concentration-time curve at a given point. Average rate = $\delta$[product]/($\delta_{t}$). Rate is always positive by convention.

Rate Law and Rate Constant: Rate = k[A]^m[B]^n, where m and n are orders (determined experimentally, NOT from stoichiometry). Overall order = m + n. The rate constant k depends on temperature but NOT on concentration. Units of k depend on order: for nth order, k has units of (concentration)^(1-n) $time^{-1}$.

Molecularity vs Order: Molecularity = number of reacting molecules in an elementary step (always a positive integer: 1, 2, or 3). Order = experimentally determined exponent in the rate law (can be zero, fractional, or negative). For elementary reactions: order = molecularity. For complex reactions: order is determined by the rate-determining step. Molecularity > 3 is not observed because simultaneous collision of >3 molecules is extremely improbable.

Zero Order Kinetics: Rate = k (independent of concentration). Integrated: [A] = [A]_0 - kt. Half-life: $\frac{t_{1}}{2}$ = [A]_0/(2k). [A] vs t is a straight line with slope = -k. Example: decomposition of NH₃ on Pt surface, photochemical reactions. Graph: [A] decreases linearly with time until [A] = 0.

First Order Kinetics: Rate = k[A]. Integrated: ln[A] = ln[A]_0 - kt, or k = (1/t)ln([A]_0/[A]) = (2.303/t)log([A]_0/[A]). Half-life: $\frac{t_{1}}{2}$ = 0.693/k (independent of initial concentration). ln[A] vs t is linear with slope = -k. Examples: radioactive decay, decomposition of N₂O₅, H₂O₂ decomposition. First order is the most commonly tested order in JEE.

Second Order Kinetics: Rate = k[A]². Integrated: 1/[A] = 1/[A]_0 + kt. Half-life: $\frac{t_{1}}{2}$ = 1/(k[A]_0). 1/[A] vs t is linear with slope = k. Half-life depends on initial concentration (doubles when [A]_0 is halved).

Pseudo-First Order Reactions: A second (or higher) order reaction that behaves as first order because one reactant is in large excess. Example: acid hydrolysis of ester (CH₃COOC₂H₅ + H₂O), where [H₂O] is essentially constant. Rate = k'[ester], where k' = k[H₂O].

Ethyl acetate (CH₃COOC₂H₅) — classic pseudo-first order hydrolysis substrate

Acetic acid — product of ethyl acetate hydrolysis

Ethanol — other product of hydrolysis

The overall reaction: CH₃COOC₂H₅ + H₂O → CH₃COOH + C₂H₅OH

Arrhenius Equation: k = A * e^(-Ea/RT). A = pre-exponential factor (frequency factor), Ea = activation energy. ln(k) = ln(A) - Ea/(RT). Plot of ln(k) vs 1/T is linear with slope = -Ea/R. Two-temperature form: ln(k2/k1) = (Ea/R)(1/T₁ - 1/T₂). For every 10 K rise near room temperature, rate roughly doubles (temperature coefficient = 2-3).

Activation Energy: The minimum energy reactant molecules must possess to undergo a successful reaction. Ea(forward) - Ea(backward) = $\delta_{H}$ (enthalpy of reaction). A catalyst lowers Ea for both forward and reverse reactions equally, without changing $\delta_{H}$ or equilibrium constant. Ea is always positive.

Collision Theory: Rate = Z * f * e^(-Ea/RT), where Z = collision frequency, f = steric factor (fraction of collisions with correct orientation). Effective collisions require both sufficient energy (>= Ea) and proper orientation. The Boltzmann distribution shows that fraction of molecules with energy >= Ea increases exponentially with temperature.

Methods for Determining Order: (1) Initial rates method: compare rates at different initial concentrations. (2) Integrated rate law method: test which plot gives a straight line ([A] vs t for zero, ln[A] vs t for first, 1/[A] vs t for second). (3) Half-life method: if $\frac{t_{1}}{2}$ is constant, order = 1; if $\frac{t_{1}}{2}$ proportional to [A]_0, order = 0; if $\frac{t_{1}}{2}$ proportional to 1/[A]_0, order = 2.

The key problem-solving concept is matching the correct integrated rate equation to the given data and applying the Arrhenius equation for temperature effects.