JEE Chemistry Chemical Kinetics Notes Set 06

Rate of Reaction

The rate of a reaction is the change in concentration of a reactant or product per unit time. For \( aA + bB \rightarrow cC + dD \), the rate of reaction is \( r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} \); dividing by the coefficient makes it the same for every species, and its unit is \( \text{mol L}^{-1}\text{ s}^{-1} \). The average rate is \( \Delta c/\Delta t \) and the instantaneous rate is the slope of the tangent to the concentration-time curve. For gases pressure may replace concentration, with unit atm s^-1.

Rate Law, Order and Molecularity

The rate law (experimental) is \( r = k[A]^m[B]^n \); m and n are the orders with respect to A and B, and \( m+n \) is the overall order (may be zero, fractional or negative). The rate constant k depends on temperature and catalyst, not on concentration, and has unit \( (\text{mol L}^{-1})^{1-n}\text{ s}^{-1} \): s^-1 for first order, L mol^-1 s^-1 for second order, mol L^-1 s^-1 for zero order. Molecularity is the number of molecules colliding in one elementary step; it is a whole number (1, 2, rarely 3) and is defined only for elementary steps, whereas order is found by experiment and applies to the overall reaction. For an elementary step the order equals the molecularity. A reaction with fractional or zero order must be complex. A pseudo-first-order reaction has a reactant in large excess whose concentration stays constant (hydrolysis of an ester in dilute acid, inversion of cane sugar).

Integrated Rate Equations

Zero order (rate = k): \( [A] = [A]_0 - kt \); \( t_{1/2} = \frac{[A]_0}{2k} \), proportional to the initial concentration; the plot of [A] against t is a straight line of slope -k. It is found in surface-catalysed reactions where the surface is saturated (decomposition of NH₃ on hot tungsten).

First order (rate = k[A]): \( k = \frac{2.303}{t}\log\frac{[A]_0}{[A]} \), \( [A] = [A]_0 e^{-kt} \), \( t_{1/2} = \frac{0.693}{k} \), independent of the initial concentration. A plot of \( \ln[A] \) against t is a straight line of slope -k. Time for 99% completion is \( \frac{2.303 \times 2}{k} \), that is, about 6.64 half-lives; \( t_{75\%} = 2t_{1/2} \). The mean (average) life is \( \tau = 1/k = 1.44\,t_{1/2} \).

Second order (rate = k[A]^2): \( \frac{1}{[A]} - \frac{1}{[A]_0} = kt \), \( t_{1/2} = \frac{1}{k[A]_0} \), inversely proportional to the initial concentration; \( t_{75\%} = 3t_{1/2} \). In general for order n (n not 1), \( t_{1/2} \propto [A]_0^{\,1-n} \), so the slope of log t_1/2 against log [A]₀ is 1 - n.

Methods of Finding the Order

Initial rate method: change one concentration at a time and compare initial rates; if doubling [A] multiplies the rate by 2^m, m is the order in A. Integrated method: test which of \( \ln[A] \), [A] or \( 1/[A] \) is linear in t. Half-life method: see how t_1/2 varies with the initial concentration. Isolation method (large excess of all but one reactant) gives pseudo-orders. Two rates at two concentrations give \( n = \ln(r_1/r_2)/\ln(c_1/c_2) \).

Following the Progress of a Reaction

For first order reactions the quantity measured can replace concentration, as long as it is proportional to the extent of reaction. Gas evolved (volume V_t and final volume V_{\( \infty \)}): \( k = \frac{2.303}{t}\log\frac{V_\infty}{V_\infty - V_t} \). Titration (acid hydrolysis of an ester, volumes V₀, V_t, V_{\( \infty \)} of alkali): \( k = \frac{2.303}{t}\log\frac{V_\infty - V_0}{V_\infty - V_t} \). Optical rotation (inversion of cane sugar, rotations r₀, r_t, r_{\( \infty \)}): \( k = \frac{2.303}{t}\log\frac{r_0 - r_\infty}{r_t - r_\infty} \). Pressure for a gaseous reaction \( A(g) \rightarrow nB(g) \): with initial pressure P_i and total pressure P_t, the pressure of A is \( P_i - x \) where \( x = \frac{P_t - P_i}{n-1} \).

Complex First Order Reactions

Parallel reactions: A forms B (k₁) and C (k₂); A decays with \( k = k_1 + k_2 \), and \( [B]/[C] = k_1/k_2 \) at all times. The overall activation energy is \( E = \frac{k_1E_1 + k_2E_2}{k_1 + k_2} \). Series reactions \( A \xrightarrow{k_1} B \xrightarrow{k_2} C \): the intermediate B reaches its maximum at \( t_{max} = \frac{\ln(k_1/k_2)}{k_1 - k_2} \). Reversible first order reactions \( A \rightleftharpoons B \): \( K = k_f/k_b \), and the approach to equilibrium is first order with rate constant \( k_f + k_b \).

Temperature Dependence: Arrhenius Equation

\( k = Ae^{-E_a/RT} \), or \( \ln k = \ln A - \frac{E_a}{RT} \); A is the frequency (pre-exponential) factor and the exponential is the fraction of collisions with energy at least E_a. A plot of ln k (or log k) against 1/T has slope \( -E_a/R \) (or \( -E_a/2.303R \)) and intercept ln A. Between two temperatures: \( \log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) \). As T tends to infinity, k tends to A. Rates of many reactions roughly double or triple for a 10 K rise (temperature coefficient 2 to 3); the rise is greater for larger E_a. For a reaction \( \Delta H = E_{a,f} - E_{a,b} \); a catalyst lowers E_a of forward and backward reactions equally, so it does not change \( \Delta H \) or the equilibrium constant, only the rate at which equilibrium is reached.

Mechanism and Rate Law

A complex reaction occurs in elementary steps; the slowest step (rate determining step) governs the rate law, and intermediates must be eliminated using fast equilibrium steps. Example: for \( 2NO + O_2 \rightarrow 2NO_2 \) with \( NO + NO \rightleftharpoons N_2O_2 \) (fast) and \( N_2O_2 + O_2 \rightarrow 2NO_2 \) (slow), rate = \( k_2K[NO]^2[O_2] \), third order, with overall \( k = k_2k_1/k_{-1} \). For \( X_2 \rightleftharpoons 2X \) (fast) followed by \( X + Y_2 \rightarrow XY + Y \) (slow), the rate is \( k[X_2]^{1/2}[Y_2] \), order 1.5. A proposed mechanism must add up to the overall equation and agree with the experimental rate law. Collision theory: only collisions with energy above the threshold and the correct orientation lead to reaction, so \( k = PZe^{-E_a/RT} \) where P is the steric factor.

Radioactive Decay

Radioactive decay is a first order process: \( N = N_0e^{-\lambda t} \), \( t_{1/2} = 0.693/\lambda \), mean life \( \tau = 1/\lambda \), activity \( A = \lambda N \). Unlike chemical reactions it is independent of temperature, pressure and the chemical state of the atom. After n half-lives the fraction left is \( (1/2)^n \). Age of a rock from the ratio of parent to daughter: \( t = \frac{t_{1/2}}{0.693}\ln\frac{N_{parent} + N_{daughter}}{N_{parent}} \) (for example, parent : daughter = 1 : 7 means 1/8 remains, three half-lives). For parallel decay to two products the total decay constant is the sum, and for a parent-daughter series the daughter number peaks when \( \lambda_1N_1 = \lambda_2N_2 \).

Quick Facts to Remember

Half-life: zero order proportional to a; first order independent of a; second order proportional to 1/a; nth order proportional to \( a^{1-n} \). Time for 75% reaction: zero order 1.5 half-lives, first order 2, second order 3. Doubling the concentration of a reactant of order n multiplies the rate by 2ⁿ. Halving the volume of a gaseous mixture multiplies each concentration by 2. For an elementary reaction rate law follows from the equation. Catalysts change the mechanism and lower E_a but do not change \( \Delta H \), \( \Delta G \) or K.