JEE Chemistry Chemical Kinetics Notes Set 04

Rate of Reaction

The rate of a reaction is the change in concentration of a reactant or product per unit time. For aA + bB → cC + dD the rate of reaction is \( r=-\frac1a\frac{d[A]}{dt}=-\frac1b\frac{d[B]}{dt}=\frac1c\frac{d[C]}{dt}=\frac1d\frac{d[D]}{dt} \): each rate of change is divided by its stoichiometric coefficient, with a negative sign for reactants. The average rate is measured over a time interval, the instantaneous rate is the slope of the concentration-time curve at one instant.
For gases the rate may be followed through partial pressures, and for reactions that change the total pressure or the volume of gas, pressure or volume measurements give the extent of reaction. Other methods include titration, conductance, optical rotation and colorimetry.

Rate Law, Order and Molecularity

The rate law (rate equation) expresses the rate as a function of concentrations, \( \text{rate}=k[A]^m[B]^n \), where m and n are determined by experiment and the overall order is m + n. The order can be zero, fractional or negative; it need not equal the stoichiometric coefficients. The rate constant k depends on temperature (and catalyst) but not on concentration. Its unit is \( (\text{mol L}^{-1})^{1-n}\ \text{s}^{-1} \): s⁻¹ for first order, mol L⁻¹ s⁻¹ for zero order and L mol⁻¹ s⁻¹ for second order.
Molecularity is the number of molecules colliding in an elementary step (1, 2 or rarely 3). It is a whole number and is defined only for elementary steps. For an elementary reaction the order equals the molecularity. A complex reaction proceeds through several steps; the slowest step (rate determining step) fixes the rate law, and intermediates are eliminated using the equilibrium steps that precede it.
Pseudo first order reactions are really bimolecular, but one reactant is in such large excess that its concentration stays constant, for example acid-catalysed hydrolysis of an ester and inversion of cane sugar.
To find the order from data, compare experiments in which one concentration changes while the others are fixed: if doubling a concentration multiplies the rate by 2ˣ, the order in that species is x.

Integrated Rate Equations

In general the half-life of an nth order reaction is proportional to \( [A]_0^{\,1-n} \). For a first order reaction the half-life is independent of the initial concentration, so equal fractions of the reactant are consumed in equal times; after n half-lives the fraction left is (1/2)ⁿ. The time for x% completion is \( t=\frac{2.303}{k}\log\frac{100}{100-x} \).
First order examples: radioactive decay, decomposition of N₂O₅ and H₂O₂. For a gas-phase reaction A → B + C followed by pressure, \( k=\frac{2.303}{t}\log\frac{2P_0}{3P_0-P_t} \). For parallel first order reactions with constants k₁ and k₂, the overall constant is k₁ + k₂ and the products form in the ratio k₁ : k₂.

Worked Example 1. A first order reaction is 75% complete in 32 minutes. Find the half-life.
Answer:
75% completion needs two half-lives, since 100% → 50% → 25% remains.
Hence t₁/₂ = 32/2 = 16 minutes, and k = 0.693/16 = 0.0433 min⁻¹.

Temperature Dependence and Collision Theory

For many reactions the rate roughly doubles or triples for every 10 K rise (temperature coefficient 2 to 3). The Arrhenius equation explains this: \( k=Ae^{-E_a/RT} \), or \( \log k=\log A-\frac{E_a}{2.303RT} \). A plot of log k against 1/T is a straight line of slope −Ea/(2.303R). Comparing two temperatures: \( \log\frac{k_2}{k_1}=\frac{E_a}{2.303R}\left(\frac{T_2-T_1}{T_1T_2}\right) \).
According to collision theory, reaction needs collisions between molecules with energy at least the threshold energy and with the proper orientation. The activation energy Ea is the minimum extra energy over the average energy of reactants. The factor e^(−Ea/RT) is the fraction of molecules with energy equal to or greater than Ea; it grows quickly with temperature, which is why the rate rises sharply, while the collision frequency rises only slightly. A is the frequency factor, related to the number of collisions with proper orientation.
A catalyst provides a different path with lower activation energy, so the rate rises. It lowers Ea for the forward and the reverse reactions equally, so it does not change ΔH or the equilibrium constant. In an exothermic reaction, E(forward) < E(reverse); ΔH = Ea(forward) − Ea(reverse).