Chemical Kinetics

Chemical kinetics answers a question thermodynamics cannot: how fast does a reaction happen? In NEET this chapter is a reliable source of numerical questions — rate laws, order determination, first-order calculations and Arrhenius-type problems appear almost every year, and the formulas are few enough to master completely.

Rate of Reaction: Average, Instantaneous and Rate Expressions

The rate of a reaction measures how quickly reactant concentration falls or product concentration builds up per unit time. If a species changes in concentration by Δ[X] over a time interval Δt, the average rate over that interval is Δ[X]/Δt, with a negative sign attached for reactants so that the rate always comes out positive. Units are usually mol L⁻¹ s⁻¹ (for gaseous reactions followed by pressure, atm s⁻¹ is also used).

Average rate is a crude number because the rate itself keeps dropping as reactants get used up. The instantaneous rate is the limiting value as Δt → 0, i.e. the derivative −d[R]/dt or +d[P]/dt. Graphically, plot concentration against time and draw a tangent at the instant of interest; the slope of that tangent gives the instantaneous rate.

Stoichiometry forces us to be careful. For a reaction

aA + bB → cC + dD

the unique rate of reaction is

Rate = −(1/a)d[A]/dt = −(1/b)d[B]/dt = +(1/c)d[C]/dt = +(1/d)d[D]/dt

So in 2N₂O₅ → 4NO₂ + O₂, the rate of disappearance of N₂O₅ is twice the rate of reaction, the rate of formation of NO₂ is four times the rate of reaction, and NO₂ appears four times as fast as O₂ does.

Factors that alter reaction rate:

• Concentration (or pressure, for gases) of reactants — more collisions per second.
• Temperature — raises the fraction of molecules with sufficient energy; rate typically roughly doubles for a 10 K rise.
• Surface area of solid reactants — powdered CaCO₃ reacts far faster than a lump.
• Catalyst — supplies an alternative path of lower activation energy.
• Nature of reactants and, for photochemical reactions, radiation.

Rate Law, Order and Molecularity

The rate law (rate equation) is the experimentally determined relationship between rate and reactant concentrations. For the general reaction above it takes the form

Rate = k[A]^x[B]^y

where x and y are the orders with respect to A and B, and the overall order is x + y. The crucial point: x and y are found by experiment and need not equal the stoichiometric coefficients a and b. They can be zero, fractional, or even negative.

Examples worth remembering:

• CHCl₃ + Cl₂ → CCl₄ + HCl: Rate = k[CHCl₃][Cl₂]^{1/2} — order 1.5, a fractional order.
• 2NO + 2H₂ → N₂ + 2H₂O: Rate = k[NO]²[H₂] — order 3, not 4.
• Decomposition of NH₃ on a hot platinum surface: essentially zero order in NH₃.
• Acid-catalysed hydrolysis of an ester: pseudo first order (discussed below).

Molecularity is a different idea: it is the number of species colliding in a single elementary step, and is therefore always a whole number (1, 2 or 3; higher values are effectively impossible because simultaneous four-body collisions are too improbable). Molecularity is meaningful only for elementary reactions; order is meaningful for both elementary and complex reactions. For a multistep reaction, the observed order is governed by the slowest (rate-determining) step, and the overall reaction has no single molecularity.

Rate constant k: obtained when all concentrations are unity; it is independent of concentration but strongly dependent on temperature (and on the catalyst). Its units follow from the rate law:

• Zero order: mol L⁻¹ s⁻¹
• First order: s⁻¹
• Second order: L mol⁻¹ s⁻¹
• n-th order: (mol L⁻¹)^{1−n} s⁻¹

This unit rule is a fast way to identify order in MCQs: if k is given in s⁻¹, the reaction is first order.

Integrated Rate Laws for Zero and First Order Reactions

Zero order. Rate = k, independent of concentration. Integrating −d[R]/dt = k:

[R] = [R]₀ − kt

A plot of [R] versus t is a straight line with slope −k and intercept [R]₀. Half-life t₁/₂ = [R]₀/2k, i.e. proportional to initial concentration. Zero order behaviour is common in surface-catalysed (heterogeneous) reactions where the catalyst surface is saturated — decomposition of HI on gold, or of NH₃ on platinum — and in some photochemical reactions.

First order. Rate = k[R]. Integration gives

k = (2.303/t) log([R]₀/[R])

equivalently ln[R] = ln[R]₀ − kt, or [R] = [R]₀e^{−kt}. A plot of log[R] against t is linear with slope −k/2.303. Half-life:

t₁/₂ = 0.693/k

which is independent of the initial concentration — the signature of first order kinetics. Radioactive decay, decomposition of N₂O₅, and inversion of sucrose all follow first order kinetics.

For a gas-phase first order reaction it is often convenient to work with pressures. If A(g) → B(g) + C(g) starting from pure A at pressure p_i, and the total pressure at time t is p_t, then partial pressure of A = 2p_i − p_t, and

k = (2.303/t) log[p_i/(2p_i − p_t)]

Pseudo first order reactions are genuinely second (or higher) order reactions that behave as first order because one reactant is present in vast excess and its concentration hardly changes. Standard examples: hydrolysis of cane sugar in dilute acid and acid hydrolysis of ethyl acetate — water is the solvent, so [H₂O] is effectively constant and the rate depends only on the ester.

A useful shortcut for first order problems: after n half-lives, the fraction left is (1/2)ⁿ. So if 75% has reacted, 25% remains, which is two half-lives; 87.5% reacted corresponds to three half-lives.

Temperature Dependence: Arrhenius Equation and Activation Energy

Reaction rates are extremely sensitive to temperature. The temperature coefficient, k at (T+10)/k at T, is typically 2–3 for reactions near room temperature.

The Arrhenius equation captures this:

k = A e^{−Ea/RT}

where A is the frequency (pre-exponential) factor, Ea the activation energy, R the gas constant and T the absolute temperature. The exponential term e^{−Ea/RT} represents the fraction of molecular collisions possessing energy equal to or greater than Ea. Taking logarithms:

log k = log A − Ea/(2.303RT)

so a plot of log k against 1/T is a straight line of slope −Ea/(2.303R) — the standard way to measure Ea. Comparing two temperatures:

log(k₂/k₁) = (Ea/2.303R)[(T₂ − T₁)/(T₁T₂)]

Conceptual points to hold on to:

1. Activation energy is the minimum extra energy reactant molecules must acquire to reach the transition state (activated complex), an unstable arrangement at the top of the energy barrier.
2. Ea for the forward reaction minus Ea for the reverse reaction equals ΔH of the reaction. Exothermic reactions therefore have a smaller reverse-direction barrier... no, the reverse: for an exothermic reaction the reverse barrier is larger by |ΔH|.
3. Raising T does not change Ea; it increases the number of molecules crossing the barrier by broadening and shifting the Maxwell–Boltzmann distribution of molecular energies to higher energies.
4. Reactions with larger Ea are more strongly accelerated by a temperature rise.
5. A catalyst provides a new path with lower Ea, speeding up forward and backward reactions equally; it does not change ΔH, ΔG, or the position of equilibrium, and it is not consumed.

Collision Theory of Bimolecular Reactions

Collision theory pictures reacting molecules as hard spheres. The number of collisions per unit volume per second is the collision frequency Z. If every collision led to reaction, gases would react explosively fast, which they do not — so two conditions must be met:

• The colliding pair must have combined energy ≥ Ea (energy barrier).
• They must be correctly oriented for bonds to break and form (orientation barrier), accounted for by the steric or probability factor P.

Hence Rate = P·Z_AB·e^{−Ea/RT}, and comparing with the Arrhenius form shows that the pre-exponential factor A is essentially P·Z_AB — a measure of how often properly oriented collisions occur. Collision theory works reasonably for simple atomic and small-molecule gas reactions but becomes crude for complex molecules, since treating a polyatomic molecule as a featureless sphere ignores internal motions.

Common Mistakes and Exam Traps

1. Equating order with stoichiometric coefficients. Order comes only from experimental data or from the rate-determining elementary step. Writing Rate = k[N₂O₅]² for 2N₂O₅ → 4NO₂ + O₂ is wrong; the reaction is first order.
2. Forgetting the stoichiometric divisor. "Rate of formation of NO₂" and "rate of reaction" differ by a factor of 4 in the N₂O₅ decomposition. Read the wording carefully and use the (1/coefficient) convention.
3. Assuming half-life is always concentration-independent. That is true only for first order. For zero order t₁/₂ ∝ [R]₀; in general t₁/₂ ∝ [R]₀^{1−n}.
4. Believing a catalyst or a temperature rise changes Ea in the same way. A catalyst lowers Ea; temperature leaves Ea untouched and instead increases the fraction of molecules above it. Also remember a catalyst cannot shift equilibrium position.
5. Mixing up log and ln forms. Using k = (1/t)log([R]₀/[R]) without the 2.303 factor is the single most common numerical error in this chapter.