Chemical kinetics and catalysis

Chemistry (B.Sc) · Physical chemistry · free study notes for exams, placements and interviews

In short

Chemical kinetics studies how fast reactions go, what controls that speed, and what sequence of elementary steps the overall equation hides. Thermodynamics says whether a reaction can happen; kinetics says whether it will happen in a useful time, which is why reactor design, drug shelf-life and catalyst selection all rest on rate data. Catalysis is the industrial lever on rate: it lowers activation energy without being consumed, and over 90% of large-scale chemical processes use a catalyst. The topic carries heavy weight in B.Sc and M.Sc physical chemistry papers, in GATE Chemistry (XL/CY) and CSIR-UGC NET, and in technical interviews at pharma (Sun, Cipla, Dr Reddy's, Divi's), petrochemical (Reliance, IOCL, HPCL), fertiliser (IFFCO, RCF) and catalyst (Johnson Matthey India, Süd-Chemie/Clariant) employers.

Explained

Rate of reaction, rate laws, order and molecularity

Rate of reaction: change of concentration per unit time. For aA + bB → cC, the unique rate = −(1/a)d[A]/dt = −(1/b)d[B]/dt = +(1/c)d[C]/dt. SI unit mol m⁻³ s⁻¹; in practice mol L⁻¹ s⁻¹.
Average rate is Δ[A]/Δt over an interval; instantaneous rate is the tangent slope at one instant. Initial rate (t → 0) is used experimentally because products do not yet interfere.
Rate law: rate = k[A]^m[B]ⁿ, determined only by experiment, never from stoichiometry.
Order: the exponent of a species in the rate law; overall order = m + n. Can be zero, fractional (chain reactions), or negative (an inhibitor, e.g. 2O₃ → 3O₂ is inhibited by O₂).
Molecularity: number of species colliding in one elementary step — always a small positive integer (1, 2, rarely 3) and defined only for elementary steps.
Units of k depend on order: zero order mol L⁻¹ s⁻¹; first order s⁻¹; second order L mol⁻¹ s⁻¹; general nth order (mol L⁻¹)¹⁻ⁿ s⁻¹.
Pseudo-order: when one reactant is in large excess its concentration is effectively constant and folds into k. Acid hydrolysis of ester or cane-sugar inversion is bimolecular but observed first order.
Methods of determining order: (1) integration/trial — fit data to each integrated equation; (2) graphical — plot the function that gives a straight line; (3) half-life method — t½ ∝ a¹⁻ⁿ; (4) differential (van't Hoff) method — log(rate) vs log(concentration), slope = order; (5) Ostwald's isolation method — vary one reactant, hold others in excess.

Integrated rate equations and half-lives

Zero order: rate independent of concentration. [A] = [A]₀ − kt; linear [A] vs t; t½ = [A]₀/2k; completes in finite time t = [A]₀/k. Seen in photochemical reactions and surface reactions on a saturated catalyst (NH₃ on tungsten).
First order: ln([A]₀/[A]) = kt, i.e. [A] = [A]₀e^−kt; plot ln[A] vs t gives slope −k. t½ = 0.693/k, independent of initial concentration; t₉₉.₉ ≈ 10 t½. All radioactive decay is first order.
Second order (equal concentrations): 1/[A] − 1/[A]₀ = kt; plot 1/[A] vs t; t½ = 1/(k[A]₀) — doubles if concentration is halved.
Second order (unequal, a ≠ b): kt = [1/(a−b)]·ln[b(a−x)/{a(b−x)}].
Third order (equal): 1/[A]² − 1/[A]₀² = 2kt; t½ = 3/(2k[A]₀²).
nth order (n ≠ 1): kt = [1/(n−1)]·{1/[A]ⁿ⁻¹ − 1/[A]₀ⁿ⁻¹}; t½ = (2ⁿ⁻¹ − 1)/{(n−1)k[A]₀ⁿ⁻¹}.
Opposing (reversible) first order: k_f + k_b obtained from approach to equilibrium; K = k_f/k_b — the kinetic route to the equilibrium constant.
Consecutive first order (A→B→C): [B] passes through a maximum at t_max = ln(k₂/k₁)/(k₂−k₁); the slower step controls product appearance.
Parallel (side) reactions: k_obs = k₁ + k₂; product ratio = k₁/k₂, the basis of kinetic control in synthesis.

Temperature dependence, activation energy and theories of reaction rate

Rate roughly doubles to trebles for every 10 K rise near room temperature; temperature coefficient = k₍T+10₎/k_T ≈ 2–3.
Arrhenius equation: k = A e^−Ea/RT. A = pre-exponential/frequency factor (same units as k); Ea = activation energy (kJ mol⁻¹); R = 8.314 J K⁻¹ mol⁻¹.
Plot ln k vs 1/T: straight line, slope = −Ea/R, intercept = ln A. Two-point form gives Ea from two rate constants.
Energy profile: reactants climb to a transition state at the top of the barrier; Ea(forward) − Ea(backward) = ΔH of reaction. A catalyst lowers both equally, so K is unchanged.
Threshold energy = energy of the activated state; Ea = threshold energy − average energy of reactants.
Collision theory (Arrhenius–Trautz–Lewis): rate = Z·e^−Ea/RT for hard spheres; corrected to rate = P·Z·e^−Ea/RT, where P is the steric/probability factor. Works for simple bimolecular gas reactions; fails badly (P ~ 10⁻⁵) for reactions of complex molecules because it ignores orientation and internal modes.
Transition state (activated complex) theory, Eyring: reactants form an activated complex in quasi-equilibrium that decomposes with frequency k_BT/h.
- k = (k_BT/h)·K‡ and in thermodynamic form k = (k_BT/h)·e^−ΔG‡/RT = (k_BT/h)·e^ΔS‡/R·e^−ΔH‡/RT.
- ΔH‡ ≈ Ea − RT for a solution/unimolecular reaction; a large negative ΔS‡ means an ordered, tightly bound complex (associative) — this interprets the steric factor physically.
Lindemann–Hinshelwood mechanism explains why unimolecular gas reactions are first order at high pressure and second order at low pressure: A + M ⇌ A* + M, A* → P; k_obs = k₂k₁[M]/(k₋₁[M] + k₂).
Steady-state approximation: for a reactive intermediate I, d[I]/dt ≈ 0 — the standard tool for deriving rate laws from mechanisms.
Rate-determining step: the slowest elementary step; the overall order follows from it plus any pre-equilibria.
Chain reactions: initiation, propagation, (branching), termination. H₂ + Br₂ gives rate = k[H₂][Br₂]^1/2/(1 + k′[HBr]/[Br₂]) — fractional order is the fingerprint of a chain. Branching chains (H₂ + O₂) give explosion limits.

Catalysis: principles and types

Catalyst: a substance that changes reaction rate by providing an alternative path of lower activation energy, and is recovered chemically unchanged (though it may change physically). Berzelius, 1836.
Characteristics: small amounts suffice; does not change ΔG, ΔH or K; cannot make a thermodynamically forbidden reaction occur; speeds forward and reverse equally; often specific; has an optimum temperature.
Positive catalysis (MnO₂ in KClO₃ decomposition), negative catalysis/inhibition (glycerol in H₂O₂ decomposition, tetraethyl lead as an anti-knock), autocatalysis (a product catalyses, e.g. Mn²⁺ in KMnO₄–oxalic acid titration — hence the sigmoid rate curve).
Homogeneous catalysis: catalyst and reactants in one phase — acid/base-catalysed ester hydrolysis, NO in the lead-chamber process, Wilkinson's catalyst and other soluble metal complexes, hydroformylation (oxo process). Advantages: high selectivity, mild conditions. Limitation: separating catalyst from product.
Heterogeneous (contact) catalysis: different phases, usually solid catalyst with gas/liquid reactants — V₂O₅ in Contact process, Fe/K₂O/Al₂O₃ in Haber process, Pt–Rh gauze in Ostwald process, Ni in hydrogenation, zeolite Y (with ZSM-5 additive) in FCC and Pt–Re/Al₂O₃ in catalytic reforming. Easy separation and continuous operation; limitation: poisoning, diffusion limitation, lower selectivity.
Mechanism on surfaces: diffusion → adsorption (chemisorption at active centres) → surface reaction → desorption → diffusion away. Rate expressions follow Langmuir–Hinshelwood kinetics; at low pressure rate ∝ p (first order), at high pressure θ → 1 and rate becomes zero order.
Promoters increase activity (Mo or K₂O with Fe in Haber); poisons destroy it (As on V₂O₅ or Pt, S and Pb on vehicle catalysts, CO on Pt at low temperature); inhibitors reduce it reversibly; catalyst support/carrier (alumina, silica, carbon) disperses the active metal and gives mechanical strength.
Acid–base catalysis: specific H₃O⁺/OH⁻ catalysis (rate depends only on pH) versus general acid–base catalysis (any Brønsted acid contributes); Brønsted catalysis law k = G·K_a^α.
Enzyme catalysis: proteins, extremely efficient (10⁶–10¹² fold), highly specific, work near 37 °C and physiological pH, with a maximum activity at optimum T and pH and denaturation beyond. Lock-and-key/induced-fit models; inhibition is competitive (K_M rises, v_max same), non-competitive (v_max falls) or uncompetitive.
Phase-transfer catalysis (quaternary ammonium salts), photocatalysis (TiO₂ for dye degradation in effluent treatment), electrocatalysis (Pt, IrO₂ in electrolysers and fuel cells), shape-selective catalysis (zeolite pore size selects reactants/products).

Experimental methods and Indian practice

Chemical methods: quench aliquots and titrate (ester hydrolysis against standard alkali; H₂O₂ against KMnO₄).
Physical methods: total pressure for gas reactions, conductometry (ionic reactions), polarimetry (sucrose inversion), spectrophotometry/UV-Vis (most common in B.Sc labs), dilatometry, pH-stat.
Fast reactions: stopped-flow, flash photolysis, relaxation (temperature-jump), NMR line-broadening; femtosecond spectroscopy for elementary steps.
Regulatory use of kinetics in India: stability testing of drugs follows ICH Q1A(R2) as adopted by CDSCO under the Drugs and Cosmetics Act, 1940 and Rules; accelerated studies (typically 40 °C/75% RH for Zone IVb, which covers India) use Arrhenius-type extrapolation to assign shelf-life/expiry. Exact storage conditions and durations are specified in the current CDSCO/ICH guidance and are revised from time to time.
Vehicle-exhaust three-way catalysts are mandated through BS-VI norms (nationwide from April 2020, Phase 2 from April 2023) notified under the Central Motor Vehicles Rules.
Reaction-rate data feed reactor sizing in chemical engineering (CSTR, PFR, batch) and hazard assessment under the Manufacture, Storage and Import of Hazardous Chemical Rules; runaway-reaction screening uses accelerating-rate calorimetry.

Key points

Rate law — experimental; order has no necessary relation to stoichiometric coefficients.
Order vs molecularity — order is experimental and can be zero/fractional; molecularity is theoretical, integral, and only for elementary steps.
First-order t½ — 0.693/k, independent of starting concentration; the only order with this property.
Zero order — rate = k, [A] falls linearly, reaction ends at t = [A]₀/k; typical of saturated surfaces and photochemistry.
Second order t½ — 1/(k[A]₀), inversely proportional to initial concentration.
Units of k — (concentration)¹⁻ⁿ(time)⁻¹; a quick check on a claimed order.
Arrhenius — ln k = ln A − Ea/RT; slope of ln k vs 1/T gives −Ea/R.
Activation energy — the minimum extra energy colliding molecules need; higher Ea means greater temperature sensitivity.
Catalyst and equilibrium — lowers Ea both ways, raises rate, leaves K, ΔG and yield unchanged.
Collision theory — rate = PZe^−Ea/RT; P corrects for orientation and is where the theory is weakest.
Eyring equation — k = (k_BT/h)e^−ΔG‡/RT; gives ΔH‡ and ΔS‡, so mechanism information.
Steady-state approximation — set d[intermediate]/dt = 0 to derive the observed rate law.
Lindemann mechanism — unimolecular reactions fall off to second order at low pressure.
Pseudo-first order — bimolecular reaction with one reactant in excess (ester or sucrose hydrolysis).
Autocatalysis — product accelerates; sigmoid concentration–time curve (KMnO₄–oxalic acid).
Promoter vs poison — promoter enhances catalyst activity; poison blocks active sites (As, S, Pb).
Michaelis–Menten — v = v_max[S]/(K_M + [S]); K_M is the substrate concentration at half v_max and inversely indicates affinity.
Enzymes — proteins with huge rate enhancement, extreme specificity, and sharp optima in temperature and pH.

Formulas and examples

Rate = −(1/a)d[A]/dt = (1/c)d[C]/dt — unit mol L⁻¹ s⁻¹
Rate = k[A]^m[B]ⁿ; overall order n = m + n (n = overall order); units of k = (mol L⁻¹)¹⁻ⁿ s⁻¹
Zero order: [A] = [A]₀ − kt; t½ = [A]₀/2k
First order: k = (2.303/t)·log([A]₀/[A]); t½ = 0.693/k (k in s⁻¹)
Second order (equal): 1/[A] = 1/[A]₀ + kt; t½ = 1/(k[A]₀); k in L mol⁻¹ s⁻¹
nth order: t½ = (2ⁿ⁻¹ − 1)/[(n−1)k[A]₀ⁿ⁻¹]
Arrhenius: k = A·e^−Ea/RT; log(k₂/k₁) = (Ea/2.303R)·(T₂−T₁)/(T₁T₂); R = 8.314 J K⁻¹ mol⁻¹
Collision theory: k = P·Z·e^−Ea/RT; Z = collision number (collisions L⁻¹ s⁻¹)
Eyring: k = (k_BT/h)·e^−ΔG‡/RT; k_B = 1.381×10⁻²³ J K⁻¹, h = 6.626×10⁻³⁴ J s
Michaelis–Menten: v = v_max[S]/(K_M + [S]); Lineweaver–Burk: 1/v = (K_M/v_max)(1/[S]) + 1/v_max
First-order half-life from titre (sucrose/ester): k = (2.303/t)·log[(r₀ − r∞)/(r_t − r∞)]

Example 1 — first-order rate constant and completion time. The first-order decomposition of N₂O₅ has k = 6.93 × 10⁻⁴ s⁻¹ at 318 K.

t½ = 0.693/k = 0.693/(6.93 × 10⁻⁴) = 1,000 s (16.7 min).
Time for 75% decomposition = 2 × t½ = 2,000 s.
Time for 90%: k = (2.303/t)log(100/10) → t = 2.303/(6.93 × 10⁻⁴) = 3,323 s (= 3.32 t½).

Example 2 — order from initial-rate data. For 2NO + O₂ → 2NO₂:

Run	[NO] mol L⁻¹	[O₂] mol L⁻¹	Initial rate mol L⁻¹ s⁻¹
1	0.10	0.10	2.5 × 10⁻³
2	0.20	0.10	1.0 × 10⁻²
3	0.10	0.20	5.0 × 10⁻³

Runs 1→2: [NO] doubled, rate × 4 → order in NO = 2.
Runs 1→3: [O₂] doubled, rate × 2 → order in O₂ = 1. Overall order = 3.
k = rate/([NO]²[O₂]) = 2.5 × 10⁻³/(0.010 × 0.10) = 2.5 L² mol⁻² s⁻¹.

Example 3 — activation energy from two temperatures. k = 2.0 × 10⁻³ s⁻¹ at 300 K and 1.6 × 10⁻² s⁻¹ at 330 K.

log(k₂/k₁) = log 8 = 0.9031.
(T₂ − T₁)/(T₁T₂) = 30/(300 × 330) = 30/99,000 = 3.0303 × 10⁻⁴ K⁻¹.
Ea = 2.303 × 8.314 × 0.9031/3.0303 × 10⁻⁴ = 17.29/3.0303 × 10⁻⁴ ≈ 5.706 × 10⁴ J mol⁻¹ = 57.1 kJ mol⁻¹.
A from k = A e^−Ea/RT at 300 K: Ea/RT = 57,060/(8.314 × 300) = 22.88; e^22.88 ≈ 8.7 × 10⁹ → A ≈ 2.0 × 10⁻³ × 8.7 × 10⁹ ≈ 1.7 × 10⁷ s⁻¹.

Example 4 — effect of a catalyst at constant temperature. A reaction with Ea = 100 kJ mol⁻¹ at 500 K is catalysed so that Ea falls to 75 kJ mol⁻¹ (A unchanged).

k_cat/k = e^ΔEa/RT = e^{25,000/(8.314 × 500)} = e^6.014 ≈ 409.
So a 25 kJ mol⁻¹ drop in Ea gives a ~400-fold speed-up, while ΔG and the equilibrium yield stay exactly the same.

Example 5 — Michaelis–Menten. An enzyme shows v_max = 2.4 × 10⁻⁵ mol L⁻¹ s⁻¹ and K_M = 5.0 × 10⁻⁴ mol L⁻¹.

At [S] = 1.0 × 10⁻⁴ M: v = (2.4 × 10⁻⁵ × 1.0 × 10⁻⁴)/(5.0 × 10⁻⁴ + 1.0 × 10⁻⁴) = 2.4 × 10⁻⁹/6.0 × 10⁻⁴ = 4.0 × 10⁻⁶ mol L⁻¹ s⁻¹ (one-sixth of v_max).
At [S] = 5.0 × 10⁻³ M (10 K_M): v = (2.4 × 10⁻⁵ × 5.0 × 10⁻³)/(5.5 × 10⁻³) = 2.18 × 10⁻⁵ mol L⁻¹ s⁻¹, i.e. 91% of v_max — saturation, effectively zero order in S.

Feature	Zero order	First order	Second order
Rate law	k	k[A]	k[A]²
Linear plot	[A] vs t	ln[A] vs t	1/[A] vs t
Units of k	mol L⁻¹ s⁻¹	s⁻¹	L mol⁻¹ s⁻¹
t½	[A]₀/2k	0.693/k	1/(k[A]₀)
t½ vs [A]₀	∝ [A]₀	independent	∝ 1/[A]₀
Example	NH₃ on hot W; photochemical	N₂O₅ decomposition, radioactive decay	2NO₂ → 2NO + O₂; saponification

Feature	Homogeneous	Heterogeneous	Enzymatic
Phase	Same as reactants	Separate (usually solid)	Aqueous, colloidal protein
Selectivity	High	Moderate	Extremely high
Conditions	Mild	Often high T, p	25–40 °C, narrow pH
Separation	Difficult	Easy (filtration/fixed bed)	Immobilisation needed
Example	H⁺ in ester hydrolysis; oxo process	Fe in Haber, V₂O₅ in Contact	Urease, invertase, amylase

Where it is used

Pharmaceutical industry: accelerated stability studies and expiry dating (first-order degradation, Arrhenius extrapolation) at Sun Pharma, Cipla, Dr Reddy's, Lupin; dissolution and release kinetics in formulation development; catalytic hydrogenation and asymmetric catalysis in API routes (Divi's, Laurus).
Petrochemicals and refining: fluid catalytic cracking, hydrocracking, catalytic reforming and isomerisation at Reliance Jamnagar, IOCL, BPCL, HPCL; catalyst-deactivation and regeneration cycles; CPCL/IOC R&D at Faridabad.
Fertiliser and heavy chemicals: Haber ammonia synthesis (IFFCO, KRIBHCO, Chambal), Contact process sulphuric acid, Ostwald nitric acid — all catalyst- and kinetics-controlled.
Environment and energy: three-way catalytic converters and SCR for BS-VI vehicles; photocatalytic and advanced-oxidation effluent treatment under CPCB/SPCB consent conditions; electrocatalysts for green hydrogen electrolysers under the National Green Hydrogen Mission.
Food, dairy and biotech: enzyme kinetics in brewing, invertase in confectionery, amylase and protease in detergents and starch processing; shelf-life prediction under FSSAI labelling requirements.
Job roles: QC/stability analyst, R&D chemist, process development chemist, catalyst scientist, reaction-engineering roles at NCL Pune, IITs, CSIR labs, and analytical roles that run UV-Vis/HPLC kinetic assays.

Interview questions

Q: How do order and molecularity differ? A: Order is the experimentally found dependence of rate on concentration and may be zero, fractional or negative; molecularity is the number of species in one elementary step and is always a small whole number.
Q: Why is the half-life of a first-order reaction independent of initial concentration? A: Because the integrated law contains only the concentration ratio, t½ = ln2/k, so any starting amount halves in the same time — the basis of radioactive dating and drug half-lives.
Q: A reaction has k with units L mol⁻¹ s⁻¹. What is its order? A: Second order overall, since units of k are (mol L⁻¹)¹⁻ⁿ s⁻¹.
Q: Does a catalyst change the equilibrium constant? A: No. It lowers the activation energy of forward and reverse steps equally, so it reaches equilibrium faster but K, ΔG and the equilibrium yield are unchanged.
Q: Why does the hydrolysis of an ester in dilute acid appear first order when it involves water as well? A: Water is present in large excess so its concentration is effectively constant and folds into the rate constant — a pseudo-first-order reaction.
Q: What does a fractional order tell you? A: That the reaction is not elementary — usually a chain mechanism or a pre-equilibrium involving dissociation, as in H₂ + Br₂ where rate ∝ [Br₂]^½.
Q: Where does collision theory fail, and how does transition state theory improve on it? A: Collision theory treats molecules as hard spheres and needs an empirical steric factor P; transition state theory treats the activated complex thermodynamically and gives P meaning through ΔS‡.
Q: Explain the steady-state approximation. A: For a highly reactive intermediate present in trace amounts, its net rate of formation is set to zero, letting you eliminate it and obtain the observed rate law from the mechanism.
Q: Why does a unimolecular gas reaction become second order at low pressure? A: Activation requires collisions; at low pressure collisional activation is slower than the decomposition of the energised molecule, so the collision step becomes rate-determining (Lindemann).
Q: What is autocatalysis and give an Indian lab example? A: A product catalyses its own reaction; in the KMnO₄–oxalic acid titration Mn²⁺ formed accelerates the reaction, so the first drops decolourise slowly and later ones fast.
Q: Distinguish a promoter, a poison and an inhibitor. A: A promoter raises a catalyst's activity (K₂O with Fe in ammonia synthesis); a poison blocks active sites irreversibly (As on V₂O₅, Pb on Pt); an inhibitor slows the reaction, usually reversibly.
Q: What does K_M mean physically? A: It is the substrate concentration giving half the maximum velocity; a low K_M means strong enzyme–substrate affinity, and K_M rises in competitive inhibition while v_max stays the same.
Q: Can a catalyst start a thermodynamically unfavourable reaction? A: No — it changes only the rate; if ΔG is positive under the given conditions the reaction will not proceed to products regardless of catalyst.

Common mistakes

Writing the rate law from the balanced equation's coefficients instead of experimental data.
Treating order and molecularity as the same thing, or quoting a fractional molecularity.
Forgetting the stoichiometric divisor, so −d[A]/dt is reported as "the" rate when a ≠ 1.
Using t½ = 0.693/k for every order; it holds only for first order.
Mixing k in min⁻¹ with t in seconds, or leaving Ea in kJ while R is in J K⁻¹ mol⁻¹ (a factor of 1,000 error).
Saying a catalyst shifts equilibrium or increases yield, rather than only the rate of approach.
Using the 2.303 factor with natural logs (it belongs with log₁₀) or omitting it with log₁₀.
In the lab, not quenching aliquots or not thermostatting the reaction mixture, so the "constant" k drifts with temperature.