Chemical kinetics

In short

Chemical kinetics is the study of how fast a chemical reaction proceeds, what the rate depends on (concentration, temperature, catalyst, pressure) and what mechanism lies behind the observed rate expression. It supplies the rate law -r_A = f(C_A, T) that every reactor sizing calculation then integrates, so no reactor volume, no catalyst loading and no runaway-safety margin can be computed without it. In Indian universities it is the opening block of Chemical Reaction Engineering I and carries 8-12 marks in GATE (CH) every year, usually as a rate-law/order problem, an Arrhenius problem or a mechanism-reduction problem. Employers test it in placement rounds at Reliance Industries, IOCL, BPCL, HPCL, ONGC, GAIL, Tata Chemicals, UPL, Aarti Industries, Deepak Nitrite, Sun Pharma, Dr. Reddy's and process-design firms such as Technip Energies, L&T Technology Services and Toyo Engineering India. It also underlies PSM/HAZOP work governed by the Manufacture, Storage and Import of Hazardous Chemical Rules, 1989 and the Static and Mobile Pressure Vessels Rules.

Explained

Rate of reaction: definitions and conventions

• Intensive rate for species A: r_A = (1/V) dN_A/dt (mol per m³ per s) - rate per unit volume of reacting fluid. This is the definition used for homogeneous reactions.
• Sign convention: r_A is negative for a reactant, so textbooks write -r_A as a positive number, "the rate of disappearance of A".
• Other bases are used when volume is not the natural measure: per mass of catalyst -r'_A (mol/kg·s), per unit interfacial area -r''_A (mol/m²·s), per unit volume of reactor -r'''_A. Mixing bases is a classic exam trap.
• Stoichiometric coupling: for aA + bB → cC + dD, (-r_A)/a = (-r_B)/b = r_C/c = r_D/d. Rates of different species are never numerically equal unless the coefficients are.
• Conversion X_A = (N_A0 - N_A)/N_A0; for constant-density systems X_A = (C_A0 - C_A)/C_A0, so C_A = C_A0(1 - X_A).
• Variable-volume (gas-phase) systems: V = V_0(1 + ε_A X_A) at constant T and P, where ε_A = y_A0 δ and δ = (d + c - b - a)/a is the change in moles per mole of A reacted. Then C_A = C_A0(1 - X_A)/(1 + ε_A X_A). With T and P changing, C_A = C_A0 (1 - X_A)/(1 + ε_A X_A) · (T_0/T)(P/P_0).
• Homogeneous vs heterogeneous: homogeneous reactions occur in one phase (liquid-phase esterification); heterogeneous ones occur at an interface (gas over a solid catalyst), where observed rates may be masked by transport. Intrinsic kinetics must be measured free of diffusion limitation.

Rate laws, order and molecularity

• Power-law rate law: -r_A = k C_A^α C_B^β; α is the order with respect to A, β with respect to B, and the overall order n = α + β.
• Order is experimental, obtained by fitting data; molecularity is the number of molecules taking part in an elementary step and is a mechanistic integer. They coincide only for elementary reactions.
• Order may be zero, fractional or negative. Zero order: rate independent of concentration (surface-saturated catalysis, photochemical reactions). Negative order: a species inhibits the reaction (product inhibition).
• Elementary reaction: the rate law follows the stoichiometry directly - A + B → products with -r_A = k C_A C_B. Non-elementary: the rate law does not match stoichiometry, e.g. H₂ + Br₂ → 2HBr with r_HBr = k C_H2 C_Br2^0.5 /(1 + k' C_HBr/C_Br2).
• Units of k: from -r_A = k C_A^n, k has units (concentration)^(1-n) (time)^(-1). For n = 0: mol/m³·s; n = 1: s⁻¹; n = 2: m³/mol·s. Writing k without units loses marks.
• Pseudo-order: when one reactant is in large excess its concentration barely changes, so it is absorbed into the constant - k' = k C_B0. Hydrolysis of an ester in excess water is second order but behaves first order (pseudo-first-order).
• Reversible reactions: A ⇌ B gives -r_A = k_f C_A - k_b C_B; at equilibrium the net rate is zero, so K_C = k_f/k_b. Thermodynamics fixes the ratio, kinetics fixes the magnitudes.

Temperature dependence

• Arrhenius equation: k = A exp(-E/RT). A is the pre-exponential (frequency) factor with the same units as k, E the activation energy (J/mol or kJ/mol), R = 8.314 J/mol·K, T absolute temperature in K.
• Linear form ln k = ln A - E/(RT): a plot of ln k against 1/T is a straight line of slope -E/R - the standard graphical question.
• Interpretation: E is the minimum energy barrier colliding molecules must cross; a high E means a strongly temperature-sensitive reaction, a low E a weakly sensitive one. Most industrial reactions have E in the range 40-250 kJ/mol.
• The thumb rule "rate doubles for every 10 K" holds only near E ≈ 50-60 kJ/mol around room temperature; it is not a general law.
• Collision theory (gas phase): k = p Z exp(-E/RT) - collision frequency Z times a steric/probability factor p times the fraction of sufficiently energetic collisions; predicts A ∝ T^0.5.
• Transition state (activated complex) theory: reactants form an activated complex in quasi-equilibrium; k = (k_B T/h) exp(ΔS‡/R) exp(-ΔH‡/RT), giving A ∝ T. Over a modest temperature range the exponential dominates, so the Arrhenius form fits both.
• Why temperature is the main handle: doubling concentration doubles a first-order rate; a 20 K rise with E = 100 kJ/mol (350 K to 370 K) multiplies it about 6.4 times. That is also why exothermic reactions can run away - an important safety point in HAZOP studies.
• Catalysts lower E by offering an alternative path; they change k, never K_C, so they cannot shift equilibrium conversion.

Integrated rate laws for constant-volume batch systems

Starting from -dC_A/dt = k C_A^n for a batch reactor of constant volume:

• Zero order: C_A = C_A0 - k t; linear in C_A vs t; reaction stops abruptly when C_A reaches zero; t_1/2 = C_A0/(2k) - proportional to initial concentration.
• First order: ln(C_A0/C_A) = k t, i.e. C_A = C_A0 e^(-kt); in terms of conversion -ln(1 - X_A) = k t; t_1/2 = ln2/k = 0.693/k - independent of initial concentration.
• Second order, single reactant (2A → P or A + B with C_A0 = C_B0): 1/C_A - 1/C_A0 = k t; X_A/(1 - X_A) = k C_A0 t; t_1/2 = 1/(k C_A0) - inversely proportional to initial concentration.
• Second order, unequal initial concentrations A + B → P, with M = C_B0/C_A0 ≠ 1: ln[(M - X_A)/(M(1 - X_A))] = C_A0 (M - 1) k t.
• nth order (n ≠ 1): C_A^(1-n) - C_A0^(1-n) = (n - 1) k t; t_1/2 = [2^(n-1) - 1] / [k (n-1) C_A0^(n-1)].
• General half-life diagnostic: t_1/2 ∝ C_A0^(1-n), so a log-log plot of half-life against initial concentration gives slope (1 - n) - a fast way to find order from two experiments.
• First-order reversible A ⇌ B: -ln[1 - X_A/X_Ae] = (k_f + k_b) t, where X_Ae is the equilibrium conversion; the slope gives the sum of constants and K_C gives the ratio, so both are recovered.
• Variable-volume batch (constant-pressure, gas phase): use V = V_0(1 + ε_A X_A); for first order, -ln(1 - X_A) = k t still holds because the first-order form cancels the volume change, while higher orders do not.

Analysis of kinetic data: how order and k are found

Three standard methods, all examined:

1. Integral method - assume an order, plot the corresponding linearised group (C_A, ln C_A, 1/C_A) against t, and accept the order that gives the best straight line; k comes from the slope. Simple and robust to scattered data, but trial-and-error and poor for fractional orders.
2. Differential method - obtain -dC_A/dt from the C_A vs t curve (graphical slopes or finite differences), then plot ln(-dC_A/dt) against ln C_A: slope = order n, intercept = ln k. Handles any order including fractional, but differentiating noisy data amplifies error.
3. Method of initial rates - measure the initial slope at several different C_A0; ln(-r_A0) vs ln C_A0 gives n. Avoids product-inhibition effects entirely, but needs many runs.

Supporting techniques:

• Half-life method: run at two or more C_A0 and use t_1/2 ∝ C_A0^(1-n).
• Method of excess (isolation): flood all but one reactant to collapse the rate law to pseudo-order in that one species, then repeat for each species.
• Experimental reactors: constant-volume batch (liquid, follow concentration or conductivity), constant-pressure batch (follow volume), differential (packed-bed) reactor with small conversion so -r_A ≈ F_A0 X / W is read directly, and a recycle/CSTR (mixed-flow) reactor where the algebraic mass balance gives the rate without any differentiation - the most accurate for intrinsic kinetics.
• Pressure-based gas data: for a constant-volume gas reaction, total pressure is linear in the extent of reaction, so P_total data converts to C_A through stoichiometry.
• Fit quality is judged by linearity/residuals, not by a single point; and a rate law valid over 10 K must never be extrapolated 100 K.

Mechanisms, rate-determining step and approximations

• A mechanism is a set of elementary steps whose sum is the overall stoichiometry; it must also reproduce the observed rate law, which is the real test.
• Intermediates (free radicals, ions, complexes, adsorbed species) appear in steps but not in the overall equation and cannot stay in the final rate law.
• Rate-determining step (RDS) approximation: if one step is far slower, the overall rate equals that step's rate with the faster steps assumed at equilibrium.
• Pseudo-steady-state hypothesis (PSSH): for a reactive intermediate present at tiny concentration, set r_intermediate ≈ 0, solve algebraically for it and substitute. This is the standard GATE method.
• Lindemann mechanism for unimolecular reactions: A + M ⇌ A* + M, A* → P gives -r_A = k1 k2 C_A C_M/(k_-1 C_M + k2). At high pressure (large C_M) it is first order; at low pressure, second order. It explains why "unimolecular" decompositions change order with pressure.
• Chain reactions: initiation → propagation → (branching) → termination; thermal cracking of naphtha and ethane in petrochemical plants, and polymerisation, follow this pattern. Branching chains underlie explosions.
• Autocatalysis: a product accelerates the reaction, -r_A = k C_A C_P; the rate-vs-conversion curve passes through a maximum, which later governs reactor choice.
• Note the limit of all of this: a mechanism consistent with the data is not proved by it - several mechanisms can give the same rate law.

Key points

• Rate of reaction - moles reacted per unit volume per unit time; always state the basis and the species.
• Rate law - the experimentally determined dependence -r_A = k(T) f(C); it is an input to reactor design, never an output.
• Order - the exponent on concentration in the fitted rate law; experimental, possibly fractional, zero or negative.
• Molecularity - number of species in an elementary step; a positive integer, equal to order only for elementary reactions.
• Units of k - (conc)^(1-n)(time)^(-1); use them to check that a quoted order is consistent.
• Elementary reaction - rate law mirrors stoichiometry; a bimolecular step is second order.
• Arrhenius equation - k = A exp(-E/RT); slope of ln k vs 1/T is -E/R.
• Activation energy - energy barrier; large E means high temperature sensitivity, not necessarily a slow reaction.
• Catalyst - lowers E and raises k for both directions; K_C and equilibrium conversion are untouched.
• Half-life - time for 50% consumption; constant for first order, ∝ C_A0 for zero order, ∝ 1/C_A0 for second order.
• First-order plot - ln C_A vs t linear; second-order - 1/C_A vs t linear; zero-order - C_A vs t linear.
• Conversion relations - C_A = C_A0(1 - X_A) only at constant volume; gas phase needs (1 + ε_A X_A).
• Expansion factor - ε_A = y_A0 δ; ignoring it is the commonest gas-phase error.
• Pseudo-first order - achieved by large excess of one reactant; k_obs = k C_B0.
• Reversible kinetics - K_C = k_f/k_b; only (k_f + k_b) is obtained from the approach-to-equilibrium slope.
• PSSH - set the net rate of a trace intermediate to zero to eliminate it from the rate expression.
• Lindemann - unimolecular reactions are first order at high pressure, second order at low pressure.
• Differential vs integral method - differential finds any order but magnifies data noise; integral is stable but trial-and-error.

Formulas and examples

Example 1 - activation energy from two rate constants. A liquid-phase reaction has k = 0.0025 s⁻¹ at 300 K and k = 0.0400 s⁻¹ at 340 K. Find E and A.

• k_2/k_1 = 0.0400/0.0025 = 16, so ln 16 = 2.7726.
• 1/T_1 - 1/T_2 = 1/300 - 1/340 = 0.00333333 - 0.00294118 = 3.9216×10⁻⁴ K⁻¹.
• E = R ln(k_2/k_1)/(1/T_1 - 1/T_2) = 8.314 × 2.7726 / 3.9216×10⁻⁴ = 23.050/3.9216×10⁻⁴ ≈ 58,776 J/mol ≈ **58.8 kJ/mol**.
• A = k_1 exp(E/RT_1) = 0.0025 × exp(58,776/(8.314×300)) = 0.0025 × exp(23.564) = 0.0025 × 1.72×10¹⁰ ≈ **4.3×10⁷ s⁻¹**.
• Check: at 340 K, E/RT = 58,776/2,826.8 = 20.792, exp(-20.792) = 9.3×10⁻¹⁰; k = 4.3×10⁷ × 9.3×10⁻¹⁰ ≈ 0.040 s⁻¹. Consistent.

Example 2 - order and k by the integral method. Batch data for A → products, constant volume, C_A0 = 2.0 mol/L:

• ln C_A vs t is not linear (drops 0.693, 0.405, 0.288, 0.223 per interval) - not first order.
• 1/C_A vs t rises by exactly 0.500 L/mol every 10 min - a straight line, so the reaction is second order.
• k = slope = 0.500/10 = **0.05 L/(mol·min)**.
• Cross-check with half-life: t_1/2 = 1/(k C_A0) = 1/(0.05 × 2.0) = 10 min, and C_A is indeed 1.0 mol/L at t = 10 min.

Example 3 - order from half-lives. For a reaction, t_1/2 = 50 min at C_A0 = 0.10 mol/L and t_1/2 = 12.5 min at C_A0 = 0.40 mol/L. Find the order and k.

• t_1/2 ∝ C_A0^(1-n): ln(12.5/50) = (1 - n) ln(0.40/0.10) → ln 0.25 = -1.3863, ln 4 = 1.3863.
• 1 - n = -1.3863/1.3863 = -1, so n = **2** (second order).
• k = 1/(t_1/2 C_A0) = 1/(50 × 0.10) = **0.20 L/(mol·min)**; check at the second run: 1/(0.20 × 0.40) = 12.5 min. Correct.

Example 4 - gas-phase variable volume. A → 3B in a constant-pressure, constant-temperature batch reactor charged with 60 mol% A and 40 mol% inert; first order, k = 0.02 min⁻¹, C_A0 = 0.030 mol/L. Find the volume ratio and C_A at 80% conversion, and the time taken.

• δ = (3 - 1)/1 = 2; y_A0 = 0.60, so ε_A = 0.60 × 2 = 1.2.
• At X_A = 0.80: V/V_0 = 1 + 1.2 × 0.80 = **1.96** (the gas nearly doubles in volume).
• C_A = 0.030 × (1 - 0.80)/1.96 = 0.030 × 0.10204 = **3.06×10⁻³ mol/L**.
• First order is unaffected by expansion: t = -ln(1 - 0.80)/k = 1.6094/0.02 = **80.5 min**.
• Had the expansion been ignored, C_A would have been quoted as 6.0×10⁻³ mol/L - a 96% error in concentration, and a large error in any higher-order rate.

Example 5 - reversible first-order reaction. A ⇌ B, liquid phase, starts with pure A. Equilibrium conversion is 0.75, and 50% conversion is reached in 8 min. Find k_f, k_b and K_C.

• K_C = X_Ae/(1 - X_Ae) = 0.75/0.25 = **3.0**.
• -ln(1 - X_A/X_Ae) = (k_f + k_b) t → -ln(1 - 0.50/0.75) = -ln(0.33333) = 1.0986.
• k_f + k_b = 1.0986/8 = 0.13733 min⁻¹.
• With k_f = 3 k_b: 4 k_b = 0.13733 → k_b = **0.0343 min⁻¹**, k_f = **0.1030 min⁻¹**.
• Note: an irreversible first-order reaction with the same k_f would need -ln(0.5)/0.1030 = 6.7 min for 50% - the reverse step slows the approach.

Comparison of order diagnostics

Where it is used

• Refining and petrochemicals: naphtha and ethane cracking, catalytic reforming, hydrotreating and alkylation kinetics at Reliance Jamnagar, IOCL Panipat/Paradip, BPCL Kochi and HMEL Bathinda - cracking severity is set from first-order coil kinetics.
• Fertilisers and bulk chemicals: ammonia synthesis, urea, sulphuric acid (contact process) and chlor-alkali plants at IFFCO, RCF, Coromandel and Tata Chemicals, where temperature-conversion trade-offs follow directly from k(T) against equilibrium.
• Pharmaceuticals and agrochemicals: API batch kinetics, impurity-formation rates and shelf-life (accelerated stability testing by Arrhenius extrapolation) at Sun Pharma, Cipla, Dr. Reddy's, UPL and Aarti Industries - documented for CDSCO and ICH-aligned filings.
• Process safety: reaction calorimetry (DSC, ARC, RC1) measures heat-release kinetics and time to maximum rate, feeding HAZOP and MAH-site risk assessment under the Manufacture, Storage and Import of Hazardous Chemical Rules, 1989; many Indian runaway incidents trace to an unrecognised exotherm's activation energy.
• Environment and utilities: BOD decay, chlorination and ozonation kinetics in effluent treatment plants designed to CPCB discharge norms; NOx reduction kinetics in SCR units on thermal power plants.
• Job roles: process engineer, R&D/kinetics engineer, catalyst development, process-simulation engineer (Aspen Plus/HYSYS reaction models), technical services, and the engineer-trainee kinetics interview at PSU refineries.

Interview questions

• Q: What is the difference between order and molecularity? A: Order is the experimentally fitted exponent on concentration and can be zero, fractional or negative; molecularity is the integer number of molecules in an elementary step. They are equal only for elementary reactions.
• Q: How do you find the units of a rate constant? A: From -r_A = k C_A^n, k has units (concentration)^(1-n)(time)^(-1) - s⁻¹ for first order, m³/(mol·s) for second order.
• Q: Why is a first-order half-life independent of initial concentration? A: Integration gives ln(C_A0/C_A) = kt, so the time to halve depends only on the ratio, not the starting value - which is also why radioactive decay has a fixed half-life.
• Q: A catalyst speeds a reaction up - does it increase equilibrium conversion? A: No. It lowers the activation energy of both forward and reverse steps equally, so K_C = k_f/k_b and the equilibrium conversion are unchanged; only the time to reach equilibrium falls.
• Q: What does a high activation energy tell you about operating the reactor? A: The rate is strongly temperature-sensitive, so small temperature excursions cause large rate changes - good for selectivity control by temperature, but risky for thermal runaway and demanding on temperature control.
• Q: What is the pseudo-steady-state hypothesis and when is it used? A: For a reactive intermediate present in very small concentration, its net rate of formation is set to zero and it is eliminated algebraically; it is used to derive non-elementary rate laws from proposed mechanisms.
• Q: Why does a "unimolecular" gas decomposition change order with pressure? A: By the Lindemann mechanism, activation needs a collision; at high pressure activation is fast and the decomposition step controls (first order), at low pressure activation controls (second order).
• Q: How is order determined when one reactant's effect is masked by others? A: By the method of excess - flood the others so only one concentration varies, giving a pseudo-order; repeat for each species and assemble the full rate law.
• Q: Which experimental reactor gives the cleanest intrinsic kinetics and why? A: A well-mixed CSTR (or recycle reactor), because the steady-state mass balance gives the rate algebraically from measured concentrations with no differentiation of data and at uniform temperature and composition.
• Q: For a gas-phase reaction A → 2B in a constant-pressure batch reactor, what changes in the analysis? A: The volume expands as V = V_0(1 + ε_A X_A), so concentration must be computed as C_A0(1 - X_A)/(1 + ε_A X_A); only a first-order rate law keeps its usual integrated form.
• Q: Can two different mechanisms give the same rate law? A: Yes - agreement with the observed rate law is necessary but not sufficient, so mechanisms need independent evidence such as intermediate detection or isotope labelling.
• Q: Why can't a rate law measured over 10 K be extrapolated 100 K? A: The dominant mechanism or rate-determining step can change (and mass-transfer or equilibrium limitations can set in), so A and E are no longer valid; the Arrhenius fit is only an interpolation within the tested range.

Common mistakes

• Assuming the rate law from the balanced equation - stoichiometry gives order only for elementary reactions, and most industrial reactions are non-elementary.
• Using C_A = C_A0(1 - X_A) for a mole-changing gas-phase reaction, i.e. forgetting ε_A and the volume change.
• Quoting k without units, or carrying mol/L and mol/m³ (and min with s) through the same calculation.
• Mixing up rates of different species: for A + 2B → C, -r_B = 2(-r_A), not -r_A.
• Substituting a temperature in °C into the Arrhenius equation, or dropping the minus sign so E comes out negative.
• Treating -dC_A/dt as the rate in a variable-volume system - the correct definition is (1/V) dN_A/dt.
• Claiming a catalyst shifts equilibrium, or that a higher-E reaction is necessarily slower at the operating temperature.
• Over-fitting: declaring an order from two or three scattered points, or differentiating noisy batch data without smoothing.

Related topics

Reactor design: batch, CSTR and PFR · Multiple reactions and selectivity · Catalysis · Non-ideal reactors and residence time distribution · Chemical reaction thermodynamics and equilibrium · Biochemical engineering basics