Multiple reactions and selectivity

In short

Real industrial reactors almost never run one clean reaction - the same reactants also form by-products through parallel (competing) and series (consecutive) paths, so the engineer's job shifts from "how fast" to "how much of the right product". Multiple-reaction analysis defines selectivity and yield, then shows how concentration level, temperature, reactant feed strategy and reactor type (batch, CSTR, PFR, semibatch, membrane) can be chosen to favour the desired reaction. It matters commercially: in partial oxidation, nitration, chlorination, alkylation and ethylene oxide manufacture a 2% selectivity gain is worth crores per year in raw material. It is a standing GATE Chemical Engineering question type (instantaneous selectivity, optimum intermediate concentration, reactor choice) and a favourite interview area for process and technology roles at Reliance, IOCL, HPCL, GAIL, Tata Chemicals, UPL, Deepak Nitrite, Aarti Industries, SRF, Dr Reddy's and process-design consultancies.

Explained

Types of multiple reactions

• Parallel (competing/simultaneous): one reactant gives two or more products by independent routes - A → D (desired), A → U (undesired). Example: ethylene + O₂ → ethylene oxide (desired) and → CO₂ + H₂O (combustion).
• Series (consecutive): the desired product is an intermediate - A → B → C. Examples: methanol → formaldehyde → CO; benzene chlorination to mono-, di-, trichlorobenzene; partial hydrogenation.
• Series-parallel (mixed): both features together, very common. Example: A + B → R, R + B → S (ethylene oxide + water → mono-, di-, triethylene glycol).
• Independent: reactions sharing no species, e.g. cracking of two different hydrocarbons in the same reactor - no selectivity problem, only conversion bookkeeping.
• Complex/network: many coupled reactions (petroleum cracking, pyrolysis) treated by lumping into a few kinetic classes.
• Reversible series/parallel adds a thermodynamic ceiling on top of the kinetic one - selectivity cannot beat equilibrium at long times.

Selectivity and yield: definitions that must be stated

• Instantaneous selectivity, S(D/U) = r_D / r_U - the ratio of formation rates at a point. This is the quantity you *manipulate* by concentration and temperature.
• Overall (integral) selectivity, S̃(D/U) = N_D / N_U (batch) or F_D / F_U (flow) - what you actually get out of the reactor.
• Instantaneous yield, Y_D = r_D / (−r_A) - moles of D formed per mole of A consumed at that instant.
• Overall yield, Ỹ_D = N_D / (N_A0 − N_A) - "yield based on reactant consumed" (operational yield).
• Yield based on reactant fed (overall/plant yield) = N_D / N_A0 = conversion × selectivity. Always say which definition you are using; Indian university papers and GATE both use more than one convention, and stoichiometric coefficients may or may not be included.
• For a CSTR at steady state the instantaneous and overall selectivities coincide, because the whole reactor sits at the exit concentration - a key exam shortcut.
• Economic selectivity sometimes beats molar selectivity: losing moles to a saleable by-product (e.g. HCl recovered as acid) is cheaper than losing them to CO₂.

Parallel reactions: controlling by concentration

Take A → D with r_D = k_D C_A^α₁ and A → U with r_U = k_U C_A^α₂. Then S(D/U) = (k_D/k_U) C_A^{α₁−α₂}.

• If α₁ > α₂ (desired reaction is of higher order in A): keep C_A high - batch or PFR/PBR, no dilution, high pressure for gases, low conversion per pass with recycle.
• If α₁ < α₂: keep C_A low - CSTR (operates at low exit concentration), dilute with inert or solvent, low pressure for gases, high conversion, or recycle a large product stream.
• Two reactants, A + B → D and A + B → U with different orders in each: four combinations arise. If D needs high C_A and low C_B, feed B slowly into a batch charge of A (semibatch) or use a membrane reactor/side-stream injection in a tubular reactor; if both must be high, use a tubular reactor with both fed at the inlet at high pressure; if both must be low, use a CSTR with dilute feed or a large recycle.
• A tubular reactor with multiple side injections mimics "high C_A, low C_B" continuously - used in nitration and chlorination trains.
• Mixing matters: for selectivity-sensitive fast reactions, micromixing (segregation at the molecular scale) changes the answer; poor mixing creates local hot, concentrated zones that feed the undesired path.

Parallel reactions: controlling by temperature

• With Arrhenius rates, S(D/U) = (A_D/A_U) exp[−(E_D − E_U)/RT] × C_A^{α₁−α₂}.
• If E_D > E_U: selectivity improves as temperature rises - run as hot as materials, catalyst stability and safety allow.
• If E_D < E_U: selectivity improves as temperature falls - run cold, accept a bigger reactor or more catalyst to recover rate.
• If E_D = E_U, temperature does not shift selectivity; only concentration and catalyst do.
• Temperature and concentration can pull in opposite directions; the real optimum is a constrained economic optimum (raw-material cost versus reactor cost and utilities).
• Progressive (ramped) temperature is often best: start hot for rate, finish cold for selectivity, or the reverse, as the activation-energy ordering dictates.

Series reactions and the maximum intermediate

For A →(k₁) B →(k₂) C, first order, constant volume:

• Batch/PFR: C_A = C_A0 e^−k₁t; C_B = C_A0 k₁ (e^−k₁t − e^−k₂t)/(k₂ − k₁) for k₁ ≠ k₂; C_C by balance.
• B passes through a maximum at t_opt = ln(k₂/k₁)/(k₂ − k₁) (for k₁ = k₂, t_opt = 1/k). The maximum value is C_B,max/C_A0 = (k₁/k₂)^[k₂/(k₂−k₁)].
• CSTR: C_B = C_A0 k₁τ /[(1 + k₁τ)(1 + k₂τ)], maximised at τ_opt = 1/√(k₁k₂), giving C_B,max/C_A0 = 1/(1 + √(k₂/k₁))².
• A PFR (or batch) always gives a higher maximum intermediate than a CSTR of the same kinetics, because back-mixing in a CSTR exposes freshly formed B to further reaction. For series reactions, plug flow wins; for parallel reactions, the winner depends on the order.
• Therefore avoid dead zones, bypass and recycle when the product is an intermediate; residence-time control (quenching) is the main lever - e.g. quenching ethylene cracker effluent in milliseconds.
• If k₂ ≫ k₁ the intermediate can never be accumulated appreciably - change the catalyst or the route, not the reactor.

Mole balances and the algorithm for multiple reactions

• Write a mole balance for every species, not for the reaction: dF_j/dV = Σ_i ν_ij r_i (PFR), dN_j/dt = Σ_i ν_ij r_i V (batch), F_j0 − F_j + Σ_i ν_ij r_i V = 0 (CSTR).
• Net rate of species j = sum of its rates in all reactions: r_j = Σ_i r_ij; relative rates within reaction i follow that reaction's own stoichiometry only.
• Use concentrations, never conversion, as variables - conversion of A is ambiguous when A appears in several reactions.
• For gas phase include the change in total moles: C_j = C_T0 (F_j/F_T) (P/P_0)(T_0/T).
• Non-isothermal case: add the energy balance with Σ(−ΔH_rx,i)(r_i) as the heat generation term - multiple reactions make multiple contributions.
• The resulting ODE/algebraic set is solved numerically (Polymath, MATLAB/Octave, Python SciPy, Aspen Plus RPlug/RCSTR); hand solutions are limited to first-order and simple cases, which is exactly what exams set.

# PFR with A -> B -> C  (first order), species balances
from scipy.integrate import solve_ivp
k1, k2, CA0, v0 = 0.5, 0.2, 2.0, 10.0   # 1/min, 1/min, mol/L, L/min
def f(V, F):
    FA, FB, FC = F
    CA, CB = FA/v0, FB/v0
    r1, r2 = k1*CA, k2*CB               # mol/(L.min)
    return [-r1*1, (r1-r2)*1, r2*1]     # dFj/dV
sol = solve_ivp(f, [0, 60], [CA0*v0, 0, 0], dense_output=True)

Reactor and process choices used to buy selectivity

• Catalyst is usually the biggest selectivity lever (shape selectivity in ZSM-5, promoted silver for ethylene oxide, selective poisons in Lindlar-type hydrogenation) - covered in the catalysis note.
• Safety and statutory angle: selectivity loss in exothermic oxidations/nitrations means extra heat release and runaway risk. Such plants are typically covered by the MSIHC Rules and, above threshold inventories, the Chemical Accidents (Emergency Planning, Preparedness and Response) Rules; hazardous-area and relief design follows the Petroleum and Explosives Safety Organisation (PESO) and OISD norms, with HAZOP/relief sizing per the plant's own standards. Occupational safety now sits under the Occupational Safety, Health and Working Conditions Code (in force from 21 November 2025, replacing the Factories Act). Exact applicability thresholds vary by state rules - check the current notification.

Key points

• Multiple reactions - two or more reactions proceeding together; product distribution, not conversion, decides profit.
• Parallel reactions - same reactant, competing products; selectivity set mainly by concentration level and temperature.
• Series reactions - desired product is an intermediate; selectivity set mainly by residence time and back-mixing.
• Instantaneous selectivity S = r_D/r_U - the design handle; overall selectivity F_D/F_U - the result.
• Yield (reactant consumed) = D formed / A reacted; yield (reactant fed) = conversion × selectivity - state which one.
• α₁ > α₂ → keep C_A high (PFR/batch, high pressure, low per-pass conversion); α₁ < α₂ → keep C_A low (CSTR, dilution).
• E_D > E_U → high temperature improves selectivity; E_D < E_U → low temperature does.
• CSTR instantaneous = overall selectivity, because the whole volume is at exit concentration.
• PFR beats CSTR for an intermediate; CSTR can beat PFR for parallel reactions needing low C_A.
• t_opt = ln(k₂/k₁)/(k₂−k₁) for maximum B in batch/PFR; τ_opt = 1/√(k₁k₂) in a CSTR.
• Semibatch / side injection creates high-C_A, low-C_B conditions that no single-feed steady reactor can.
• Write a balance per species, with net rate r_j = Σ r_ij; never use conversion as the variable.
• Mixing and hot spots destroy selectivity in fast exothermic systems - micromixing matters.
• Selectivity-conversion trade-off - selectivity usually falls as conversion rises; the optimum is economic, with recycle cost included.
• Unreacted reactant is recoverable; a by-product is often not - hence low per-pass conversion with recycle is a standard industrial answer.
• Catalyst choice typically outranks reactor choice as a selectivity lever.
• Economic selectivity counts by-product value and disposal cost, not just moles.

Formulas and examples

Example 1 - Parallel reactions: which reactor? A → D, r_D = 0.08 C_A² ; A → U, r_U = 0.20 C_A (rates in mol/(L·min) with C_A in mol/L, so k_D = 0.08 L/(mol·min) and k_U = 0.20 min⁻¹). Feed C_A0 = 4 mol/L.

• S(D/U) = (0.08/0.20) C_A²⁻¹ = 0.4 C_A. Desired reaction is higher order → keep C_A high → PFR/batch, no dilution.
• At C_A = 4: S = 1.6 (62% of A consumed goes to D). At C_A = 0.5: S = 0.20 (17%).
• Instantaneous yield of D at C_A = 4: Y_D = r_D/(r_D + r_U) = (0.08×16)/(0.08×16 + 0.20×4) = 1.28/(1.28 + 0.80) = 0.615.
• At C_A = 0.5: Y_D = 0.02/(0.02 + 0.10) = 0.167. Lesson: run at low per-pass conversion and recycle A.

Example 2 - CSTR selectivity and sizing with two reactions Same kinetics, CSTR, v₀ = 10 L/min, exit C_A = 2 mol/L.

• r_D = 0.08 × 4 = 0.32 mol/(L·min); r_U = 0.20 × 2 = 0.40 mol/(L·min); −r_A = 0.72 mol/(L·min).
• S̃(D/U) = 0.32/0.40 = 0.8 (equals the instantaneous value in a CSTR).
• τ = (C_A0 − C_A)/(−r_A) = (4 − 2)/0.72 = 2.78 min → V = 2.78 × 10 = 27.8 L.
• F_D = r_D·V = 0.32 × 27.8 = 8.89 mol/min; F_U = 0.40 × 27.8 = 11.1 mol/min; F_A0 = 40 mol/min.
• X_A = 0.5, yield on A fed = 8.89/40 = 0.222; yield on A consumed = 8.89/20 = 0.444.

Example 3 - Series reaction, batch versus CSTR A →(k₁ = 0.50 min⁻¹) B →(k₂ = 0.20 min⁻¹) C, C_A0 = 2 mol/L.

• Batch/PFR: t_opt = ln(0.20/0.50)/(0.20 − 0.50) = ln(0.4)/(−0.30) = (−0.9163)/(−0.30) = 3.05 min.
• C_B,max = 2 × (0.5/0.2)^[0.2/(0.2−0.5)] = 2 × 2.5^−0.6667 = 2 × 0.5429 = 1.086 mol/L → 54.3% of A₀.
• Check: C_A = 2e^{−0.5×3.054} = 2 × 0.2172 = 0.434; C_B = 2×0.5(e^−1.527 − e^−0.611)/(0.2−0.5) = 1.0(0.2172 − 0.5428)/(−0.3) = 1.085 mol/L ✓.
• CSTR: τ_opt = 1/√(0.5×0.2) = 1/0.3162 = 3.16 min; C_B,max = 2/(1 + √(0.2/0.5))² = 2/(1 + 0.6325)² = 2/2.665 = 0.751 mol/L (37.5%).
• PFR gives 1.086 versus 0.751 mol/L - about 45% more intermediate for essentially the same mean residence time.

Example 4 - Temperature choice from activation energies A → D (E_D = 120 kJ/mol) and A → U (E_U = 80 kJ/mol), both first order in A, A_D/A_U = 0.01. Compare 350 K and 450 K.

• At 350 K: exponent = −(40,000)/(8.314 × 350) = −13.75 → e^−13.75 = 1.07×10⁻⁶ → S = 0.01 × 1.07×10⁻⁶ ≈ 1.07×10⁻⁸ (hopeless).
• At 450 K: exponent = −(40,000)/(8.314 × 450) = −10.69 → e^−10.69 = 2.27×10⁻⁵ → S = 2.27×10⁻⁷.
• Selectivity rises about 21-fold for a 100 K rise because E_D > E_U; the ratio S(450)/S(350) = exp[(40,000/8.314)(1/350 − 1/450)] = exp(3.055) = 21.2. Run as hot as the catalyst and metallurgy permit.

Example 5 - Two-reactant parallel system and feed strategy A + B → D, r_D = k_D C_A^0.5 C_B² ; A + B → U, r_U = k_U C_A^1.5 C_B.

• S(D/U) = (k_D/k_U) C_B / C_A → needs low C_A and high C_B.
• Right choice: a semibatch reactor with B charged and A fed slowly, or a CSTR fed with large excess B, or a tubular reactor with A injected along the length into a B-rich stream.
• With k_D/k_U = 2: at C_A = 0.2, C_B = 3 mol/L → S = 2 × 3/0.2 = 30. If instead C_A = 3, C_B = 0.2 → S = 2 × 0.2/3 = 0.133 - the same chemistry, 225 times worse, purely from feed strategy.

Where it is used

• Petrochemicals: ethylene oxide (silver catalyst, selectivity ~80-90% against total combustion), steam-cracker contact time and quench design, propylene oxidation to acrolein/acrylic acid - Reliance Jamnagar/Dahej, HMEL, GAIL Pata, Indian Oil Panipat and Paradip.
• Agrochemicals and dyes/intermediates: nitration, chlorination, sulphonation, where mono- versus di-substitution is pure series-parallel selectivity - Aarti Industries, Deepak Nitrite, UPL, Atul, Vinati Organics, mostly semibatch with slow addition.
• Pharmaceuticals and fine chemicals: regio- and chemoselectivity in API routes, impurity profiles controlled under GMP (CDSCO/Schedule M); continuous-flow reactors adopted to replace long batch holds - Dr Reddy's, Divi's, Sun Pharma.
• Fertilizers and inorganics: ammonia oxidation to NO (not N₂) in nitric acid plants, SO₂ oxidation trains, formaldehyde from methanol - RCF, GSFC, Tata Chemicals.
• Polymers: branching, chain transfer and cross-linking are competing reactions that fix product grade - covered quantitatively in the polymer reaction engineering note.
• Job roles: process engineer and technology/R&D engineer (yield-improvement projects), catalyst evaluation and pilot-plant engineer, process-safety engineer sizing relief for runaway side reactions, simulation engineer building Aspen/gPROMS kinetic models; GATE-based PSU recruitment (IOCL, BPCL, HPCL, GAIL, ONGC, NFL) tests this directly.

Interview questions

• Q: What is the difference between selectivity and yield? A: Selectivity compares the desired product with the undesired one (r_D/r_U or F_D/F_U); yield compares the desired product with the reactant - either per mole reacted or per mole fed. Yield on feed = conversion × selectivity.
• Q: For A → D (second order in A) and A → U (first order in A), which reactor and why? A: A PFR or batch reactor at high C_A, since S = (k_D/k_U)C_A rises with concentration; also avoid dilution, raise pressure if gaseous, and keep per-pass conversion low with recycle.
• Q: Why does instantaneous selectivity equal overall selectivity in a CSTR? A: A perfectly mixed CSTR has the entire volume at exit conditions, so the local rate ratio is the same everywhere and equals the ratio of exit molar flows.
• Q: For A → B → C, why does a PFR give more B than a CSTR? A: In a PFR every element spends the same time, so B can be harvested at its optimum age; a CSTR's back-mixing keeps some B in the vessel far longer, converting it to C.
• Q: How do you choose the reactor temperature when selectivity and rate conflict? A: Compare activation energies - if E_desired > E_undesired go hot, otherwise go cold - then optimise economically against reactor volume, catalyst life and utility cost; a temperature profile often beats a single temperature.
• Q: Why is low per-pass conversion with recycle so common industrially? A: Selectivity usually falls as reactant concentration falls, and unreacted reactant can be separated and recycled, whereas a by-product is usually a permanent loss plus a disposal cost.
• Q: When would you use a semibatch reactor for selectivity? A: When the desired reaction needs one reactant high and the other low - charge the one that must stay high and add the other slowly; it also limits heat release in exothermic nitrations and chlorinations.
• Q: What goes wrong if you write mole balances in terms of conversion for multiple reactions? A: Conversion of a species consumed by several reactions is ambiguous and the stoichiometric table breaks down; you must write a balance per species with net rates r_j = Σ r_ij.
• Q: How does mixing affect selectivity? A: For fast reactions with different orders, imperfect micromixing creates local pockets of high concentration and temperature that favour one path, so observed selectivity differs from the ideally mixed prediction; impeller design, feed-nozzle location and Damköhler number matter.
• Q: Give an industrial example where selectivity, not conversion, is the design objective. A: Ethylene oxide - deep oxidation to CO₂ is thermodynamically favoured, so the plant runs low per-pass conversion with recycle and a promoted silver catalyst to maximise selectivity.
• Q: If k₂ ≫ k₁ in A → B → C, can you still get a good yield of B? A: No - B reacts away as fast as it forms; the maximum possible C_B is tiny whatever the reactor, so you must change the catalyst, temperature or synthetic route.
• Q: What is a membrane reactor's role in selectivity? A: It can distribute one reactant along the length (keeping its concentration low) or remove a product continuously, shifting both equilibrium and the competing-rate balance towards the desired path.

Common mistakes

• Quoting "selectivity" or "yield" without saying which definition - instantaneous or overall, per mole fed or per mole reacted - and losing marks even with correct numbers.
• Writing mole balances per *reaction* or in terms of conversion of A instead of one balance per *species* with net rates.
• Assuming a PFR is always better - true for an intermediate in a series reaction, false for a parallel system that needs low C_A.
• Using t_opt = ln(k₂/k₁)/(k₂−k₁) for a CSTR, or τ_opt = 1/√(k₁k₂) for a batch reactor - the two expressions are not interchangeable.
• Forgetting that total moles change in gas-phase reactions, so concentrations must be computed from F_j/F_T with pressure and temperature corrections.
• Raising temperature to "speed things up" without checking the activation-energy ordering - this often destroys selectivity and can trigger runaway in exothermic side reactions.
• Treating a by-product as free: ignoring separation, effluent-treatment and disposal costs, and the extra heat duty from side reactions.

Related topics

• Chemical kinetics · Reactor design: batch, CSTR and PFR · Non-ideal reactors and residence time distribution · Catalysis · Polymer reaction engineering · Process safety and hazard analysis