A higher activation energy Ea in the Arrhenius equation, k = A*exp(-Ea/RT), means the rate constant k is

more sensitive to changes in temperature

always smaller at every temperature

more sensitive to changes in temperature

independent of temperature entirely

equal to the pre-exponential factor A

The integral method of interpreting kinetic data is preferred over the differential method chiefly because it

avoids taking a numerical derivative of noisy experimental data

never requires assuming a reaction order in advance

avoids taking a numerical derivative of noisy experimental data

works only for zero-order reactions

requires no concentration-time data at all

In the Michaelis-Menten rate law, v = Vmax*[S]/(Km + [S]), the constant Km represents

the substrate concentration at which the rate is half of Vmax

the maximum possible reaction rate

the substrate concentration at which the rate is half of Vmax

the enzyme concentration required for saturation

the reaction's activation energy

Which of the following are true of the tanks-in-series and dispersion models of a non-ideal reactor's RTD?

Each fits the measured RTD using a single adjustable parameter.

The tanks-in-series model approaches ideal PFR behaviour as the number of tanks goes to infinity.

The dispersion model approaches ideal PFR behaviour as its dispersion coefficient goes to zero.

Each fits the measured RTD using a single adjustable parameter.

The tanks-in-series model approaches ideal PFR behaviour as the number of tanks goes to infinity.

The dispersion model approaches ideal PFR behaviour as its dispersion coefficient goes to zero.

Both models require measuring the reactor's internal geometry directly, never a tracer test.

An exothermic CSTR with cooling can show up to three steady states where heat generation equals heat removal; the middle steady state is

always the safest one to operate at

identical to the low-temperature steady state

reached only when the reactor is isothermal

A large Thiele modulus for a porous catalyst pellet signals that the observed reaction rate is limited chiefly by

internal pore diffusion, which cannot keep up with the surface reaction rate

external mass transfer to the pellet's outer surface only

internal pore diffusion, which cannot keep up with the surface reaction rate

the reaction's activation energy alone

the reactor's residence time distribution

Sintering, fouling/coking and poisoning are three distinct mechanisms of

reactor selectivity improvement

Selectivity in a multiple-reaction system is best described as

the rate of desired product formation relative to undesired, which often depends on conditions differently than conversion does

identical to overall conversion in every case

the rate of desired product formation relative to undesired, which often depends on conditions differently than conversion does

always maximised by maximising conversion

Kinetics, Ideal and Non-Ideal Reactors, and Catalysis

Section 7 of the GATE Chemical Engineering (CH) paper — the discipline that decides how big a reactor has to be and how it should be shaped. Start with rate theories and homogeneous kinetics, and how kinetic data actually gets interpreted from raw concentration-versus-time measurements; then the three ideal reactors (batch, CSTR, PFR) and how single and multiple reactions, and enzyme kinetics, play out in each; then what happens once a real reactor stops behaving ideally — residence time distribution and non-isothermal operation; and finally heterogeneous catalysis, where diffusion into a porous catalyst and its slow decline in activity become part of the rate expression itself.

1. Theories of reaction rates, homogeneous kinetics, and interpreting kinetic data

Collision theory pictures a reaction happening when molecules collide with enough energy and the right orientation, giving the Arrhenius relation k = A*exp(-Ea/RT); transition-state theory refines this by proposing every reaction proceeds through a high-energy activated complex, and it is the height of this energy barrier, Ea, that Arrhenius' equation actually measures — a higher Ea means the rate constant is more sensitive to temperature, not merely smaller. Rate laws for elementary reactions follow directly from the stoichiometry (order equals the molecularity), but MOST real reactions are non-elementary, so the rate law's order and form have to be found from data, not written down from the balanced equation on sight.

🧠 Integral method versus differential method, in one line each

The differential method plots the measured rate itself (or an estimate of it from slopes of a concentration-time curve) against concentration and reads the order off a log-log slope; the integral method instead GUESSES an order, integrates the assumed rate law to get concentration as an explicit function of time, and checks whether the data actually falls on the straight line that integrated form predicts — the integral method is more robust to noisy data because it never needs a numerical derivative of experimental points.

2. Ideal reactors, single and multiple reactions, and enzyme kinetics

The three ideal reactor models
Reactor	Mixing assumption	Conversion for the same volume and rate law
Batch / PFR	No back-mixing along the reaction coordinate (time for batch, length for PFR)	Higher, for a positive-order reaction
CSTR	Perfectly mixed — reactor contents are uniform, equal to the outlet	Lower, for a positive-order reaction, at the same volume

A CSTR needs MORE volume than a PFR for the same conversion of a positive-order reaction because it operates entirely at the LOW, outlet concentration (and hence low rate), while a PFR's rate is high near the inlet and only falls gradually — this is exactly why reactors in series (CSTRs approaching PFR behaviour as the number of tanks grows large) and reactor networks are chosen the way they are. For multiple reactions run together, selectivity (rate of desired product formation relative to undesired) usually depends on concentration and temperature differently than overall conversion does, so a reactor and operating condition are often chosen to maximise selectivity rather than raw conversion alone; enzyme kinetics follows the same logic with its own rate law, Michaelis-Menten, v = Vmax*[S]/(Km + [S]), which is first order in substrate at low [S], saturates to zero order at high [S], and where Km is the substrate concentration at half of Vmax.

3. Non-ideal reactors: RTD and single-parameter models, and non-isothermal reactors

The residence time distribution (RTD), measured with a tracer pulse or step input, describes how long different fluid elements actually spend inside a real reactor — a spread that no ideal PFR (a single fixed residence time for everyone) or ideal CSTR (an exponential spread, but for a specific, known reason) fully captures once channelling, dead zones or bypassing enter the picture. Single-parameter models fit that measured RTD with one adjustable number: the tanks-in-series model uses an equivalent number of ideal CSTRs in series, and the dispersion model adds one axial-dispersion term onto an otherwise ideal PFR — both collapse back to their ideal limit as the parameter is pushed to its extreme (infinite tanks, or zero dispersion).

⚠️ An exothermic reactor can have three steady states, and the middle one is unstable

For a non-isothermal, exothermic CSTR with cooling, plotting heat generation (rises steeply, then plateaus as conversion saturates) against heat removal (a straight line in temperature) can intersect at up to THREE steady states — a low-temperature (low-conversion), an unstable middle, and a high-temperature (high-conversion, possibly runaway) point. The middle intersection is unstable: any small perturbation grows rather than decays, which is exactly why an industrial exothermic reactor is designed and controlled to sit at the low or the high steady state deliberately, never near the middle one.

4. Heterogeneous catalytic kinetics, diffusion effects and catalyst deactivation

A heterogeneous catalytic reaction on a solid surface proceeds through a sequence: external mass transfer of reactant to the catalyst particle, internal (pore) diffusion into it, adsorption onto an active site, surface reaction, desorption of product, then diffusion and mass transfer back out — and the observed overall rate can be limited by ANY one of these steps, not necessarily the surface reaction itself. When internal pore diffusion is slow relative to the surface reaction, reactant is consumed before it reaches the catalyst's interior, and the effectiveness factor eta (observed rate divided by the rate if the whole catalyst were exposed to bulk conditions) falls below 1 — a large Thiele modulus (a dimensionless group comparing reaction rate to diffusion rate inside the pore) signals exactly this diffusion-limited regime.

Catalyst deactivation occurs by three broad mechanisms: sintering (thermally driven loss of active surface area at high temperature), fouling/coking (a physical deposit, often carbon, blocking active sites and pores), and poisoning (a strongly adsorbing impurity occupying active sites irreversibly or near-irreversibly) — the three call for different remedies (lower operating temperature, periodic regeneration/burn-off, and feed purification, respectively).
Because deactivation is usually slow compared to the reaction itself, it is tracked with its own separate rate expression (activity a, declining with time on stream by its own kinetics) layered on top of the intrinsic reaction rate — the same separation-of-timescales idea that lets a reactor be designed against fresh-catalyst kinetics first and a catalyst-replacement or regeneration schedule second.

Key takeaways

Arrhenius' equation measures the activation energy barrier of transition-state theory; a higher Ea means the rate constant is MORE sensitive to temperature, not simply a smaller rate.
A CSTR needs more volume than a PFR for the same conversion of a positive-order reaction, because a CSTR operates entirely at the low, outlet concentration rather than the higher concentrations a PFR sees near its inlet.
Tanks-in-series and dispersion models each fit a real reactor's measured RTD with a single adjustable parameter, and both collapse to their ideal-reactor limit as that parameter is pushed to its extreme.
An exothermic CSTR can have three steady states; the middle one is unstable, which is why real reactors are designed and controlled to sit deliberately at the low or high state.
A large Thiele modulus signals a diffusion-limited catalytic reaction, with effectiveness factor below 1; sintering, fouling/coking and poisoning are three distinct deactivation mechanisms needing three distinct remedies.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

A higher activation energy Ea in the Arrhenius equation, k = A*exp(-Ea/RT), means the rate constant k is
1. always smaller at every temperature
2. more sensitive to changes in temperature
3. independent of temperature entirely
4. equal to the pre-exponential factor A
Show answer
Answer: B — more sensitive to changes in temperature
Ea sits in the exponent divided by RT, so a larger Ea makes k respond much more steeply to a given change in T — a high-Ea reaction's rate can change by orders of magnitude over a modest temperature range, which is exactly what makes Ea, not A, the number that matters most for temperature sensitivity.
The integral method of interpreting kinetic data is preferred over the differential method chiefly because it
1. never requires assuming a reaction order in advance
2. avoids taking a numerical derivative of noisy experimental data
3. works only for zero-order reactions
4. requires no concentration-time data at all
Show answer
Answer: B — avoids taking a numerical derivative of noisy experimental data
The differential method needs the rate itself, usually estimated by numerically differentiating a noisy concentration-time curve, which amplifies noise; the integral method instead integrates an assumed rate law analytically and checks the fit directly against measured concentrations, sidestepping that numerical differentiation entirely.
For a positive-order reaction and the same reactor volume, a CSTR achieves ______ conversion than a PFR because it operates entirely at ______ concentration.
1. lower; the low, outlet
2. higher; the high, inlet
3. the same; an average
4. lower; the high, inlet
Show answer
Answer: A — lower; the low, outlet
A CSTR's contents are uniform and equal to its outlet, so the whole reactor volume reacts at the low outlet concentration (and hence low rate); a PFR's concentration and rate are high near the inlet and fall only gradually, so a PFR achieves higher conversion for the same volume.
In the Michaelis-Menten rate law, v = Vmax*[S]/(Km + [S]), the constant Km represents
1. the maximum possible reaction rate
2. the substrate concentration at which the rate is half of Vmax
3. the enzyme concentration required for saturation
4. the reaction's activation energy
Show answer
Answer: B — the substrate concentration at which the rate is half of Vmax
Setting [S] = Km in the rate law gives v = Vmax*Km/(Km+Km) = Vmax/2, which is exactly the defining property of Km — a low Km means the enzyme reaches half its maximum rate at low substrate concentration, i.e. it has high apparent affinity for that substrate.
Which of the following are true of the tanks-in-series and dispersion models of a non-ideal reactor's RTD?
1. Each fits the measured RTD using a single adjustable parameter.
2. The tanks-in-series model approaches ideal PFR behaviour as the number of tanks goes to infinity.
3. The dispersion model approaches ideal PFR behaviour as its dispersion coefficient goes to zero.
4. Both models require measuring the reactor's internal geometry directly, never a tracer test.
Show answer
Answer: A — Each fits the measured RTD using a single adjustable parameter.; B — The tanks-in-series model approaches ideal PFR behaviour as the number of tanks goes to infinity.; C — The dispersion model approaches ideal PFR behaviour as its dispersion coefficient goes to zero.
The fourth statement is false: both models are fitted to a measured tracer RTD (pulse or step input), not to internal geometry directly — that measured curve is exactly what the single parameter (tank count, or dispersion coefficient) is adjusted to match. The first three describe the models and their ideal limits correctly.
An exothermic CSTR with cooling can show up to three steady states where heat generation equals heat removal; the middle steady state is
1. always the safest one to operate at
2. unstable — any small perturbation grows rather than decays
3. identical to the low-temperature steady state
4. reached only when the reactor is isothermal
Show answer
Answer: B — unstable — any small perturbation grows rather than decays
The middle intersection of the heat-generation and heat-removal curves is a point where any small temperature fluctuation pushes the reactor away rather than back toward it, making it unstable in practice; industrial reactors are designed and controlled to operate deliberately at one of the two stable (low or high) steady states instead.
A large Thiele modulus for a porous catalyst pellet signals that the observed reaction rate is limited chiefly by
1. external mass transfer to the pellet's outer surface only
2. internal pore diffusion, which cannot keep up with the surface reaction rate
3. the reaction's activation energy alone
4. the reactor's residence time distribution
Show answer
Answer: B — internal pore diffusion, which cannot keep up with the surface reaction rate
The Thiele modulus directly compares the intrinsic surface-reaction rate to the rate of diffusion into the pellet's pores; a large value means reaction consumes reactant faster than diffusion can resupply the pellet's interior, leaving the core underused and pushing the effectiveness factor eta below 1.
Sintering, fouling/coking and poisoning are three distinct mechanisms of
1. reactor start-up
2. catalyst deactivation
3. residence time distribution
4. reactor selectivity improvement
Show answer
Answer: B — catalyst deactivation
All three describe different physical routes by which a solid catalyst's activity declines over time — thermally driven loss of surface area, physical blockage of sites and pores by a deposit, and irreversible occupation of active sites by a strongly adsorbing impurity — and each calls for a different operating or maintenance remedy.
Selectivity in a multiple-reaction system is best described as
1. identical to overall conversion in every case
2. the rate of desired product formation relative to undesired, which often depends on conditions differently than conversion does
3. always maximised by maximising conversion
4. independent of reactor type
Show answer
Answer: B — the rate of desired product formation relative to undesired, which often depends on conditions differently than conversion does
Selectivity and conversion answer different questions — how much desired product forms per unit of undesired, versus how much reactant is consumed overall — and because they can respond to temperature, concentration and reactor choice differently, a process is often deliberately run below maximum conversion to protect selectivity.
A first-order irreversible reaction has rate constant k = 0.2 per minute. In an ideal batch reactor, what is the half-life t(1/2) in minutes (t(1/2) = ln(2)/k, taking ln(2) as approximately 0.693)?
Numerical answer — type the value.
Show answer
Answer: 3.465
t(1/2) = ln(2)/k = 0.693/0.2 = 3.465 minutes. This half-life is independent of the starting concentration for a first-order reaction, which is exactly the property that makes 'half-life' a meaningful, concentration-independent number only for first-order kinetics.