The Gold Standard of Zeolite Catalysis | Why Product Escape and Consecutive Reaction Can Change Selectivity Without Changing the Initial Branch

Imagine a molecule-equivalent of an intermediate B newly formed inside a porous catalyst. Two possible events follow. It can leave the modelled region as B, or it can react there to form C. The molecule has already reached the initial branch point; our question concerns its subsequent fate.

We count conserved material units rather than assume that every real chemical reaction preserves molecule number. In this teaching network one unit of B becomes one unit of C on the chosen conserved basis. A real hydrocarbon network needs an appropriate carbon or mass account with the actual stoichiometry.

The distinction matters when comparing samples. An outlet rich in C could reflect faster secondary conversion, slower escape, different initial formation, or an omitted process. A model becomes useful when it identifies which of those changes have been supplied as assumptions and which are actually inferred from measurements.

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Let B(t) denote the amount of retained intermediate in one modelled region. Let E(t) be cumulative B exported from that region, and R(t) cumulative secondary-product formation. We initially treat C formation as an accounted outcome, without asserting that an instrument measures C immediately.

Supply a constant formation flux F into B, with units of conserved amount per time. The region starts empty, so B(0), E(0) and R(0) are zero. The boundary includes the point where B exits. Material outside that boundary does not return in this version of the model.

Defining the boundary prevents a common confusion: product formation inside the material and product arrival at an external detector need not be simultaneous. Our first ledger records formation of C. If transport of C to the detector is significant, a later model must include that retained population and its own export flux.

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Let ke be the escape coefficient and kr the secondary-reaction coefficient, both positive and with units of inverse time. The escape flux is keB and the reaction flux is krB. A coefficient represents an instantaneous rate per retained amount under the model, not a guarantee that every molecule leaves after a fixed waiting time.

Both coefficients are constant. The region is represented by one retained population; there are no separate spatial gradients, age classes or accessibility classes. No saturation, reverse conversion, return transport or additional reaction is included. Formation F is supplied independently rather than solved from an upstream reactant equation.

These restrictions make the competition transparent. They also limit the interpretation. In particular, ke is not automatically a molecular diffusion coefficient. A diffusion coefficient has dimensions of length squared per time, and a spatial diffusion problem generally produces more structure than a single constant escape hazard.

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Over a short time interval, the retained amount gains F times the interval and loses the escape and reaction fluxes times that interval. Dividing by the interval and taking its limit gives dB/dt = F−(ke+kr)B.

Define q = ke+kr. The equation becomes dB/dt+qB = F. Multiplying by exp(qt) and integrating from zero to t gives B(t) = F[1−exp(−qt)]/q for the initially empty region.

The result has two interpretable parts. F/q is the stationary retained amount, and 1/q is the relaxation time. At small times B is approximately Ft, because losses have had little time to accumulate. At long times B approaches F/q, while both output processes continue. A stationary retained population therefore does not mean that individual molecules have stopped moving or reacting.

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The exported-B flux is Je(t) = keF[1−exp(−qt)]/q. The secondary-formation flux is Jr(t) = krF[1−exp(−qt)]/q. Their ratio is ke/kr whenever B is positive.

The fraction of the two instantaneous outcome fluxes assigned to B export is Je/(Je+Jr) = ke/q. It is constant in this restricted model, even while the retained amount and both fluxes rise from zero. This is an outcome fraction among material currently leaving the retained-B population by either event.

It is not the same as exported B divided by all B formed so far. Early in the experiment, some of that formed material is still retained. Keeping the denominators visible will let us distinguish a stable branching fraction from a cumulative recovery that changes with the observation window.

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Integrating Je from zero to t gives E(t) = keF[t−(1−exp(−qt))/q]/q. Replacing ke by kr gives R(t). Both start at zero and increase monotonically for positive coefficients.

At very short times, E is approximately keFt²/2 and R is approximately krFt²/2. Their quadratic start contrasts with the linear initial growth B≈Ft. An initially empty region must first accumulate intermediate before a substantial outgoing flux can develop.

At long times, E and R grow approximately linearly. Their slopes approach keF/q and krF/q. The finite offset reflects initial filling. Extrapolating that late-time line back to the origin would miss the transient and can misdescribe a short experiment. The full expressions preserve both the stationary competition and the material that has not yet reached either outcome.

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Adding B, E and R cancels the exponential terms and gives B(t)+E(t)+R(t)=Ft. The right side is the total supplied amount of intermediate. The left side places every supplied unit in exactly one accounted population or cumulative outcome.

This equality is stronger than obtaining a plausible-looking curve. It checks the boundary, the units and the integration. If a calculation gives E+R greater than Ft while B remains positive, the model implementation has created material. If it gives a persistent deficit without a retained population, it has silently omitted material.

A useful reporting habit is to present this ledger beside any outlet fraction. The missing fraction of formed B may be retained rather than destroyed or assigned to an unobserved product. The ledger does not establish that a real experiment has only these three accounts; it states what must be true if this teaching network is adequate.

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Choose F = 10 amount units per time unit, ke = 3 per time unit and kr = 2 per time unit. Then q = 5 per time unit. At t = 0.4 time units, qt = 2 and exp(−2) is approximately 0.135335.

The retained amount is B = 2(1−exp(−2)), approximately 1.729329. Export is E = 6[0.4−(1−exp(−2))/5], approximately 1.362402. Secondary formation is R = 4[0.4−(1−exp(−2))/5], approximately 0.908268.

Their sum is exactly four before rounding, equal to Ft. Export accounts for sixty percent of the completed outcomes E+R, but only approximately 34.06 percent of all four supplied units. Both fractions are correct. The difference is the retained intermediate, which has not yet reached either accounted outcome by this observation time.

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The completed-outcome fraction is E/(E+R) = ke/q. The finite-time export yield is E/(Ft). The retained fraction is B/(Ft). In the original example, these quantities have different values because their denominators answer different questions.

Write τ = qt. The finite-time export yield is E/(Ft) = (ke/q)[1−(1−exp(−τ))/τ]. The bracket is less than one at finite positive τ. It approaches zero as τ approaches zero and approaches one as τ grows without bound.

Thus an apparent improvement in finite-time exported yield can result from waiting longer even when neither coefficient changes. A comparison between samples observed for different dimensionless times should not be interpreted immediately as a change in intrinsic chemistry. First determine whether the experiment compares completed outcomes, total input, or an accumulated detector signal with another delay.

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Keep F = 10 and kr = 2, but compare different ke values. The eventual exported fraction ke/(ke+2) increases as escape becomes faster. For ke = 1 the fraction is one-third; for ke = 3 it is three-fifths; for ke = 6 it is three-quarters.

No initial formation branch was changed in this comparison. The secondary-reaction coefficient also stayed fixed. Faster escape simply reduces the opportunity for the retained intermediate to undergo that secondary event under the supplied hazard model.

This calculation supports a bounded conclusion: transport-related escape can affect observed product distributions without a changed initial branch. It does not establish that this is the explanation for a real catalyst comparison. Changing particle structure may simultaneously change accessibility, adsorption, active sites and formation flux. The conditional comparison tells us what to measure independently before assigning a cause.

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Now retain ke = 3 and increase kr from two to six. The eventual exported fraction falls from three-fifths to one-third. The stationary retained amount also changes from F/5 to F/9.

This gives a useful diagnostic distinction. Increasing kr and decreasing ke can both reduce the exported-B fraction, but they need not produce the same retained amount or relaxation time. A measurement that observes only the outcome fraction discards that potentially distinguishing information.

The direction of a selectivity change therefore does not identify the direction of every underlying rate change. Even with just two competing processes, an outlet fraction is a relationship between rates. It is sensible to describe a change as consistent with increased secondary conversion while preserving the alternative explanation of reduced escape until another observation separates them.

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Compare the original pair ke = 3, kr = 2 with ke = 6, kr = 4. Both give ke/q = 0.6. The second pair has q = 10 rather than five, so its relaxation time is halved.

For the same formation flux, the second stationary retained amount is F/10 rather than F/5. At a fixed clock time, its finite-time export yield is also different because qt differs. A constant completed-outcome fraction does not imply identical retained populations or transient export curves.

This example exposes a structural information limit. Any common positive multiplier applied to ke and kr leaves their branching fraction unchanged. Measuring that fraction more precisely cannot determine the common scale. An independent time scale, a retained amount with known formation flux, or another appropriately calibrated dynamic observation is needed to recover more than the ratio.

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Suppose the model is justified and separate observations establish the completed-outcome fraction s and the relaxation time τrel. Then q = 1/τrel, ke = sq and kr = (1−s)q.

For s = 0.6 and τrel = 0.2 time units, q = 5, ke = 3 and kr = 2. The inverse recovers the original coefficients. It relies on two independent pieces of information: a relative outcome and an absolute time scale.

The interpretation remains conditional. A multiexponential relaxation, detector delay or time-varying formation flux can make a fitted single time constant describe something other than q. Before inserting a fitted number into this inverse, state which population was measured and which equation its relaxation is assumed to satisfy. Parameter recovery should follow an adequate observation model rather than a convenient curve shape.

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Replace constant formation by an initial pulse: B(0)=B0 and no further formation. The retained population is B(t)=B0 exp(−qt). Cumulative export is E(t)=B0ke[1−exp(−qt)]/q, and secondary formation has kr in place of ke.

The ledger is now B+E+R=B0, not Ft. As time grows, the eventual outcomes approach B0ke/q and B0kr/q. The same competing fraction appears, while the transient differs from the initially empty, continuously fed model.

A pulse and a continuous feed therefore provide different time traces even when the underlying hazards are identical. Confusing their input histories can create a false rate discrepancy. Name the preparation and input before comparing fitted constants, and reconstruct the matching ledger rather than borrowing an expression from a different experimental sequence.

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Suppose the constant formation flux is stopped at time ts. Just before stopping, the retained amount is Bs=F[1−exp(−qts)]/q. For later times, B(t)=Bs exp[−q(t−ts)].

Additional export after stopping is keBs[1−exp(−q(t−ts))]/q. Additional secondary formation has kr in place of ke. The final amounts include both the outcomes accumulated during feeding and those produced from the retained inventory afterward.

A detector signal continuing after feed termination does not automatically establish a new production pathway. Retained material can continue to leave or react in the existing network. Conversely, a long signal tail might also reflect transport of another population or instrument response. The model offers a candidate account to test, not permission to assign every observed tail to the same retained species.

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If collection ends soon after a pulse or feed stop, part of the material remains inside the region. A lower collected amount can therefore reflect incomplete observation rather than a lower eventual escaped fraction.

For an isolated pulse observed until qt = ln 2, half of B0 remains. Exported B is then sB0/2 and secondary formation is (1−s)B0/2. Among the completed outcomes the fraction is still s, while exported B divided by the initial pulse is s/2.

This exact halfway case makes the denominator issue easy to test. If a report describes s/2 as an eventual product fraction, it has mixed a finite collection window with an infinite-time outcome. Practical measurements need a justified collection boundary and a quantified or bounded retained inventory. Waiting for an arbitrarily long time is not a universal experimental remedy when the material itself changes during observation.

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Our R account records conversion to C. Suppose C can remain in the region and leave with coefficient kc. Introduce retained C(t), with dC/dt=krB−kcC, and cumulative C export G(t), with dG/dt=kcC.

The physical ledger becomes B+C+E+G=Ft for the continuously supplied case. R is then an internal cumulative conversion count equal to C+G under this irreversible network, not an extra material population to add again.

A time-resolved detector ratio involving E and G can differ from the ratio E and R, especially while C is accumulating. This extension identifies why immediate measurement of a reaction outcome is an assumption. It also warns against double counting: adding retained C, exported C and cumulative C formation as three separate conserved accounts would count the same converted material twice.

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For a pulse B0 with q=kc, the retained B is B0 exp(−qt). Solving the C equation gives C(t)=krB0t exp(−qt). This finite expression replaces formulas that appear to contain a division by kc−q when the two rates approach equality.

The C population initially grows, reaches a maximum and then decays as supply from B dwindles and C escapes. Its peak occurs at t=1/q in this equal-rate pulse case. The time of the C detector maximum would require examining kcC and the observation response rather than assuming it coincides with initial B formation.

This limit is a useful algebraic check and an interpretive one. A delayed product maximum does not necessarily imply a delayed switch-on of the chemical reaction. Sequential population filling and emptying can produce the delay with constant coefficients. The full material ledger remains the check on any proposed explanation.

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A constant escape hazard corresponds to an exponential waiting-time model in the absence of reaction. Real transport can involve a broader or more structured distribution of path lengths and waiting times. Two systems with the same mean residence time need not have the same probability of escaping before reaction.

For a supplied nonreactive residence time T and an independent reaction hazard kr, the survival to escape is exp(−krT). If T varies, the eventual escaped fraction is the average of that exponential over the residence distribution. A single average residence time does not generally determine that average survival.

This extension explains a boundary of the constant-hazard model rather than silently replacing its coefficients with a spatial diffusivity. The correct transport representation depends on the physical question and evidence. A more detailed model should earn its added complexity by explaining an observable that the simpler model cannot adequately describe.

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Take a mean nonreactive residence time of one time unit and kr = 1 per time unit. If every residence time is exactly one, survival to escape is exp(−1), approximately 0.3679.

Now let half the supplied material have residence time zero and half have residence time two. The mean is still one, but mean survival is [1+exp(−2)]/2, approximately 0.5677. The distribution is idealised; the zero-time component represents an immediate bypass within this mathematical example.

The larger survival is consistent with the convexity of exp(−krT), so averaging residence times before exponentiating changes the predicted outcome. This is not evidence that every broad residence distribution improves every catalytic objective. It isolates one surviving-intermediate calculation under a stated independent reaction law. The useful lesson is to preserve the distribution when the observable depends nonlinearly on it.

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For one newly formed unit, the probability of remaining without either event until time t is exp(−qt). The probability density for escape at that time is ke exp(−qt). Integrating it from zero to infinity gives ke/q.

This derivation reaches the same eventual fraction as the population balance. It supplies an independent interpretation of why the two hazards add in the survival exponent. Neither event can occur while the unit remains in the retained-B state; the first event ends that state.

The probability picture also makes independence assumptions visible. Constant competing hazards describe a particular memoryless model. If a molecule’s escape propensity depends on its age, location or prior adsorption history, the same simple density need not apply. Agreement between the probability and population calculations checks the model’s internal consistency, not its adequacy for every physical pore network.

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A controlled comparison should distinguish changes in formation flux, secondary chemistry, escape behaviour, observation window and detector response. Keeping one nominal composition or framework code unchanged does not prove that all these quantities are constant.

For example, a changed crystal size can affect transport while also being associated with differences in synthesis history or accessible site inventory. A changed acid-site density can alter reaction and the populations that occupy pores. The model’s one-variable comparisons are deliberately cleaner than many real material comparisons.

A defensible conclusion therefore states both the observed change and the controls supporting its interpretation. An unchanged initial branch is a supplied condition in our examples. To establish it experimentally requires evidence about primary product formation or a suitable independent mechanism analysis, rather than an outlet fraction alone.

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Bickel Rogers and Gounder’s 2024 study, Assessing the influences of kinetics and intrazeolite diffusion on propene oligomerization product selectivity in MFI zeolites, independently varies material properties to assess coupled kinetic and transport effects. Its institutional abstract emphasises the difficulty of separating those influences in product selectivity.

That is the contextual reason for keeping our formation, reaction and escape assumptions separate. The study does not supply the fictional coefficients used here, and our one-population constant-hazard model is not its complete diffusion–reaction treatment. The source record was checked on 25 October 2026.

Read the primary research for its experimental design and mechanistic scope. Use the present model to practise a narrower inference: which combinations of competing rates can an outlet fraction identify, and which additional measurements would be needed to separate them?

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A pulse supplies B0=12 conserved units to an initially empty model region. Let ke=1 and kr=3 per time unit. No further formation occurs. Observe the system at t=ln 2/4 time units.

Find the retained amount, cumulative export and secondary formation. Calculate both the exported fraction among completed outcomes and the exported fraction of the supplied pulse. Then double both coefficients while keeping the pulse and observation time fixed. Predict which fraction remains unchanged and which quantities change.

Finally suppose a separate relaxation measurement establishes q=8 in the doubled-rate experiment. Recover ke and kr from the completed-outcome fraction. Explain why an outcome fraction without that time-scale observation would permit infinitely many common-scaled coefficient pairs. Try the problem from the ledger and survival law before reading the answer.

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Initially q=4 and qt=ln 2, so half of the pulse remains: B=6. The completed outcomes total six. Because ke/q=1/4, export is E=1.5 and secondary formation is R=4.5. The ledger is 6+1.5+4.5=12.

The completed-outcome exported fraction is 1/4, while export divided by the pulse is 1/8. Doubling the rates gives q=8 and exp(−qt)=1/4 at the same observation time. The retained amount is now three, completed outcomes total nine, export is 2.25 and secondary formation is 6.75. The completed-outcome fraction stays 1/4; the finite-time exported fraction becomes 3/16.

With q=8 independently known, ke=(1/4)8=2 and kr=(3/4)8=6. The same branching fraction with no time scale would identify only their ratio, not these particular magnitudes.

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Suppose the objective is a long-time escaped fraction of at least 0.8 under the constant-hazard model, with kr held fixed. The condition ke/(ke+kr)≥0.8 rearranges to ke≥4kr.

That inequality concerns eventual competition. It does not guarantee that at least eighty percent of all material supplied during a short continuous-feed experiment has already been exported. The finite-time yield includes the filling factor 1−(1−exp(−qt))/(qt), which remains below one.

Nor is this mathematical target a process recommendation. Changing a real material or operating condition can affect both coefficients, the upstream formation network and other constraints. A target calculation becomes useful when its denominator, time horizon and fixed quantities are stated explicitly. Otherwise it can provide a precise numerical answer to a question different from the intended one.

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For s=ke/(ke+kr), the derivative with respect to ke at fixed kr is kr/q², while the derivative with respect to kr at fixed ke is −ke/q². Both follow directly from differentiating the ratio.

The logarithmic sensitivities are d ln s/d ln ke=1−s and d ln s/d ln kr=−(1−s). At high escaped fraction, small proportional changes in either coefficient have a relatively small proportional effect on s within the local model. Near a desired fraction of one, a large rate ratio may be required for a modest further fraction increase.

These derivatives describe controlled mathematical perturbations. An observed selectivity trend with temperature or crystal size cannot be assigned to either derivative until the accompanying changes in both coefficients and other model inputs are addressed. Sensitivity analysis clarifies conditional dependence; it does not replace an experimental causal comparison.

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Suppose independent evidence supports 2≤ke≤4 and 1≤kr≤3 in the same inverse-time units, with no further relation between the two coefficients. The escaped fraction increases with ke and decreases with kr. Therefore its allowed range is from 2/(2+3)=0.4 to 4/(4+1)=0.8.

If the coefficient uncertainties are correlated, these opposite corners may not both be feasible. For example, a known fixed q would restrict the possible pairs to a line rather than the whole rectangular range. Use the actual joint constraint when it is available.

This bounded calculation is often more informative than choosing midpoints and reporting a single fraction without qualification. It also keeps a distinction between uncertainty about the parameters and the structural ambiguity of an outlet fraction that identifies only a ratio even when that fraction is measured exactly.

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Repeated measurements can reduce random uncertainty in an outcome fraction, subject to the actual dependence between observations. They cannot remove the common-scale ambiguity of ke and kr when no time-scale information is supplied.

Likewise, a low-residual fit to one exported-yield curve should be checked against the known input history and measurement response. An unknown formation flux can trade off with amplitude; an unmodelled detector delay can trade off with a fitted relaxation; extra retained populations can change the shape.

A useful verification sequence is to identify what exact observables could determine under the selected equations, then propagate measurement uncertainty, then test whether the equations adequately represent the physical experiment. Narrow numerical parameter intervals are meaningful only within the model and calibration assumptions used to produce them.

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If F is independently known and a genuine stationary retained amount Bstar is measured, the balance gives q=F/Bstar. Combined with the completed-outcome fraction s, this supplies ke=sF/Bstar and kr=(1−s)F/Bstar. It is another route to the same conditional inverse without observing the full relaxation curve.

For the original model, F=10 and Bstar=2 imply q=5. With s=0.6, the recovered coefficients are three and two. A retained amount measured before the stationary regime cannot be inserted into this expression: B(t) is smaller than F/q, so treating it as stationary overestimates q.

At the original observation time t=0.4, B is approximately1.7293. Dividing ten by that value gives approximately5.7826 rather than five. This discrepancy is not a change in chemistry. It comes from applying a stationary relation to a transient population. To justify the inverse, establish that formation and outcome fluxes balance and that the retained amount has stabilised within the relevant uncertainty. A nearly flat detector trace alone may describe a different population or response.

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If kr approaches zero with ke positive, the eventual exported fraction approaches one. During a finite continuous-feed interval, some intermediate still remains inside, so exported amount need not yet equal supplied amount. If ke approaches zero with kr positive, export vanishes while retained B and secondary formation still close the balance.

If both coefficients approach zero, the model becomes dB/dt=F, so B=Ft and both outcomes vanish. The expression F[1−exp(−qt)]/q has that finite limit even though direct substitution produces zero divided by zero. A stable numerical implementation can use a small-argument exponential routine or the series expansion rather than interpret the apparent singularity as a physical divergence.

These limits provide independent checks on units and meaning. They also keep an eventual-outcome fraction separate from finite-time material recovery. At exactly zero total hazard, there are no completed outcomes, so their branching fraction is undefined rather than a measured zero or one.

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Does slower escape always reduce the desired product yield?

Only if the desired outcome is escaped B and the other supplied assumptions remain valid. If C is desired, a secondary conversion can be useful instead. State the objective and the reaction network before judging a transport change.

Is ke the same as a measured diffusion coefficient?

No. It is an inverse-time coefficient in a one-population escape model. Connecting it to spatial diffusion requires additional geometry, boundary and transport assumptions. A numerical similarity does not establish that connection.

Can a changed outlet fraction prove that acid sites changed?

Not by itself. The example changes the fraction through escape while keeping the secondary-reaction coefficient and formation branch fixed. Real acid-site changes need independent characterisation and a compatible mechanism analysis.

Does unchanged selectivity mean the catalyst response is unchanged?

No. Common scaling of both hazards preserves the completed-outcome fraction while changing relaxation, retained amount and finite-time export. Examine the observable relevant to the actual decision.

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For the material foundation, return to the Zeolite Catalysis subject guide. For a spatial transport calculation with a different observable, read why diffusion can change an apparent activation energy. Its slab effectiveness model should not be confused with the constant escape hazard introduced here.

For active-site evidence, use bounding internal and external site counts when probe access is incomplete. For a different kinetic cause of a changed rate, read why increasing a reactant can lower a shared-site reaction rate. These routes distinguish a missing denominator, competitive occupancy and post-formation escape rather than treating all three as the same explanation.

Connect the habit of closing a model before interpreting its outputs through The Gold Standard of Education and the How X Works hub. A strong explanation identifies where the intermediate is counted, which competing events are modelled, what the observation window includes and which independent evidence supports the inferred rates.

For practice, explain the same ledger aloud using one retained population, two outcome counters and a clearly stated observation boundary.

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