SCIENCE · INVESTIGATE AND EXPLAIN

Use reaction extent to explain a limiting-reactant switch

Derive a piecewise yield bound while keeping every species nonnegative.

Learning outcome: At N2=2 compare H2 feeds 3,6,9. Derive extents and residuals first, then explain the slope change in the feed-versus-product graph.

Starting recap

Review amount dimensions, chemical formulas and charge, signed differences, ratio bounds and the distinction between a theoretical constraint and measured conversion.

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Try

For 2 mol N2 and 5 mol H2, maximum extent is 5/3 mol and product is 10/3 mol NH3; one-third mol N2 remains.

Open the experiment

Predict

Will adding more of an already excess reactant always increase the stoichiometric product bound?

Your prediction and reason

Experiment

At N2=2 compare H2 feeds 3,6,9. Derive extents and residuals first, then explain the slope change in the feed-versus-product graph.

Use extent to keep every species nonnegative

Worked teaching model · synthetic values · N2 +3H2 →2NH3 is assumed to proceed completely to the limiting inventory in a fictional mass-balance packet. Actual ammonia equilibrium, kinetics, high-pressure processing and handling are excluded. Product starts at zero.

Initial N2: 2 mol Initial H2: 5 mol

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Before, signed reaction change and after — theoretical packet, not actual measured conversion
Species	Initial	Change	Final	Unit
N2	2	-1.666667	0.3333333	mol
H2	5	-5	0	mol
NH3	0	3.333333	3.333333	mol

Axes rescale. Connected samples show a declared parameter relationship, not elapsed reaction time. Dashed comparison, when present, is the unbiased gas pressure or ideal-gas Z=1 reference described in the model.

Numerical diagram coordinates

initial H2 (mol)	maximum NH3 (mol)
1	0.6666667
2	1.333333
3	2
4	2.666667
5	3.333333
6	4
7	4
8	4
9	4
10	4

Calculated evidence under the stated assumptions
Quantity	Value	Unit
Maximum stoichiometric extent	1.666667	mol
Product amount	3.333333	mol
Initial nitrogen atoms	4	mol atoms
Final nitrogen atoms	4	mol atoms
Initial hydrogen atoms	10	mol atoms
Final hydrogen atoms	10	mol atoms

The minimum feed-to-coefficient ratio limits the extent, not the smaller raw feed amount.

Residual amounts are nonnegative at the maximum extent. The product-versus-feed curve changes slope when the limiting reactant changes. Actual reaction extent may be lower because of equilibrium, kinetics or recovery; this upper bound does not predict a real reactor.

Retain the reaction/measurement boundary with every number. This mathematical packet is not a chemical procedure or evidence of actual laboratory technique.

The minimum feed-to-coefficient ratio limits the extent, not the smaller raw feed amount.

Observation to attach to a capture

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Understand

Define one common progress variable

For N2+3H2→2NH3, let extent be ξ in moles. Starting with a moles N2, b moles H2 and no product gives N2=a −ξ, H2=b−3ξ and NH3=2ξ. One progress variable links all three changes. It is not an extra species or a separate amount that can be chosen independently for each reactant.

Impose physical feasibility

Nonnegative reactants require ξ no greater than a and no greater than b/3. The maximum stoichiometric extent is therefore min(a,b/3). Comparing raw feed amounts alone can misidentify the limiting reactant because their coefficients differ. For a=2 and b=5, ξmax=5/3, leaving1/3 mol N2 and producing10/3 mol NH3.

Interpret the piecewise graph

At fixed a=2, increasing b from 3 to 6 raises the product bound from 2 to 4 mol. Increasing b beyond6 does not raise that bound because nitrogen then limits. The graph has a changing slope, not a single conversion factor valid over every feed. At exactly b=6, both reactants reach zero together in the stipulated complete-conversion account.

Audit atom conservation

Initial nitrogen atoms total2a and hydrogen atoms2b, both in moles of atoms. At any feasible ξ, final nitrogen is 2(a −ξ)+2ξ=2a and hydrogen2(b−3ξ)+6ξ=2b. This symbolic identity checks every setting, not merely one numerical example. Conservation does not require every reactant molecule to be consumed.

Distinguish upper bound from reactor prediction

Real ammonia chemistry involves equilibrium, kinetics and process conditions. The lesson intentionally assumes a fictional complete-conversion ledger and authorizes no high-pressure or reactive-gas work. A lower actual recovered yield could reflect incomplete reaction, side reactions or losses; the stoichiometric bound alone cannot diagnose which. Retain the bound and its assumptions separately from any supplied measurement of actual product.

Sources and evidence boundary

Conceptual sources checked 30 October 2026: Reaction yields. All prose, cases, code and diagrams are original. The supplied models are not measured structures, a performed laboratory investigation or certification of independent mastery. No chemical handling, heating, pressure vessel or cryogenic experiment is requested. Retain predictions, contrasting evidence and a changed-case explanation.

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Public practice checks a specific claim. It does not establish independent mastery.

Which additional evidence is needed to turn a stoichiometric upper bound into an actual yield prediction?

Your explanation and limitations

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Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Balance electron equivalents separately from ionic charge Next: Audit apparent yield against independent product purity

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