The limiting reagent, also called the limiting reactant, is the reactant that runs out first and so caps the product. Neither the smallest mass nor the fewest moles reliably finds it; the smallest moles divided by the coefficient does. We swept 25 two-reactant textbook reactions across mass ratios from 1:10 to 10:1. For hydrogen with chlorine, the smaller-mass guess was wrong in every mixture in that range where hydrogen weighed less. A yield hides more: OpenStax's copper example reports 77.3%, yet on our arithmetic at most 21.8% of the reactant mass ends up as copper.
Hydrogen and chlorine fool the smaller-mass test over a 35-fold range of mixtures
H₂ + Cl₂ → 2HCl uses 2.016 g of hydrogen per 70.90 g of chlorine, a mass ratio of 1 : 35.2 on CIAAW's abridged atomic weights. So 1 g of hydrogen with 10 g of chlorine runs out of chlorine first: 0.14 mol of Cl₂ against 0.50 mol of H₂.
The guess fails anywhere between equal masses and 35.2 times as much chlorine. For any pair it goes wrong between a mass ratio of 1 and the equation's own, so the further that sits from 1, the wider the trap. Light hydrogen made this the worst of our 25. Over 2,001 log-spaced mass ratios from 1:10 to 10:1, we counted where each shortcut named the wrong reactant. The range is our choice; a wider one moves every percentage and no window.
| Reaction | Mass ratio in the equation | Smaller mass wrong | Fewer moles wrong |
|---|---|---|---|
| H₂ + Cl₂ | 1 : 35.2 | 50.0% | 0% |
| 2H₂ + O₂ | 1 : 7.9 | 44.9% | 5.0% |
| 3Si + 2N₂ | 1.5 : 1 | 8.9% | 8.8% |
| Fe₂O₃ + 3CO | 1.9 : 1 | 13.9% | 23.8% |
| Pb(NO₃)₂ + 2KI | 1 : 1.00 | 0.0% | 15.0% |
Textbooks warn against comparing grams; the count is arithmetic anyone can repeat.
Mole counts mislead whenever the coefficients differ
In OpenStax Chemistry 2e Example 4.12, 2.00 g of silicon and 1.50 g of nitrogen give 0.0712 mol of Si and 0.0535 mol of N₂. Nitrogen has fewer moles, yet silicon runs out: 3Si + 2N₂ → Si₃N₄ takes three silicon for every two nitrogen, and 0.0712 ÷ 3 is smaller than 0.0535 ÷ 2.
The 8 of our reactions with equal coefficients never fooled the fewer-moles guess. Each of the other 17 had a failure window, and in 8 of the 25 it did worse than mass, as with Fe₂O₃ + 3CO. The IUPAC Green Book, §2.10.1, defines the extent of reaction ξ through nⱼ = nⱼ,₀ + νⱼξ. The smallest moles ÷ coefficient is the largest ξ the amounts allow, a rule deduced from that definition rather than printed in it.
Percent yield flatters a reaction with a ceiling set by its equation
Percent yield is actual over theoretical yield, times 100. Example 4.13 reacts 1.274 g of copper(II) sulfate with "excess zinc metal" and recovers 0.392 g of copper, 77.3%. The ceiling comes from the equation alone: 1.274 g at 159.62 g/mol is 0.007981 mol, enough for 0.5072 g of copper at 63.55 g/mol.
Our reaction yield and limiting reagent calculator could not take this example as written until 3 November 2026: it accepted moles only, refused a single reactant and could not mark one in excess, and its FAQ called a high yield a sign of an efficient, clean reaction. The revision of that date takes grams, a single reactant, or a reactant ticked "In excess". Its "One reactant in excess" example loads 1.274 g of CuSO₄ at 159.62 g/mol, zinc in excess, product coefficient 1, product molar mass 63.55 g/mol and actual yield 0.392 g, and returns a theoretical yield of 0.5072 g and a percent yield of 77.3%. It does not compute atom economy; those figures are ours.
Copper displacement disguises a low atom economy behind its yield
Atom economy asks what share of the reactants' mass can end up in the wanted product at all. For CuSO₄ + Zn → Cu + ZnSO₄ it is 28.2% on CIAAW's 2024 abridged values: 63.546 g of copper per 224.98 g of reactants. On our arithmetic, 77.3% of 28.2% is 21.8%, and that holds only if the zinc were exactly stoichiometric; with excess zinc the share is lower. Of our hand-picked 30 reactions, 11 reach 100% by making one product.
Barry Trost named the idea in 1991; his abstract in Science describes reactions "economical in atom count" and gives no percentage formula. A 2018 article by Machado in Química argues that the number came first from Roger Sheldon, in industry. That is one researcher's reading.
The US EPA's 1997 Greener Synthetic Pathways Award credits the BHC ibuprofen process with "approximately 80 percent atom utilization" in three catalytic steps, against under 40 percent for the six-step route it replaced. Our own balanced sum of the three steps' overall change gives 77.45%.
High yields mask the waste per kilogram as atom economy falls
On his E factor page, Sheldon says he introduced the measure in the late 1980s as "the mass ratio of waste to desired product". It counts the yield, and he generally left water out. His order-of-magnitude bands run from under 0.1 kg of waste per kg of product in oil refining to 25 and above in pharmaceuticals.
With exactly stoichiometric reactants and every unrecovered gram counted as waste, E is at least 1 ÷ (atom economy × yield) − 1, on our derivation.
| Atom economy | Yield | Lowest E (kg waste per kg product) |
|---|---|---|
| 100% | 50% | 1.00 |
| 50% | 95% | 1.11 |
| 30% | 95% | 2.51 |
A 95% yield at 50% atom economy wastes more mass than a 50% yield at 100%. Solvents and work-up sit outside this floor, so a real E factor is higher.
Near-stoichiometric mixtures trick a correct table of rounded masses
Singapore's 2027 G3 Chemistry syllabus, K324, uses whole numbers with chlorine at 35.5; the 2026 H2 Chemistry syllabus, 9729, one decimal place. Against CIAAW's abridged values, K324's moved the theoretical yield per gram by a median 0.072% over our 30 reactions, and by 1.05% for Mg + 2HCl.
CIAAW's ranges and stated uncertainties, taken at full width, moved no answer by more than 0.155%. The school table shifted the answer further in 29 of 30 reactions; lead nitrate, with lead's wide interval, was the exception. Example 4.13 gives 77.28% on CIAAW's values and 76.92% on K324's. Our guides cover the elements IUPAC gives a range and the mole since 2019.
Rounding can change which reactant limits only near the stoichiometric ratio. In a mixture we constructed, 1.000 g of magnesium with 3.02 g of hydrogen chloride, K324's masses make hydrogen chloride run out first and CIAAW's make magnesium. The printed table is still the right one for its exam.
K324 and the 2026 O-Level Chemistry syllabus, 6092, both say "calculations involving the idea of limiting reactants may be set" and ask for % yield. The 2026 O-Level Science syllabus 5086 says yield calculations "are not required". The word yield does not appear in 9729 or 9476, the two H2 syllabuses. None of the five mentions atom economy.
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