# Stoichiometry **Stoichiometry** is the part of [[Chemistry|chemistry]] that calculates the quantities of reactants and products in a [[Chemical_reaction|chemical reaction]] from the balanced [[Chemical_equation|chemical equation]] and the [[Mole_(unit)|molar]] masses of the substances involved. It rests on two ideas: that a balanced equation states the ratio in which molecules react, and that the [[Conservation_of_mass|conservation of mass]] lets those ratios be carried from the molecular scale to the laboratory scale through the mole. A stoichiometric calculation converts a mass to an [[Amount_of_substance|amount of substance]], applies the mole ratio the equation gives, and converts back to a mass, a volume of gas, or a concentration.[^os-af-ch7] On the Chemistry flagship the article serves the section *Mole and amount of substance* (Part II — Modern principles › Matter): the mole is the unit that makes the arithmetic possible, and stoichiometry is the arithmetic. In the microsim below the reader sets the amounts of [[Hydrogen|hydrogen]] and [[Oxygen|oxygen]] fed to the reaction `2H₂ + O₂ → 2H₂O`, in moles or in grams, and the mole-ratio bars show which reagent runs out first, how much water forms, and how much of the other reagent is left over; the equation the sim answers is `n_product = n_limiting × (coefficient of product / coefficient of limiting reagent)`. A second preset, `2KClO₃ → 2KCl + 3O₂`, reads the oxygen produced as litres at [[Standard_temperature_and_pressure|standard temperature and pressure]] through `V = nRT/P`, with 22.41 L for each mole of gas.[^averill-stp] ## Etymology The word was coined by the German chemist Jeremias Benjamin Richter, whose *Anfangsgründe der Stöchyometrie oder Meßkunst chymischer Elemente* ("Rudiments of stoichiometry, or the art of measuring the chemical elements") appeared in three parts from 1792.[^richter1792] Richter built it from the Greek *stoicheion*, "element", and *metron*, "measure", and used it for the modern purpose: finding the fixed mass ratios in which acids and bases neutralise one another. The term therefore predates [[John_Dalton|Dalton's]] atomic theory of 1808, which supplied the explanation for the ratios Richter had measured.[^dalton1808] ## Definitions A balanced equation such as `2H₂ + O₂ → 2H₂O` can be read at two scales. At the molecular scale it says that two molecules of hydrogen combine with one [[Molecule|molecule]] of oxygen to give two molecules of water. At the laboratory scale it says that two moles of hydrogen combine with one mole of oxygen to give two moles of water, because a mole of anything contains the same number of entities. Since 2019 the mole has been defined by fixing the [[Avogadro_constant|Avogadro constant]] at exactly 6.02214076 × 10²³ per mole, so one mole is that many specified entities, whether [[Atom|atoms]], molecules, [[Ion|ions]] or [[Electron|electrons]].[^bipm-si] The molar mass `M` of a substance, in grams per mole, is numerically the same as the mass of one molecule in unified atomic mass units, which lets a chemist weigh out a known number of molecules: for [[Zinc|zinc]], 65.409 g contains one mole of atoms.[^ball-atomicmass] Stoichiometry then names three things: the stoichiometric coefficients, which are the numbers in front of the formulas; the stoichiometric ratio between two substances, which is the ratio of their coefficients; and a stoichiometric mixture, in which the reactants are present in exactly that ratio and are consumed together. Any other mixture has one reagent in excess and one that limits.[^os-af-ch7] ## Converting grams to moles Every stoichiometric calculation begins by converting what can be weighed into what the equation counts, `n = m/M`.[^averill-nmM] One gram of zinc is `1.00 g ÷ 65.409 g/mol = 0.0153 mol`,[^ball-atomicmass] and because each zinc atom that dissolves in [[Acid|acid]] releases one molecule of hydrogen, `Zn + 2HCl → ZnCl₂ + H₂`, the same 0.0153 mol is the amount of hydrogen produced. The molar mass of a compound is the sum of the [[Atomic_mass|atomic masses]] in its formula: [[Sodium|sodium]] azide, NaN₃, has `M = 22.99 + 3 × 14.01 = 65.01 g/mol`, so the 5.00 g charge of an automobile airbag inflator is `5.00/65.01 = 0.0769 mol NaN₃`.[^averill-airbag] The conversion runs the other way at the end, `m = n × M`, and it is the only place where the identity of the substance enters; everything in between is counting. Two habits keep the counting honest: carry units, so that grams divided by grams per mole visibly yield moles, and write the balanced equation before touching a balance, because the [[Chemical_formula|formula]] fixes the molar mass but only the equation fixes the ratio. ## Molar proportion The coefficients of a balanced equation give the molar proportions of the substances, and those proportions are the conversion factors of stoichiometry. For `2H₂ + O₂ → 2H₂O` the proportions are 2 : 1 : 2, so the amount of oxygen needed for a given amount of hydrogen is `n(O₂) = n(H₂) × 1/2`, the amount of water formed from that hydrogen is `n(H₂O) = n(H₂) × 2/2`, and so on for any pair.[^os-af-ch7] Mass proportions differ, because the substances have different molar masses: 4.03 g of hydrogen (two moles) consumes 32.00 g of oxygen (one mole), a mass ratio of nearly 1 : 8, the kind of ratio Richter-era chemists measured without knowing why. The microsim draws these proportions as bars. Each reagent's bar is scaled by its coefficient, so that stoichiometric amounts give bars of equal height; when the reader feeds more of one reagent than the other can use, its bar stands above the line and the excess is visible before any arithmetic is done. Scaling every coefficient by 0.5 gives `H₂ + ½O₂ → H₂O`, the same reaction written for one mole of water and the form [[Thermochemistry|thermochemistry]] prefers. ## Determining amount of product The amount of a product is the amount of the reagent that is consumed completely, multiplied by the ratio of coefficients: `n(product) = n(reagent) × (coefficient of product / coefficient of reagent)`. In the airbag reaction `2NaN₃ → 2Na + 3N₂`, the 0.0769 mol of sodium azide gives `0.0769 × 3/2 = 0.115 mol N₂`, which the Portal Book's worked example collects at 22 °C and 762 mmHg over [[Water|water]] and finds to occupy 2.85 L once the water's own [[Vapor_pressure|vapour pressure]] of 19.84 mmHg is subtracted from the total.[^averill-airbag] The volume inflates the bag; the stoichiometry sizes the charge. ### Further examples Jacques Charles's hydrogen balloon of 1783 held 31,150 L of hydrogen, which the Portal Book takes as being made by the reaction of iron with sulfuric acid at a stated [[Pressure|pressure]] and [[Temperature|temperature]], `Fe + H₂SO₄ → FeSO₄ + H₂`. At 30 °C and 745 mmHg the gas is `n = PV/RT = (745/760 atm × 31,150 L)/(0.08206 L·atm·K⁻¹·mol⁻¹ × 303 K) = 1.23 × 10³ mol`, and because the equation is one-to-one, the same 1.23 × 10³ mol of [[Iron|iron]] was needed: 68.6 kg of it.[^averill-fe] The example runs the stoichiometric chain backwards, from a required product to the reagent to be bought. Working forwards, 1.00 g of zinc in excess hydrochloric acid gives 0.0153 mol of hydrogen, which at 30 °C and a total pressure of 760 mmHg over water occupies 0.397 L.[^averill-zn] ## Stoichiometric ratio The stoichiometric ratio of two substances is the ratio of their coefficients, and a mixture is at stoichiometric ratio when the reagents are supplied in that proportion. For `2H₂ + O₂ → 2H₂O` the ratio is 2 mol H₂ per 1 mol O₂; in mass it is `2 × 2.016 : 31.998 = 1 : 7.94`. Supplying the reagents in any other proportion leaves one of them over, and the ratio between what was supplied and what was needed is the single number that tells a chemist which reagent that will be.[^os-af-ch7] In [[Combustion|combustion]] the ratio is called the stoichiometric air-to-fuel ratio, and an engine running "lean" or "rich" is running above or below it; the same idea sets the [[Molar_concentration|concentrations]] mixed in a [[Titration|titration]]. ## Limiting reagent and percent yield When reagents are not supplied in stoichiometric ratio, the one that is used up first is the [[Limiting_reagent|limiting reagent]], and it alone determines how much product can form; the others are in excess and some of each remains.[^os-af-ch7] The test is to divide the amount of each reagent by its coefficient and take the smallest quotient: `n(H₂)/2` against `n(O₂)/1`. That quotient, the number of times the equation "runs", multiplied by any coefficient gives the amount of that substance consumed or produced, and the difference between what was supplied and what was consumed is the leftover. The microsim makes this the reader's own experiment. Two sliders set `n(H₂)` and `n(O₂)`, or the equivalent grams; the sim computes the two quotients, names the limiting reagent, and draws three bars: the water formed, the excess reagent remaining, and, faintly, the amount of the limiting reagent that would have been needed to use up the excess. Dragging one slider through the stoichiometric point flips which reagent limits, and the product bar changes slope there because it now tracks the other input. Nothing in the sim knows about hydrogen or oxygen beyond their coefficients and molar masses, which is why the same code serves the potassium chlorate preset. The theoretical yield is the amount of product the limiting reagent allows. The actual yield is what is isolated, and the percent yield is `100 × actual/theoretical`.[^os-af-ch7] Yields fall short because side reactions consume reagent, because reactions reach [[Chemical_equilibrium|equilibrium]] before completion, and because product is lost in isolation; a yield above 100 % signals a wet or impure product, not a breach of conservation of mass. ### Example Feed the sim 3.00 mol of hydrogen and 2.00 mol of oxygen. The quotients are `3.00/2 = 1.50` for hydrogen and `2.00/1 = 2.00` for oxygen, so hydrogen limits and the equation runs 1.50 times. Water formed: `1.50 × 2 = 3.00 mol`, which is `3.00 × 18.015 = 54.0 g`. Oxygen consumed: `1.50 × 1 = 1.50 mol`, leaving `2.00 − 1.50 = 0.50 mol`, or 16.0 g. The mass check closes: `6.05 g H₂ + 64.0 g O₂ = 70.0 g` in, and `54.0 g H₂O + 16.0 g O₂ = 70.0 g` out. Switching the sliders to grams and feeding 10.0 g of hydrogen with 64.0 g of oxygen reverses the outcome: `10.0/2.016 = 4.96 mol H₂` and `64.0/31.998 = 2.00 mol O₂`, the quotients are 2.48 and 2.00, oxygen now limits, 4.00 mol (72.1 g) of water forms and 0.96 mol (1.94 g) of hydrogen is left. The same 64 g of oxygen gave a different amount of water because the other input changed which reagent was in charge. If 68.0 g of water is then isolated, the percent yield is `100 × 68.0/72.1 = 94 %`. ## Different stoichiometries in competing reactions The same reagents can react by more than one balanced equation, and which one dominates depends on the proportions supplied. Methane burns to carbon dioxide when oxygen is plentiful, `CH₄ + 2O₂ → CO₂ + 2H₂O`, but to carbon monoxide when it is short, `2CH₄ + 3O₂ → 2CO + 4H₂O`; the two equations need two and one-and-a-half moles of [[Oxygen|oxygen]] per mole of methane respectively, so a burner fed below the first ratio produces both products, and a fire in a closed room drifts toward the second.[^os-af-ch7] Iron with chlorine gives FeCl₂ or FeCl₃ by the same logic, and a limiting-reagent calculation done with the wrong equation gives the wrong answer with the right arithmetic. Stoichiometry therefore begins with the chemistry: the equation is a claim about what actually happens, and competing pathways must be settled by [[Chemical_kinetics|kinetics]] or [[Chemical_thermodynamics|thermodynamics]] before the counting starts. ## Stoichiometric coefficient and stoichiometric number The stoichiometric coefficient is the positive number written in front of a formula in a balanced equation. The stoichiometric number `ν_i` of a species is the coefficient given a sign: negative for reactants, positive for products, so that the reaction `2H₂ + O₂ → 2H₂O` has `ν(H₂) = −2`, `ν(O₂) = −1` and `ν(H₂O) = +2`.[^iupac-gold] The sign convention allows a reaction to be written as a single sum, `Σ ν_i B_i = 0`, and gives the extent of reaction `ξ` a clean definition: as a reaction proceeds, the change in the amount of every species is `dn_i = ν_i·dξ`, so one value of `ξ` describes the whole mixture.[^iupac-gold] In the example above, "runs 1.50 times" is precisely `ξ = 1.50 mol`, and the leftover oxygen is `n₀(O₂) + ν(O₂)·ξ = 2.00 − 1.50`. The limiting reagent is the species for which `n₀,i/|ν_i|` is smallest, because that is the extent at which its amount would reach zero. ## Stoichiometry matrix When a system contains several reactions at once, the stoichiometric numbers are collected into a matrix `N` with one row per species and one column per reaction, entry `ν_ij` being the stoichiometric number of species `i` in reaction `j`. For the sim's two presets, taken together as a system of five species and two reactions, the rows for H₂, O₂, H₂O, KClO₃ and KCl are `(−2, 0)`, `(−1, +3)`, `(+2, 0)`, `(0, −2)` and `(0, +2)`, and oxygen is the one species that appears in both columns, consumed by the first reaction and produced by the second. The amounts then evolve as `n = n₀ + N·ξ`, with `ξ` now a vector of extents, and rates as `dn/dt = N·v`, where `v` is the vector of reaction rates.[^iupac-gold] The matrix form is what makes stoichiometry a piece of [[Linear_algebra|linear algebra]]: its left null space gives the conserved quantities (here the totals of each element), and metabolic and atmospheric models with thousands of reactions are handled the same way, one column at a time. ## Gas stoichiometry For a gas, the amount is most easily obtained not from a mass but from a volume through the [[Ideal_gas_law|ideal gas law]], `PV = nRT`, with `R = 0.082057 L·atm·K⁻¹·mol⁻¹`.[^averill-pvnrt] At standard temperature and pressure, 273.15 K and 1 atm, one mole of any ideal gas occupies 22.41 L, and the sim's second preset uses that number to convert the oxygen it counts into litres.[^averill-stp] The Portal Book's worked case is the decomposition of [[Potassium|potassium]] chlorate, `2KClO₃ → 2KCl + 3O₂`: if 1.34 g of KCl is formed, that is `1.34/74.55 = 0.0180 mol KCl`, which required 0.0180 mol, or 2.20 g, of KClO₃ and released `0.0180 × 3/2 = 0.0270 mol O₂`, which is 0.863 g and, at STP, `0.0270 × 22.41 = 0.604 L`, the book's 604 mL.[^averill-kclo3] The sim reproduces those three numbers from the single input of 1.34 g. Away from STP the full law is used. The same book asks how much oxygen the [[Contact_process|contact process]] needs to make one ton of sulfuric acid at 295 K and 0.980 atm: the acid is `9.07 × 10⁵ g/98.08 g·mol⁻¹ = 9250 mol`, the overall stoichiometry `S + 3/2 O₂ + H₂O → H₂SO₄` calls for `1.39 × 10⁴ mol O₂`, and `V = nRT/P = 3.43 × 10⁵ L` of oxygen at those conditions.[^averill-h2so4] Because equal volumes of ideal gases hold equal amounts at the same T and P, the coefficients of a gas-phase reaction are also volume ratios, which is how [[Gay-Lussac's_law|Gay-Lussac's]] combining volumes were read before the mole existed. ## Stoichiometric air-to-fuel ratios of common fuels The stoichiometric air-to-fuel ratio of a fuel is the mass of air that supplies exactly the oxygen its complete combustion equation requires, divided by the mass of the [[Fuel|fuel]]. Dry air is about 21 % oxygen by volume with a mean molar mass near 29 g/mol,[^averill-air] so its oxygen mass fraction is `0.21 × 32.00/29 = 0.232` and the ratio is `AFR = (mass of O₂ required per mass of fuel)/0.232`. The table gives the results for pure fuels, each derived from its balanced equation with that air composition (derived; the second decimal shifts with the exact air composition and molar masses used). | Fuel | Combustion equation | O₂ per unit mass of fuel | Stoichiometric AFR (by mass) | |---|---|---|---| | Hydrogen | H₂ + ½O₂ → H₂O | 7.94 | 34 | | Methane | CH₄ + 2O₂ → CO₂ + 2H₂O | 3.99 | 17.2 | | Propane | C₃H₈ + 5O₂ → 3CO₂ + 4H₂O | 3.63 | 15.7 | | Octane | C₈H₁₈ + 12½O₂ → 8CO₂ + 9H₂O | 3.50 | 15.1 | | Ethanol | C₂H₅OH + 3O₂ → 2CO₂ + 3H₂O | 2.08 | 9.0 | | Carbon | C + O₂ → CO₂ | 2.66 | 11.5 | The ordering is the chemistry of the fuel. Hydrogen needs the most air per gram because it is the lightest fuel per mole of oxygen consumed; ethanol needs the least because it already carries an oxygen atom, and [[Carbon|carbon]] sits between. An [[Internal_combustion_engine|engine]] management system uses the stoichiometric ratio of the actual blend as its reference: the ratio of the supplied mixture to the stoichiometric one, `λ`, is held at 1 for a three-way [[Catalysis|catalyst]] to work, and departures from it are the lean and rich regimes of the competing-reaction section above. ## See also - [[Mole_(unit)]] - [[Amount_of_substance]] - [[Avogadro_constant]] - [[Molar_concentration]] - [[Law_of_multiple_proportions]] - [[Limiting_reagent]] - [[Chemical_equation]] - [[Conservation_of_mass]] - [[Ideal_gas_law]] - [[Combustion]] ## References [^os-af-ch7]: Flowers, Paul; Neth, Edward; Robinson, William, et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 7, "Stoichiometry of Chemical Reactions", pp. 313–362 (reaction stoichiometry, limiting reactant, percent yield; page to pin). Portal Book 051. https://openstax.org/details/books/chemistry-atoms-first-2e [^averill-stp]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 905 (one mole of an ideal gas occupies 22.41 L at 273.15 K and 1 atm). Portal Book 050. https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^richter1792]: Richter, Jeremias Benjamin (1792–1794). *Anfangsgründe der Stöchyometrie oder Meßkunst chymischer Elemente*. Three parts. Breslau and Hirschberg: Johann Friedrich Korn. [^dalton1808]: Dalton, John (1808). *A New System of Chemical Philosophy*, Part I. Manchester: R. Bickerstaff. [^bipm-si]: Bureau International des Poids et Mesures (2019). *The International System of Units (SI)*, 9th edition, §2.3.1 (definition of the mole; Avogadro constant fixed at 6.02214076 × 10²³ mol⁻¹). https://www.bipm.org/en/publications/si-brochure [^ball-atomicmass]: Ball, David W. (2011). *Introductory Chemistry*. §3.3 "Masses of Atoms and Molecules", pp. 120–124 (atomic mass of zinc 65.409 u; molecular masses as sums of atomic masses). Portal Book 056. https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^averill-nmM]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 912 (`n = m/M`). Portal Book 050. [^averill-airbag]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 934–935 (Example 14: 5.00 g NaN₃ → 0.115 mol N₂, collected at 22 °C and 762 mmHg over water, P_H₂O = 19.84 mmHg, V = 2.85 L). Portal Book 050. [^averill-fe]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 906–907 and 932 (Charles's balloon, 31,150 L of H₂; iron required at 30 °C and 745 mmHg, 68.6 kg). Portal Book 050. [^averill-zn]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 935–936 (1.00 g Zn + HCl at 30 °C and 760 mmHg over water → 0.397 L H₂). Portal Book 050. [^iupac-gold]: IUPAC. *Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book"), entries "stoichiometric number" and "extent of reaction". https://goldbook.iupac.org [^averill-pvnrt]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 904 (`PV = nRT`; R = 0.082057 L·atm·K⁻¹·mol⁻¹ = 8.3145 J·K⁻¹·mol⁻¹). Portal Book 050. [^averill-kclo3]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 937–938 (KClO₃ decomposition: 1.34 g KCl formed from 2.20 g KClO₃; 0.863 g O₂, 604 mL at STP). Portal Book 050. [^averill-h2so4]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 931–932 (oxygen for 1.00 ton of H₂SO₄ at 295 K and 0.980 atm: 9250 mol acid, 1.39 × 10⁴ mol O₂, 3.43 × 10⁵ L). Portal Book 050. [^averill-air]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 946–947 (average molar mass of air about 29 g/mol). Portal Book 050. ## External links - [Chemistry: Atoms First 2e](https://openstax.org/details/books/chemistry-atoms-first-2e), OpenStax — Chapter 7, the open text behind Portal Book 051 - [The International System of Units (SI), 9th edition](https://www.bipm.org/en/publications/si-brochure), BIPM — the 2019 definition of the mole - [IUPAC Gold Book](https://goldbook.iupac.org) — "stoichiometric number", "extent of reaction" - For the pair's other external links, see the Wikipedia article's *External links* section. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Stoichiometry.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Stoichiometry* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Stoichiometry.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Stoichiometry.html" data-title="Stoichiometry"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Stoichiometry) : [Wikitube](https://en.wikitube.io/wiki/Stoichiometry) · pinned revision [1372736584](https://en.wikipedia.org/w/index.php?oldid=1372736584) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K13 · sim pending (matter/Stoichiometry).*