Stoichiometry (sometimes called reaction stoichiometry to distinguish it from composition stoichiometry) is the calculation of quantitative (measurable) relationships of the reactants and products in chemical reactions (chemical equations). A calculation is a deliberate process for transforming one or more inputs into one or more results. ... A scale for measuring mass A quantitative property is one that exists in a range of magnitudes, and can therefore be measured. ... A reactant or reagent is any substance initially present in a chemical reaction. ... For other uses, see Chemical reaction (disambiguation). ... A chemical equation is a symbolic representation of a chemical reaction. ...

Etymology

"Stoichiometry" is derived from the Greek words στοιχειον (stoikheion, meaning element) and μετρον (metron, meaning measure.) In patristic Greek, the word Stoichiometria was used by Nicephorus to refer to the number of line counts of the canonical books of the New Testament and some of the Apocrypha. . Bön . Hinduism (Tattva) and Buddhism (MahābhÅ«ta) Prithvi / Bhumi — Earth Ap / Jala — Water Vayu / Pavan — Air / Wind Agni / Tejas — Fire Akasha — Aether . ... Measurement is the estimation of the magnitude of some attribute of an object, such as its length or weight, relative to a unit of measurement. ... The Church Fathers or Fathers of the Church are the early and influential theologians and writers in the Christian Church, particularly those of the first five centuries of Christian history. ... Nicephorus III or Saint Nicephorus (ca. ... A biblical canon is a list of Biblical books which establishes the set of books which are considered to be authoritative as scripture by a particular Jewish or Christian community. ... This article is about the Christian scriptures. ... Apocrypha (from the Greek word , meaning those having been hidden away[1]) are texts of uncertain authenticity or writings where the authorship is questioned. ...

Definition

Stoichiometry rests upon the law of conservation of mass, the law of definite proportions (i.e., the law of constant composition) and the law of multiple proportions. In general, chemical reactions combine in definite ratios of chemicals. Since chemical reactions can neither create nor destroy matter, nor transmute one element into another, the amount of each element must be the same throughout the overall reaction. For example, the amount of element X on the reactant side must equal the amount of element X on the product side. The law of conservation of mass states that the mass of a system of substances will always remain constant, regardless of the processes acting inside the system. ... In chemistry, the law of definite proportions and also the elements states that a chemical compound always contains exactly the same proportion of elements by mass. ... One of the fundamental observations of modern chemistry, the law of definite proportions states that, in a pure compound, the elements combine in definite proportions to each other. ... In chemistry, the law of multiple proportions is one of the most basic laws of stoichiometry. ... Nuclear transmutation is the conversion of one chemical element or isotope into another, which occurs through nuclear reactions. ...

Stoichiometry is often used to balance chemical equations. For example, the two diatomic gases, hydrogen and oxygen, can combine to form a liquid, water, in an exothermic reaction, as described by the following equation: A space-filling model of the diatomic molecule dinitrogen, N2. ... This article is about the chemistry of hydrogen. ... This article is about the chemical element and its most stable form, or dioxygen. ... In chemistry, an exothermic reaction is one that releases heat . ...

The term stoichiometry is also often used for the molar proportions of elements in stoichiometric compounds. For example, the stoichiometry of hydrogen and oxygen in H₂O is 2:1. In stoichiometric compounds, the molar proportions are whole numbers (that is what the law of definite proportions is about). The mole (symbol: mol) is the SI base unit that measures an amount of substance. ...

Compounds for which the molar proportions are not whole numbers are called non-stoichiometric compounds. Non-stoichiometric compounds are chemical compounds with an elemental composition that cannot be represented by a ratio of well-defined natural numbers, and are therefore in violation of the law of definite proportions. ...

Stoichiometry is not only used to balance chemical equations but also used in conversions, i.e., converting from grams to moles, or from grams to milliliters. For example, to find the number of moles in 2.00 g of NaCl, one would do the following:

In the above example, when written out in fraction form, the units of grams form a multiplicative identity, which is equivalent to one (g/g=1), with the resulting amount of moles (the unit that was needed), is shown in the following equation,

Stoichiometry is also used to find the right amount of reactants to use in a chemical reaction. An example is shown below using the thermite reaction, In chemistry, the reactants are the substances that exist at the start of a chemical reaction. ... For other uses, see Chemical reaction (disambiguation). ... A thermit mixture using Iron Oxide A thermite reaction is a type of aluminothermic reaction in which aluminium metal is oxidized by the oxide of another metal, most commonly iron oxide. ...

So, to completely react with 85.0 grams of iron (III) oxide, 28.7 grams of aluminum are needed.

Different stoichiometries in competing reactions

Often, more than one reaction is possible given the same starting materials. The reactions may differ in their stoichiometry. For example, the methylation of benzene (C₆H₆) may produce singly-methylated (C₆H₅CH₃), doubly-methylated (C₆H₄(CH₃)₂), or still more highly-methylated (C₆H_{6 − n}(CH₃)_n) products, as shown in the following example, Methylation is a term used in the chemical sciences to denote the attachment or substitution of a methyl group on various substrates. ... For benzine, see petroleum ether. ...

In this example, which reaction takes place is controlled in part by the relative concentrations of the reactants. For other uses, see Concentration (disambiguation). ...

Stoichiometric coefficient

The stoichiometric coefficient in a chemical reaction system of the i–th component is defined as For other uses, see Chemical reaction (disambiguation). ... For other uses, see System (disambiguation). ...

or

where N_i is the number of molecules of i, and ξ is the progress variable or extent of reaction (Prigogine & Defay, p. 18; Prigogine, pp. 4–7; Guggenheim, p. 37 & 62). The extent of reaction can be regarded as a real (or hypothetical) product, one molecule of which is produced each time the reaction event occurs. 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

The stoichiometric coefficient ν_i represents the degree to which a chemical species participates in a reaction. The convention is to assign negative coefficients to reactants (which are consumed) and positive ones to products. However, any reaction may be viewed as "going" in the reverse direction, and all the coefficients then change sign (as does the free energy). Whether a reaction actually will go in the arbitrarily-selected forward direction or not depends on the amounts of the substances present at any given time, which determines the kinetics and thermodynamics, i.e., whether equilibrium lies to the right or the left. The thermodynamic free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... Water and steam are two different forms of the same chemical substance A chemical substance is a material with a definite chemical composition. ... In physical chemistry, chemical kinetics or reaction kinetics is the study of reaction rates in a chemical reaction. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... A burette, an apparatus for carrying out acid-base titration, is an important part of equilibrium chemistry. ...

If one contemplates actual reaction mechanisms, stoichiometric coefficients will always be integers, since elementary reactions always involve whole molecules. If one uses a composite representation of an "overall" reaction, some may be rational fractions. There are often chemical species present that do not participate in a reaction; their stoichiometric coefficients are therefore zero. Any chemical species that is regenerated, such as a catalyst, also has a stoichiometric coefficient of zero. In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... It has been suggested that this article or section be merged into Catalysis. ...

The simplest possible case is an isomerism In chemistry, isomers are molecules with the same chemical formula and often with the same kinds of chemical bonds between atoms, but in which the atoms are arranged differently (analogous to a chemical anagram). ...

in which ν_B = 1 since one molecule of B is produced each time the reaction occurs, while ν_A = −1 since one molecule of A is necessarily consumed. In any chemical reaction, not only is the total mass conserved, but also the numbers of atoms of each kind are conserved, and this imposes a corresponding number of constraints on possible values for the stoichiometric coefficients. Of course, only a small subset of the possible atomic rearrangements will occur. The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. ... For other uses, see Atom (disambiguation). ... The Periodic Table redirects here. ... “Superset” redirects here. ...

There are usually multiple reactions proceeding simultaneously in any natural reaction system, including those in biology. Since any chemical component can participate in several reactions simultaneously, the stoichiometric coefficient of the i–th component in the k–th reaction is defined as This article is about the physical universe. ... For the song by Girls Aloud see Biology (song) Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, speech lit. ... In general, a things components are its parts; the things that compose it. ...

so that the total (differential) change in the amount of the i–th component is

.

Extents of reaction provide the clearest and most explicit way of representing compositional change, although they are not yet widely used.

With complex reaction systems, it is often useful to consider both the representation of a reaction system in terms of the amounts of the chemicals present { N_i } (state variables), and the representation in terms of the actual compositional degrees of freedom, as expressed by the extents of reaction { ξ_k }. The transformation from a vector expressing the extents to a vector expressing the amounts uses a rectangular matrix whose elements are the stoichiometric coefficients [ ν_i k ]. In thermodynamics, state variables, state parameters or thermodynamic variables describe the momentary condition of a system. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

The maximum and minimum for any ξ_k occur whenever the first of the reactants is depleted for the forward reaction; or the first of the "products" is depleted if the reaction as viewed as being pushed in the reverse direction. This is a purely kinematic restriction on the reaction simplex, a hyperplane in composition space, or N‑space, whose dimensionality equals the number of linearly-independent chemical reactions. This is necessarily less than the number of chemical components, since each reaction manifests a relation between at least two chemicals. The accessible region of the hyperplane depends on the amounts of each chemical species actually present, a contingent fact. Different such amounts can even generate different hyperplanes, all of which share the same algebraic stoichiometry. The largest and the smallest element of a set are called extreme values, or extreme records. ... Kinematics (Greek κινειν,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... A hyperplane is a concept in geometry. ... 2-dimensional renderings (ie. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ...

In accord with the principles of chemical kinetics and thermodynamic equilibrium, every chemical reaction is reversible, at least to some degree, so that each equilibrium point must be an interior point of the simplex. As a consequence, extrema for the ξ's will not occur unless an experimental system is prepared with zero initial amounts of some products. In physical chemistry, chemical kinetics or reaction kinetics is the study of reaction rates in a chemical reaction. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

The number of physically-independent reactions can be even greater than the number of chemical components, and depends on the various reaction mechanisms. For example, there may be two (or more) reaction paths for the isomerism above. The reaction may occur by itself, but faster and with different intermediates, in the presence of a catalyst.

The (dimensionless) "units" may be taken to be molecules or moles. Moles are most commonly used, but it is more suggestive to picture incremental chemical reactions in terms of molecules. The N's and ξ's are reduced to molar units by dividing by Avogadro's number. While dimensional mass units may be used, the comments about integers are then no longer applicable. 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... For other uses, see Mass (disambiguation). ...

Stoichiometry matrix

In complex reactions, stoichiometries are often represented in a more compact form called the stoichiometry matrix. The stoichiometry matrix is denoted by the symbol, .

If a reaction network has n reactions and m participating molecular species then the stoichiometry matrix will have corresponding n columns and m rows.

For example, consider the system of reactions shown below:

S₁ → S₂

5S₃ + S₂ → 4S₃ + 2S₂

S₃ → S₄

S₄ → S₅

This systems comprises four reactions and five different molecular species. The stoichiometry matrix for this system can be written as:

where the rows correspond to S₁, S₂, S₃, S₄ and S₅, respectively. Note that the process of converting a reaction scheme into a stoichiometry matrix can be a lossy transformation, for example, the stoichiometries in the second reaction simplify when included in the matrix. This means that it is not always possible to recover the original reaction scheme from a stoichiometry matrix.

Often the stoichiometry matrix is combined with the rate vector, v to form a compact equation describing the rates of change of the molecular species:

Gas stoichiometry

Gas stoichiometry is the quantitative relationship between reactants and products in a chemical reaction when it is employed for reactions that produce gases. Gas stoichiometry applies when the gases produced are assumed to be ideal, and the temperature, pressure, and volume of the gases are all known. Often, but not always, the standard temperature and pressure (STP) are taken as 0°C and 1 atmosphere and used as the conditions for gas stoichiometric calculations. For other uses, see Chemical reaction (disambiguation). ... -1... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... Temperature and air pressure can vary from one place to another on the Earth, and can also vary in the same place with time. ...

Gas stoichiometry calculations solve for the unknown volume or mass of a gaseous product or reactant. For example, if we wanted to calculate the volume of gaseous NO₂ produced from the combustion of 100 g of NH₃, by the reaction: For other uses, see Volume (disambiguation). ... For other uses, see Mass (disambiguation). ...

4NH₃ (g) + 7O₂ (g) → 4NO₂ (g) + 6H₂O (l)

we would carry out the following calculations:

There is a 1:1 molar ratio of NH₃ to NO₂ in the above balanced combustion reaction, so 5.871 mol of NO₂ will be formed. We will employ the ideal gas law to solve for the volume at 0 °C (273.15 K) and 1 atmosphere using the gas law constant of R = 0.08206 L · atm · K^-1 · mol^-1 : Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... The gas constant (also known as the molar, universal, or ideal gas constant, usually denoted by symbol R) is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. ...

PV	= nRT
V

Gas stoichiometry often involves having to know the molar mass of a gas, given the density of that gas. The ideal gas law can be re-arranged to obtain a relation between the density and the molar mass of an ideal gas: Molar mass is the mass of one mole of a chemical element or chemical compound. ... For other uses, see Density (disambiguation). ... For other uses, see Density (disambiguation). ... Molar mass is the mass of one mole of a chemical element or chemical compound. ...

    and    

and thus:

This article is about pressure in the physical sciences. ... For other uses, see Volume (disambiguation). ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... For other uses, see Temperature (disambiguation). ...

Stoichiometric air-fuel ratios of common fuels

Methods to solving stoichiometry problems

To use the following methods, you must first determine the molar mass of the reagents and the products, and balance the reaction. Using the known masses of compounds in the reaction, calculate the number of moles there are of each known. Then determine which chemical is the limiting reagent. In chemistry, the limiting reagent, or also called the limiting reactant, is the chemical that determines how far the reaction will go before the chemical in question gets used up, causing the reaction to stop. ...

One method has been commonly taught in various text books. Like equivalent weight, it is the amount of an element that reacts, or is involved in reaction with, 1 mole of electrons. When choosing primary standards in analytical chemistry, compounds with higher "equivalent weights" are, in general, more desirable because weighing errors are reduced or minimized. For example, hydrogen, with atomic weight 1.008 and valence of 1, has an equivalent weight of 1.008. Oxygen assumes a valence of 2 and has an atomic weight of 15.9994, so it has an equivalent weight of 7.9997. Equivalent weight is the atomic weight of an element or radical divided by the valence it assumes in a chemical compound. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... A primary standard in chemistry is a reliable, readily quantified substance. ... This article does not cite any references or sources. ... This article is about the chemistry of hydrogen. ... This article is about the chemical element and its most stable form, or dioxygen. ...

Calculations

A simple equation with moles and the coefficient number of limiting reagents and products, known as the Moum method, will give the number of moles of the unknown quite simply.

This can be re-arranged to give the Lecce method:

See also

• Equivalence ratio

Air-fuel ratio (AFR) is the mass ratio of air to fuel present during combustion. ...

References

1. ^ North American Mfg. Co.: "North American Combustion Handbook", 1952

• Ilya Prigogine & R. Defay, translated by D.H. Everett; Chapter IV (1954). Chemical Thermodynamics. Longmans, Green & Co. Exceptionally clear on the logical foundations as applied to chemistry; includes non-equilibrium thermodynamics. 
• Ilya Prigogine (1967). Thermodynamics of Irreversible Processes, 3rd ed.. Interscience: John Wiley & Sons. A simple, concise monograph.  Library of Congress Catalog No. 67-29540
• E.A. Guggenheim (1967). Thermodynamics: An Advanced Treatment for Chemists and Physicists, 5th ed.. North Holland; John Wiley & Sons (Interscience). A remarkably astute treatise.  Library of Congress Catalog No. 67-20003
• Zumdahl, Steven S. Chemical Principles. Houghton Mifflin, New York, 2005, pp 148-150.

Non-equilibrium thermodynamics is a branch of thermodynamics concerned with studying time-dependent thermodynamic systems, irreversible transformations and open systems. ...

External links

• University of Plymouth :Engine Combustion primer
• Carnegie Mellon's ChemCollective :Free Stoichiometry Tutorials