![Combustion reaction of methane](https://upload.wikimedia.org/wikipedia/commons/7/7c/Combustion_reaction_of_methane.jpg)
Stoichiometry refers to the relationship between the quantities of
reactant
In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
s and
products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
before, during, and following
chemical reaction
A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s.
Stoichiometry is founded on the
law of conservation of mass
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
where the total mass of the reactants equals the total mass of the products, leading to the insight that the relations among quantities of reactants and products typically form a ratio of positive integers. This means that if the amounts of the separate reactants are known, then the amount of the product can be calculated. Conversely, if one reactant has a known quantity and the quantity of the products can be empirically determined, then the amount of the other reactants can also be calculated.
This is illustrated in the image here, where the balanced equation is:
:
Here, one
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
of
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...
reacts with two molecules of
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
gas to yield one molecule of
carbon dioxide
Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
and two molecules of
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
. This particular chemical equation is an example of complete combustion. Stoichiometry measures these quantitative relationships, and is used to determine the amount of products and reactants that are produced or needed in a given reaction. Describing the quantitative relationships among substances as they participate in chemical reactions is known as ''reaction stoichiometry''. In the example above, reaction stoichiometry measures the relationship between the quantities of methane and oxygen that react to form carbon dioxide and water.
Because of the well known relationship of
moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
* The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
*Abraham Moles, French engin ...
to
atomic weights
Relative atomic mass (symbol: ''A''; sometimes abbreviated RAM or r.a.m.), also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a give ...
, the ratios that are arrived at by stoichiometry can be used to determine quantities by weight in a reaction described by a balanced equation. This is called ''composition stoichiometry''.
''Gas stoichiometry'' deals with reactions involving gases, where the gases are at a known temperature, pressure, and volume and can be assumed to be
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
es. For gases, the volume ratio is ideally the same by the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, but the mass ratio of a single reaction has to be calculated from the
molecular mass
The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
es of the reactants and products. In practice, due to the existence of
isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers) ...
s,
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
es are used instead when calculating the mass ratio.
Etymology
The term ''stoichiometry'' was first used by
Jeremias Benjamin Richter
Jeremias Benjamin Richter (; 10 March 1762 – 4 May 1807) was a German chemist. He was born at Hirschberg in Silesia, became a mining official at Breslau in 1794, and in 1800 was appointed assessor to the department of mines and chemist to the ...
in 1792 when the first volume of Richter's ''Stoichiometry or the Art of Measuring the Chemical Elements'' was published. The term is derived from the
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
words ''stoicheion'' "element" and ''metron'' "measure". In
patristic
Patristics or patrology is the study of the early Christian writers who are designated Church Fathers. The names derive from the combined forms of Latin ''pater'' and Greek ''patḗr'' (father). The period is generally considered to run from ...
Greek, the word ''Stoichiometria'' was used by
Nicephorus to refer to the number of line counts of the
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
New Testament
The New Testament grc, Ἡ Καινὴ Διαθήκη, transl. ; la, Novum Testamentum. (NT) is the second division of the Christian biblical canon. It discusses the teachings and person of Jesus, as well as events in first-century Christ ...
and some of the
Apocrypha
Apocrypha are works, usually written, of unknown authorship or of doubtful origin. The word ''apocryphal'' (ἀπόκρυφος) was first applied to writings which were kept secret because they were the vehicles of esoteric knowledge considered ...
.
Definition
A stoichiometric amount or stoichiometric ratio of a
reagent
In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
is the optimum amount or ratio where, assuming that the reaction proceeds to completion:
# All of the reagent is consumed
# There is no deficiency of the reagent
# There is no excess of the reagent.
Stoichiometry rests upon the very basic laws that help to understand it better, i.e.,
law of conservation of mass
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
, the
law of definite proportions
In chemistry, the law of definite proportions, sometimes called Proust's law, or law of constant composition states that a given
chemical compound always contains its component elements in fixed ratio (by mass) and does not depend on its source an ...
(i.e., the
law of constant composition
In chemistry, the law of definite proportions, sometimes called Proust's law, or law of constant composition states that a given
chemical compound always contains its component elements in fixed ratio (by mass) and does not depend on its source an ...
), the
law of multiple proportions
In chemistry, the law of multiple proportions states that if two elements form more than one compound, then the ratios of the masses of the second element which combine with a fixed mass of the first element will always be ratios of small whole ...
and the
law of reciprocal proportions The law of reciprocal proportions also called law of equivalent proportions or law of permanent ratios is one of the basic laws of stoichiometry.
It relates the proportions in which elements combine across a number of different elements. It was fir ...
. 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 number of atoms of a given element X on the reactant side must equal the number of atoms of that element on the product side, whether or not all of those atoms are actually involved in a reaction.
Chemical reactions, as macroscopic unit operations, consist of simply a very large number of
elementary reaction
An elementary reaction is a chemical reaction in which one or more chemical species react directly to form products in a single reaction step and with a single transition state. In practice, a reaction is assumed to be elementary if no reaction in ...
s, where a single molecule reacts with another molecule. As the reacting molecules (or moieties) consist of a definite set of atoms in an integer ratio, the ratio between reactants in a complete reaction is also in integer ratio. A reaction may consume more than one molecule, and the stoichiometric number counts this number, defined as positive for products (added) and negative for reactants (removed).
The unsigned coefficients are generally referred to as the stoichiometric coefficients.
Each element has an
atomic mass
The atomic mass (''m''a or ''m'') is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) – equivalently, unified atomic mass unit (u). 1&nbs ...
, and considering molecules as collections of atoms, compounds have a definite
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
. By definition, the molar mass of carbon-12 is 12 g/mol. The number of molecules per mole in a substance is given by the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
. Thus, to calculate the stoichiometry by mass, the number of molecules required for each reactant is expressed in moles and multiplied by the molar mass of each to give the mass of each reactant per mole of reaction. The mass ratios can be calculated by dividing each by the total in the whole reaction.
Elements in their natural state are mixtures of
isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers) ...
s of differing mass; thus,
atomic mass
The atomic mass (''m''a or ''m'') is the mass of an atom. Although the SI unit of mass is the kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (symbol: Da) – equivalently, unified atomic mass unit (u). 1&nbs ...
es and thus molar masses are not exactly integers. For instance, instead of an exact 14:3 proportion, 17.04 kg of ammonia consists of 14.01 kg of nitrogen and 3 × 1.01 kg of hydrogen, because natural nitrogen includes a small amount of nitrogen-15, and natural hydrogen includes hydrogen-2 (
deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
).
A stoichiometric reactant is a reactant that is consumed in a reaction, as opposed to a
catalytic reactant, which is not consumed in the overall reaction because it reacts in one step and is regenerated in another step.
Converting grams to moles
Stoichiometry is not only used to balance chemical equations but also used in conversions, i.e., converting from grams to moles using
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
as the conversion factor, or from grams to milliliters using
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
. For example, to find the
amount
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
of NaCl (sodium chloride) in 2.00 g, 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 in moles (the unit that was needed), as shown in the following equation,
:
Molar proportion
Stoichiometry is often used to balance chemical equations (reaction stoichiometry). For example, the two
diatomic
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
gases,
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
and
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
, can combine to form a liquid, water, in an
exothermic reaction
In thermochemistry, an exothermic reaction is a "reaction for which the overall standard enthalpy change Δ''H''⚬ is negative." Exothermic reactions usually release heat. The term is often confused with exergonic reaction, which IUPAC defines ...
, as described by the following equation:
:2 + → 2
Reaction stoichiometry describes the 2:1:2 ratio of hydrogen, oxygen, and water molecules in the above equation.
The molar ratio allows for conversion between moles of one substance and moles of another. For example, in the reaction
:2 + 3 → 2 + 4
the amount of water that will be produced by the combustion of 0.27 moles of is obtained using the molar ratio between and of 2 to 4.
:
The term stoichiometry is also often used for the
molar proportions of elements in stoichiometric compounds (composition stoichiometry). For example, the stoichiometry of hydrogen and oxygen in H
2O is 2:1. In stoichiometric compounds, the molar proportions are whole numbers.
Determining amount of product
Stoichiometry can also be used to find the quantity of a product yielded by a reaction. If a piece of solid
copper
Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...
(Cu) were added to an aqueous solution of
silver nitrate
Silver nitrate is an inorganic compound with chemical formula . It is a versatile precursor to many other silver compounds, such as those used in photography. It is far less sensitive to light than the halides. It was once called ''lunar caustic' ...
(AgNO
3), the
silver
Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...
(Ag) would be replaced in a
single displacement reaction
A single-displacement reaction, also known as single replacement reaction or exchange reaction, is a chemical reaction in which one element is replaced by another in a compound.
It can be represented generically as:
:A + BC -> AC + B
where eithe ...
forming aqueous
copper(II) nitrate
Copper(II) nitrate describes any member of the family of inorganic compounds with the formula Cu( NO3)2(H2O)x. The hydrates are blue solids. Anhydrous copper nitrate forms blue-green crystals and sublimes in a vacuum at 150-200 °C. Common hy ...
(Cu(NO
3)
2) and solid silver. How much silver is produced if 16.00 grams of Cu is added to the solution of excess silver nitrate?
The following steps would be used:
# Write and balance the equation
# Mass to moles: Convert grams of Cu to moles of Cu
# Mole ratio: Convert moles of Cu to moles of Ag produced
# Mole to mass: Convert moles of Ag to grams of Ag produced
The complete balanced equation would be:
: + 2 → + 2
For the mass to mole step, the mass of copper (16.00 g) would be converted to moles of copper by dividing the mass of copper by its
molecular mass
The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
: 63.55 g/mol.
:
Now that the amount of Cu in moles (0.2518) is found, we can set up the mole ratio. This is found by looking at the coefficients in the balanced equation: Cu and Ag are in a 1:2 ratio.
:
Now that the moles of Ag produced is known to be 0.5036 mol, we convert this amount to grams of Ag produced to come to the final answer:
:
This set of calculations can be further condensed into a single step:
:
Further examples
For
propane
Propane () is a three-carbon alkane with the molecular formula . It is a gas at standard temperature and pressure, but compressible to a transportable liquid. A by-product of natural gas processing and petroleum refining, it is commonly used a ...
(C
3H
8) reacting with
oxygen gas
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
(O
2), the balanced chemical equation is:
:
The mass of water formed if 120 g of propane (C
3H
8) is burned in excess oxygen is then
:
Stoichiometric ratio
Stoichiometry is also used to find the right amount of one
reactant
In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
to "completely" react with the other reactant in a
chemical reaction
A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
– that is, the stoichiometric amounts that would result in no leftover reactants when the reaction takes place. An example is shown below using the
thermite reaction
Thermite () is a pyrotechnic composition of metal powder and metal oxide. When ignited by heat or chemical reaction, thermite undergoes an exothermic reduction-oxidation (redox) reaction. Most varieties are not explosive, but can create brief ...
,
:
This equation shows that 1 mole of and 2 moles of
aluminum
Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. It has ...
will produce 1 mole of
aluminium oxide and 2 moles of
iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in f ...
. So, to completely react with 85.0 g of (0.532 mol), 28.7 g (1.06 mol) of aluminium are needed.
:
Limiting reagent and percent yield
The limiting reagent is the reagent that limits the amount of product that can be formed and is completely consumed when the reaction is complete. An excess reactant is a reactant that is left over once the reaction has stopped due to the limiting reactant being exhausted.
Consider the equation of roasting
lead(II) sulfide
Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cut, le ...
(PbS) in oxygen (O
2) to produce
lead(II) oxide
Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula Pb O. PbO occurs in two polymorphs: litharge having a tetragonal crystal structure, and massicot having an orthorhombic crystal structure. Modern ap ...
(PbO) and
sulfur dioxide
Sulfur dioxide (IUPAC-recommended spelling) or sulphur dioxide (traditional Commonwealth English) is the chemical compound with the formula . It is a toxic gas responsible for the odor of burnt matches. It is released naturally by volcanic activ ...
(SO
2):
:2 + 3 → 2 + 2
To determine the theoretical yield of lead(II) oxide if 200.0 g of lead(II) sulfide and 200.0 g of oxygen are heated in an open container:
:
:
Because a lesser amount of PbO is produced for the 200.0 g of PbS, it is clear that PbS is the limiting reagent.
In reality, the actual yield is not the same as the stoichiometrically-calculated theoretical yield. Percent yield, then, is expressed in the following equation:
:
If 170.0 g of lead(II) oxide is obtained, then the percent yield would be calculated as follows:
:
Example
Consider the following reaction, in which
iron(III) chloride
Iron(III) chloride is the inorganic compound with the formula . Also called ferric chloride, it is a common compound of iron in the +3 oxidation state. The anhydrous compound is a crystalline solid with a melting point of 307.6 °C. The col ...
reacts with
hydrogen sulfide
Hydrogen sulfide is a chemical compound with the formula . It is a colorless chalcogen-hydride gas, and is poisonous, corrosive, and flammable, with trace amounts in ambient atmosphere having a characteristic foul odor of rotten eggs. The unde ...
to produce
iron(III) sulfide
Iron(III) sulfide, also known as ferric sulfide or sesquisulfide (), is one of the several binary iron sulfides. It is a solid, black powder that degrades at ambient temperature.
Reactions
() decays at a temperature over 20 °C into iron ...
and
hydrogen chloride
The compound hydrogen chloride has the chemical formula and as such is a hydrogen halide. At room temperature, it is a colourless gas, which forms white fumes of hydrochloric acid upon contact with atmospheric water vapor. Hydrogen chloride ga ...
:
:
The stoichiometric masses for this reaction are:
:324.41 g FeCl
3, 102.25 g H
2S, 207.89 g Fe
2S
3, 218.77 g HCl
Suppose 90.0 g of FeCl
3 reacts with 52.0 g of H
2S. To find the limiting reagent and the mass of HCl produced by the reaction, we change the above amounts by a factor of 90/324.41 and obtain the following amounts:
:90.00 g FeCl
3, 28.37 g H
2S, 57.67 g Fe
2S
3, 60.69 g HCl
The limiting reactant (or reagent) is FeCl
3, since all 90.00 g of it is used up while only 28.37 g H
2S are consumed. Thus, 52.0 − 28.4 = 23.6 g H
2S left in excess. The mass of HCl produced is 60.7 g.
Note: By looking at the stoichiometry of the reaction, one might have guessed FeCl
3 being the limiting reactant; three times more FeCl
3 is used compared to H
2S (324 g vs 102 g).
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
In the chemical sciences, methylation denotes the addition of a methyl group on a substrate, or the substitution of an atom (or group) by a methyl group. Methylation is a form of alkylation, with a methyl group replacing a hydrogen atom. These t ...
of
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, ...
(C
6H
6), through a
Friedel–Crafts reaction
The Friedel–Crafts reactions are a set of reactions developed by Charles Friedel and James Crafts in 1877 to attach substituents to an aromatic ring. Friedel–Crafts reactions are of two main types: alkylation reactions and acylation reactions ...
using
AlCl3 as a catalyst, may produce singly methylated (C
6H
5CH
3), doubly methylated (C
6H
4(CH
3)
2), or still more highly methylated (C
6H
6−''n''(CH
3)
''n'') products, as shown in the following example,
:C
6H
6 + CH
3Cl → C
6H
5CH
3 + HCl
:C
6H
6 + 2 CH
3Cl → C
6H
4(CH
3)
2 + 2 HCl
:C
6H
6 + ''n'' CH
3Cl → C
6H
6−''n''(CH
3)
''n'' + ''n'' HCl
In this example, which reaction takes place is controlled in part by the relative
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
s of the reactants.
Stoichiometric coefficient and stoichiometric number
In lay terms, the ''stoichiometric coefficient'' of any given component is the number of molecules and/or
formula units
In chemistry, a formula unit is the empirical formula of any ionic or covalent network solid compound used as an independent entity for stoichiometric calculations. It is the lowest whole number ratio of ions represented in an ionic compound. Ex ...
that participate in the reaction as written. A related concept is the ''stoichiometric number'' (using IUPAC nomenclature), wherein the stoichiometric coefficient is multiplied by +1 for all products and by −1 for all reactants.
For example, in the reaction , the stoichiometric number of CH
4 is −1, the stoichiometric number of O
2 is −2, for it would be +1 and for H
2O it is +2.
In more technically precise terms, the stoichiometric number in a
chemical reaction
A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
of the ''i''th component is defined as
:
or
:
where
is the number of
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s of ''i'', and
is the progress variable or
extent of reaction In physical chemistry and chemical engineering, extent of reaction is a quantity that measures the extent to which the reaction has proceeded. Often, it refers specifically to the value of the extent of reaction when equilibrium has been reached. It ...
.
The stoichiometric number
represents the degree to which a chemical species participates in a reaction. The convention is to assign negative numbers to ''reactants'' (which are consumed) and positive ones to ''products'', consistent with the convention that increasing the extent of reaction will correspond to shifting the composition from reactants towards products. However, any reaction may be viewed as going in the reverse direction, and in that point of view, would change in the negative direction in order to lower the system's Gibbs 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
Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to:
Science and medicine
* Kinetics (physics), the study of motion and its causes
** Rigid body kinetics, the study of the motion of rigid bodies
* Chemical ki ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, i.e., whether
equilibrium lies to the ''right'' or the ''left'' of the initial state,
In
reaction mechanism
In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs.
A chemical mechanism is a theoretical conjecture that tries to describe in detail what takes place at each stage of ...
s, stoichiometric coefficients for each step are always
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, since elementary reactions always involve whole molecules. If one uses a composite representation of an overall reaction, some may be
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. 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
Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
, also has a stoichiometric coefficient of zero.
The simplest possible case is an
isomerization
In chemistry, isomerization or isomerisation is the process in which a molecule, polyatomic ion or molecular fragment is transformed into an isomer with a different chemical structure. Enolization is an example of isomerization, as is tautomeriz ...
:A → B
in which since one molecule of B is produced each time the reaction occurs, while since one molecule of A is necessarily consumed. In any chemical reaction, not only is the total
mass conserved but also the numbers of
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s of each
kind are conserved, and this imposes corresponding constraints on possible values for the stoichiometric coefficients.
There are usually multiple reactions proceeding simultaneously in any
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
reaction system, including those in
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
. Since any chemical component can participate in several reactions simultaneously, the stoichiometric number of the ''i''th component in the ''k''th reaction is defined as
:
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 (
state variables
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
), and the representation in terms of the actual compositional
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, as expressed by the extents of reaction . The transformation from a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
expressing the extents to a vector expressing the amounts uses a rectangular
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
whose elements are the stoichiometric numbers .
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
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
restriction on the reaction
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
in composition space, or ''N''‑space, whose
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
ality 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 sharing the same algebraic stoichiometry.
In accord with the principles of
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
and
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
, every chemical reaction is ''reversible'', at least to some degree, so that each equilibrium point must be an
interior point
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the ...
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.
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
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s or
moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
* The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
*Abraham Moles, French engin ...
. 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 the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
. While dimensional
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
units may be used, the comments about integers are then no longer applicable.
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 N.
If a reaction network has ''n'' reactions and ''m'' participating molecular species then the stoichiometry matrix will have correspondingly ''m'' rows and ''n'' columns.
For example, consider the system of reactions shown below:
:S
1 → S
2
:5 S
3 + S
2 → 4 S
3 + 2 S
2
:S
3 → S
4
:S
4 → S
5
This system comprises four reactions and five different molecular species. The stoichiometry matrix for this system can be written as:
:
where the rows correspond to S
1, S
2, S
3, S
4 and S
5, 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, and the species vector, x to form a compact equation describing the rates of change of the molecular species:
:
Gas stoichiometry
''Gas stoichiometry'' is the quantitative relationship (ratio) between reactants and products in a
chemical reaction
A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
with reactions that produce
gases
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
. Gas stoichiometry applies when the gases produced are assumed to be
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
, and the temperature, pressure, and volume of the gases are all known. The ideal gas law is used for these calculations. Often, but not always, the
standard temperature and pressure
Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union o ...
(STP) are taken as 0 °C and 1 bar and used as the conditions for gas stoichiometric calculations.
Gas stoichiometry calculations solve for the unknown
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
or
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of a gaseous product or reactant. For example, if we wanted to calculate the volume of gaseous NO
2 produced from the combustion of 100 g of NH
3, by the reaction:
:
we would carry out the following calculations:
:
There is a 1:1 molar ratio of NH
3 to NO
2 in the above balanced combustion reaction, so 5.871 mol of NO
2 will be formed. We will employ the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
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:
:
Gas stoichiometry often involves having to know the
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
of a gas, given the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of that gas. The ideal gas law can be re-arranged to obtain a relation between the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and the
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
of an ideal gas:
:
and
and thus:
:
where:
*''P'' = absolute gas
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
*''V'' = gas
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
*''n'' = amount (measured in
moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
* The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
*Abraham Moles, French engin ...
)
*''R'' = universal ideal gas law constant
*''T'' = absolute gas
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
*''ρ'' = gas density at ''T'' and ''P''
*''m'' = mass of gas
*''M'' = molar mass of gas
Stoichiometric air-to-fuel ratios of common fuels
In the
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combusti ...
reaction, oxygen reacts with the fuel, and the point where exactly all oxygen is consumed and all fuel burned is defined as the stoichiometric point. With more oxygen (overstoichiometric combustion), some of it stays unreacted. Likewise, if the combustion is incomplete due to lack of sufficient oxygen, fuel remains unreacted. (Unreacted fuel may also remain because of slow combustion or insufficient mixing of fuel and oxygen – this is not due to stoichiometry). Different hydrocarbon fuels have different contents of carbon, hydrogen and other elements, thus their stoichiometry varies.
Note that oxygen makes up only 20.95% of the volume of air, and only 23.20% of its mass. The air-fuel ratios listed below are much higher than the equivalent oxygen-fuel ratios, due to the high proportion of inert gasses in the air.
Gasoline engines can run at stoichiometric air-to-fuel ratio, because gasoline is quite volatile and is mixed (sprayed or carburetted) with the air prior to ignition. Diesel engines, in contrast, run lean, with more air available than simple stoichiometry would require. Diesel fuel is less volatile and is effectively burned as it is injected.
See also
*
Non-stoichiometric compound
In chemistry, non-stoichiometric compounds are chemical compounds, almost always solid inorganic compounds, having elemental composition whose proportions cannot be represented by a ratio of small natural numbers (i.e. an empirical formula); mos ...
References
*Zumdahl, Steven S. ''Chemical Principles''. Houghton Mifflin, New York, 2005, pp 148–150.
* Internal Combustion Engine Fundamentals, John B. Heywood
External links
Engine Combustion primerfrom the University of Plymouth
Free Stoichiometry Tutorialsfrom Carnegie Mellon's ChemCollective
Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coëfficients and stoichiometry.
Reaction Stoichiometry Calculatora comprehensive free online reaction stoichiometry calculator.
Stoichiometry Plusa stoichiometry calculator and more for Android.
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Chemical reaction engineering