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# Stoichiometry

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).

## Etymology

"Stoichiometry" derives from the Greek words στοιχειον stoikheion (Grammatical translation) ("element") and metriā ("measure," from metron). 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.

## 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.

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:

$2H_2 + O_2 \rightarrow 2H_2O\,$

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).

Compounds for which the molar proportions are not whole numbers are called non-stoichiometric compounds.

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

$\frac{2.00 \mbox{ g NaCl}}{58.44 \mbox{ g NaCl mol}^{-1}} = 0.034 \ mol$

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), as shown in the following equation,

$\left(\frac{2.00 \mbox{ g NaCl}}{1}\right)\left(\frac{1 \mbox{ mol NaCl}}{58.44 \mbox{ g NaCl}}\right) = 0.034\ mol$

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,

$Fe_2O_3 + 2Al \rightarrow Al_2O_3 + 2Fe$

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

$\left(\frac{85.0 \mbox{ g }Fe_2O_3}{1}\right)\left(\frac{1 \mbox{ mol }Fe_2 O_3}{159.7 \mbox{ g }Fe_2 O_3}\right)\left(\frac{2 \mbox{ mol }Al}{1 \mbox{ mol }Fe_2 O_3}\right)\left(\frac{27.0 \mbox{ g }Al}{1 \mbox{ mol }Al}\right) = 28.7 \mbox{ g }Al$

## 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 (C6H6 − n(CH3)n) products, as shown in the following example,

$C_6H_6 + \quad CH_3Cl \rightarrow C_6H_5CH_3 + HCl\,$
$C_6H_6 + 2\mbox{ }CH_3Cl \rightarrow C_6H_4(CH_3)_2 + 2HCl\,$
$C_6H_6 + n\mbox{ }CH_3Cl \rightarrow C_6H_{6-n}(CH_3)_n + nHCl\,$

In this example, which reaction takes place is controlled in part by the relative concentrations of the reactants.

## Stoichiometric coefficient

The stoichiometric coefficient in a chemical reaction system of the i–th component is defined as

$\nu_i = \frac{dN_i}{d\xi} \,$

or

$dN_i = \nu_i d\xi \,$

where Ni 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.

The stoichiometric coefficient νi 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".

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 which do not participate in a reaction; their stoichiometric coefficients are therefore zero. Any chemical species which is regenerated, such as a catalyst, also has a stoichiometric coefficient of zero.

The simplest possible case is an isomerism

$A \iff B$

in which νB = 1 since one molecule of B is produced each time the reaction occurs, while νA = −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, 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.

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

$\nu_{ik} = \frac{\partial N_i}{\partial \xi_k} \,$

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

$dN_i = \sum_k \nu_{ik} d\xi_k \,$ .

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 { Ni } (state variables), and the representation in terms of the actual compositional degrees of freedom, as expressed by the extents of reaction { ξk }. The transformation from a vector expressing the extents to a vector expressing the amounts uses a rectangular matrix whose elements are the stoichiometric coefficients [ νi k ].

The maximum and minimum for any ξk 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.

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. Consequently, 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 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.

## Stoichiometry Matrix

In complex reactions, stoichiometries are often represented in a more compact form called the stoichiometry matrix. Traditionally the stoichiometry matrix is denoted by the symbol, $\mathbf{N}$.

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:

$\mathbf{N} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & -1 & -1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$

where the rows correspond to S1, S2, S3, S3 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 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:

$\frac{d\mathbf{S}}{dt} = \mathbf{N} \cdot \mathbf{v}$

## 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.

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:

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

we would carry out the following calculations:

$100 \ \mbox{g}\,NH_3 \cdot \frac{1 \ \mbox{mol}\,NH_3}{17.034 \ \mbox{g}\,NH_3} = 5.871 \ \mbox{mol}\,NH_3\$

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 :

PV = nRT $= \frac{nRT}{P} = \frac{5.871 \cdot 0.08206 \cdot 273.15}{1} = 131.597 \ \mbox{L}\,NO_2$

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:

$\rho = \frac{m}{V}$     and     $n = \frac{m}{M}$

and thus:

$\rho = \frac {M P}{R\,T}$
P where: = absolute gas pressure = gas volume = number of moles = universal ideal gas law constant = absolute gas temperature = gas density at T and P = mass of gas = molar mass of gas

## Stoichiometric air-fuel ratios of common fuels

Fuel By weight By volume [1] Percent fuel by weight
Gasoline 14.7 : 1 - 6.8%
Natural Gas 17.2 : 1 9.7  : 1 5.8%
Propane (LP) 15.5 : 1 23.9 : 1 6.45%
Ethanol 9 : 1 - 11.1%
Methanol 6.4 : 1 - 15.6%
Hydrogen 34 : 1 2.39 : 1 2.9%
Diesel 14.6 : 1 - 6.8%

## 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.

### The Milberg method

The Milberg method is useful in gravimetric stoichiometry problems. 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 generally 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.

### 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.

$\frac{ \mbox{ Moles of Limiting Reagent}}{ \mbox{ Coefficient of Limiting Reagent}} = \frac{ \mbox{ Moles of Product}}{ \mbox{ Coefficient of Product}}$

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

$\mbox{Moles of Limiting Reagent} \times \frac{ \mbox{ Coefficient of Product}}{ \mbox{ Coefficient of Limiting Reagent}} = \mbox{Moles of Product}$

• Chemical reaction
• Equivalence ratio

## 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.