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Stoichiometry () is the relationships between the masses of and products before, during, and following chemical reactions.
Stoichiometry is based on the law of conservation of mass; the total mass of reactants must equal the total mass of products, so the relationship between reactants and products must 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 unbalanced equation is:
Here, one molecule of methane reacts with two molecules of oxygen gas to yield one molecule of carbon dioxide and two molecules of liquid water. This particular chemical equation is an example of complete combustion. The numbers in front of each quantity are a set of stoichiometric coefficients which directly reflect the molar ratios between the products and reactants. 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: for every mole of methane combusted, two moles of oxygen are consumed, one mole of carbon dioxide is produced, and two moles of water are produced.
Because of the well known relationship of moles to atomic weights, 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 solely involving gases, where the gases are at a known temperature, pressure, and volume and can be assumed to be . For gases, the volume ratio is ideally the same by the ideal gas law, but the mass ratio of a single reaction has to be calculated from the of the reactants and products. In practice, because of the existence of , are used instead in calculating the mass ratio.
Stoichiometry rests upon the very laws that help to understand it better, i.e., law of conservation of mass, the law of definite proportions (i.e., the law of constant composition), the law of multiple proportions and the law of reciprocal 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 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 involved in a reaction.
Chemical reactions, as macroscopic unit operations, consist of many elementary reactions, where a single molecule reacts with another molecule. As the reacting Molecule (or Formula unit or Ion association) consist of a definite set of Atom in an integer ratio, the ratio between reactants in a complete reaction is also in an 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 Reagent (removed). The unsigned coefficients are generally referred to as the stoichiometric coefficients.
Each Chemical element has an atomic mass (usually given as an average in the form of the standard atomic weight), and considering molecules as collections of atoms, every compound has a molecular mass (if molecular) or formula mass (if non-molecular), which when expressed in daltons is numerically equal to the molar mass in Gram/mol. By definition, the atomic mass of carbon-12 is exactly 12 Da, making its molar mass 12 g/mol. The number of Molecular entity per mole in a substance is given by the Avogadro constant, exactly since the 2019 revision of the SI. 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 of differing mass; thus, atomic masses and thus molar masses are not exactly integers. For instance, instead of an exact 14:3 proportion, 17.031 g of ammonia consists of 14.007 g of nitrogen and 3 × 1.008 g of hydrogen, because natural nitrogen includes a small amount of nitrogen-15, and natural hydrogen includes hydrogen-2 (deuterium).
A stoichiometric reactant is a reactant that is consumed in a reaction, as opposed to a catalysis, which is not consumed in the overall reaction because it reacts in one step and is regenerated in another step.
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,
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
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 is 2:1. In stoichiometric compounds, the molar proportions are whole numbers.
The following steps would be used:
The complete balanced equation would be:
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 molar mass: 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:
The mass of water formed if 120 g of propane () is burned in excess oxygen is then
This equation shows that 1 mole of and 2 moles of aluminium will produce 1 mole of aluminium oxide and 2 moles of iron. So, to completely react with 85.0 g of (0.532 mol), 28.7 g (1.06 mol) of aluminium are needed.
Consider the equation of roasting lead(II) sulfide (PbS) in oxygen () to produce lead(II) oxide (PbO) and sulfur dioxide ():
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:
The stoichiometric masses for this reaction are:
Suppose 90.0 g of reacts with 52.0 g of . 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:
The limiting reactant (or reagent) is , since all 90.00 g of it is used up while only 28.37 g are consumed. Thus, 52.0 − 28.4 = 23.6 g left in excess. The mass of HCl produced is 60.7 g.
By looking at the stoichiometry of the reaction, one might have guessed being the limiting reactant; three times more is used compared to (324 g vs 102 g).
In this example, which reaction takes place is controlled in part by the relative of the reactants.
For example, in the reaction , the stoichiometric number of is −1, the stoichiometric number of is −2, for it would be +1 and for it is +2.
In more technically precise terms, the stoichiometric number in a chemical reaction system of the i-th component is defined as
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 and thermodynamics, i.e., whether equilibrium lies to the right or the left of the initial state,
In reaction mechanisms, stoichiometric coefficients for each step are always , since elementary reactions always involve whole molecules. If one uses a composite representation of an overall reaction, some may be rational number 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.
The simplest possible case is an isomerization
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 of each periodic table are conserved, and this imposes corresponding constraints on possible values for the stoichiometric coefficients.
There are usually multiple reactions proceeding simultaneously in any nature reaction system, including those in biology. 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
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), and the representation in terms of the actual compositional degrees of freedom, as expressed by the extents of reaction . The transformation from a vector space expressing the extents to a vector expressing the amounts uses a rectangular matrix whose elements are the stoichiometric numbers .
The extreme value 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 kinematics restriction on the reaction simplex, a hyperplane in composition space, or N‑space, whose 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 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 ξs 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 or moles. Moles are most commonly used, but it is more suggestive to picture incremental chemical reactions in terms of molecules. The Ns and ξs are reduced to molar units by dividing by the Avogadro constant. While dimensional mass units may be used, the comments about integers are then no longer applicable.
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:
This system comprises four reactions and five different molecular species. The stoichiometry matrix for this system can be written as:
-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 respectively. 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, the biochemical systems equation, describing the rates of change of the molecular species:
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 produced from the combustion of 100 g of , by the reaction:
There is a 1:1 molar ratio of to in the above balanced combustion reaction, so 5.871 mol of 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 constant of R = 0.08206 L·atm·K−1·mol−1:
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:
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 | 14.7 : 1 | 6.9% | ||
| Natural gas | 14.5 : 1 | 9.7 : 1 | 6.9% | |
| Propane (Liquid propane) | 15.67 : 1 | 23.9 : 1 | 6.45% | |
| Ethanol | 9 : 1 | 11.1% | ||
| Methanol | 6.47 : 1 | 15.6% | ||
| N-Butanol | 11.2 : 1 | 8.2% | ||
| Hydrogen | 34.3 : 1 | 2.39 : 1 | 2.9% | |
| Diesel fuel | 14.5 : 1 | 6.8% | ||
| Methane | 17.23 : 1 | 9.52 : 1 | 5.5% | |
| Acetylene | 13.26 : 1 | 11.92 : 1 | 7.0% | |
| Ethane | 16.07 : 1 | 16.68 : 1 | 5.9% | |
| Butane | 15.44 : 1 | 30.98 : 1 | 6.1% | |
| Pentane | 15.31 : 1 | 38.13 : 1 | 6.1% |
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.
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