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A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow that points towards the products to show the direction of the reaction. The chemical formulas may be symbolic, structural (pictorial diagrams), or intermixed. The coefficients next to the symbols and formulas of entities are the absolute values of the stoichiometric numbers. The first chemical equation was diagrammed by Jean Beguin in 1615.
Structure
A chemical equation (see an example below) consists of a list of reactants (the starting substances) on the left-hand side, an arrow symbol, and a list of products (substances formed in the chemical reaction) on the right-hand side. Each substance is specified by its chemical formula, optionally preceded by a number called stoichiometric coefficient. The coefficient specifies how many entities (e.g. molecules) of that substance are involved in the reaction on a molecular basis. If not written explicitly, the coefficient is equal to 1. Multiple substances on any side of the equation are separated from each other by a plus sign.
As an example, the equation for the reaction of hydrochloric acid with sodium can be denoted:
- 2HCl + 2Na → 2NaCl + H2
Given the formulas are fairly simple, this equation could be read as "two H-C-L plus two N-A yields two N-A-C-L and H two." Alternately, and in general for equations involving complex chemicals, the chemical formulas are read using IUPAC nomenclature, which could verbalise this equation as "two hydrochloric acid molecules and two sodium atoms react to form two formula units of sodium chloride and a hydrogen gas molecule."
Reaction types
Different variants of the arrow symbol are used to denote the type of a reaction:
net forward reaction reaction in both directions equilibrium stoichiometric relation resonance (not a reaction)
State of matter
To indicate physical state of a chemical, a symbol in parentheses may be appended to its formula: (s) for a solid, (l) for a liquid, (g) for a gas, and (aq) for an aqueous solution. This is especially done when one wishes to emphasize the states or changes thereof. For example, the reaction of aqueous hydrochloric acid with solid (metallic) sodium to form aqueous sodium chloride and hydrogen gas would be written like this:
- 2HCl(aq) + 2Na(s) → 2NaCl(aq) + H2(g)
That reaction would have different thermodynamic and kinetic properties if gaseous hydrogen chloride were to replace the hydrochloric acid as a reactant:
- 2HCl(g) + 2Na(s) → 2NaCl(s) + H2(g)
Alternately, an arrow without parentheses is used in some cases to indicate formation of a gas ↑ or precipitate ↓. This is especially useful if only one such species is formed. Here is an example indicating that hydrogen gas is formed:
- 2HCl + 2Na → 2 NaCl + H2 ↑
Catalysis and other conditions
If the reaction requires energy, it is indicated above the arrow. A capital Greek letter delta (Δ) or a triangle (△) is put on the reaction arrow to show that energy in the form of heat is added to the reaction. The expression hν is used as a symbol for the addition of energy in the form of light. Other symbols are used for other specific types of energy or radiation.
Similarly, if a reaction requires a certain medium with certain specific characteristics, then the name of the acid or base that is used as a medium may be placed on top of the arrow. If no specific acid or base is required, another way of denoting the use of an acidic or basic medium is to write H+ or OH− (or even "acid" or "base") on top of the arrow. Specific conditions of the temperature and pressure, as well as the presence of catalysts, may be indicated in the same way.
Notation variants
The standard notation for chemical equations only permits all reactants on one side, all products on the other, and all stoichiometric coefficients positive. For example, the usual form of the equation for dehydration of methanol to dimethylether is:
- 2 CH3OH → CH3OCH3 + H2O
Sometimes an extension is used, where some substances with their stoichiometric coefficients are moved above or below the arrow, preceded by a plus sign or nothing for a reactant, and by a minus sign for a product. Then the same equation can look like this:
Such notation serves to hide less important substances from the sides of the equation, to make the type of reaction at hand more obvious, and to facilitate chaining of chemical equations. This is very useful in illustrating multi-step reaction mechanisms. Note that the substances above or below the arrows are not catalysts in this case, because they are consumed or produced in the reaction like ordinary reactants or products.
Another extension used in reaction mechanisms moves some substances to branches of the arrow. Both extensions are used in the example illustration of a mechanism.
Use of negative stoichiometric coefficients at either side of the equation (like in the example below) is not widely adopted and is often discouraged.
Balancing chemical equations
Because no nuclear reactions take place in a chemical reaction, the chemical elements pass through the reaction unchanged. Thus, each side of the chemical equation must represent the same number of atoms of any particular element (or nuclide, if different isotopes are taken into account). The same holds for the total electric charge, as stated by the charge conservation law. An equation adhering to these requirements is said to be balanced.
A chemical equation is balanced by assigning suitable values to the stoichiometric coefficients. Simple equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a system of linear equations.
Balanced equations are usually written with smallest natural-number coefficients. Yet sometimes it may be advantageous to accept a fractional coefficient, if it simplifies the other coefficients. The introductory example can thus be rewritten as
In some circumstances the fractional coefficients are even inevitable. For example, the reaction corresponding to the standard enthalpy of formation must be written such that one molecule of a single product is formed. This will often require that some reactant coefficients be fractional, as is the case with the formation of lithium fluoride:
Inspection method
4 + 2 O
2 → CO
2 + 2 H
2O, a coefficient of 2 must be placed before the oxygen gas on the reactants side and before the water on the products side in order for, as per the law of conservation of mass, the quantity of each element does not change during the reaction
This chemical equation is being balanced by first multiplying H3PO4 by four to match the number of P atoms, and then multiplying H2O by six to match the numbers of H and O atoms.
The method of inspection can be outlined as setting the most complex substance's stoichiometric coefficient to 1 and assigning values to other coefficients step by step such that both sides of the equation end up with the same number of atoms for each element. If any fractional coefficients arise during this process, the presence of fractions may be eliminated (at any time) by multiplying all coefficients by their lowest common denominator.
- Example
Balancing of the chemical equation for the complete combustion of methane
is achieved as follows:
- A coefficient of 1 is placed in front of the most complex formula (CH4):
- The left-hand side has 1 carbon atom, so 1 molecule of CO2 will balance it. The left-hand side also has 4 hydrogen atoms, which will be balanced by 2 molecules of H2O:
- Balancing the 4 oxygen atoms of the right-hand side by 2 molecules of O2 yields the equation
- The coefficients equal to 1 are omitted, as they do not need to be specified explicitly:
- It is wise to check that the final equation is balanced, i.e. that for each element there is the same number of atoms on the left- and right-hand side: 1 carbon, 4 hydrogen, and 4 oxygen.
System of linear equations
For each chemical element (or nuclide or unchanged moiety or charge) i, its conservation requirement can be expressed by the mathematical equation
where
- aij is the number of atoms of element i in a molecule of substance j (per formula in the chemical equation), and
- sj is the stoichiometric coefficient for the substance j.
This results in a homogeneous system of linear equations, which are readily solved using mathematical methods. Such system always has the all-zeros trivial solution, which we are not interested in, but if there are any additional solutions, there will be infinite number of them. Any non-trivial solution will balance the chemical equation. A "preferred" solution is one with whole-number, mostly positive stoichiometric coefficients sj with greatest common divisor equal to one.
Example
Let us assign variables to stoichiometric coefficients of the chemical equation from the previous section and write the corresponding linear equations:
All solutions to this system of linear equations are of the following form, where r is any real number:
The choice of r = 1 yields the preferred solution,
which corresponds to the balanced chemical equation:
Matrix method
The system of linear equations introduced in the previous section can also be written using an efficient matrix formalism. First, to unify the reactant and product stoichiometric coefficients sj, let us introduce the quantity
called stoichiometric number, which simplifies the linear equations to
where J is the total number of reactant and product substances (formulas) in the chemical equation.
Placement of the values aij at row i and column j of the composition matrix
- A =
and arrangement of the stoichiometric numbers into the stoichiometric vector
- ν =
allows the system of equations to be expressed as a single matrix equation:
- Aν = 0
Like previously, any nonzero stoichiometric vector ν, which solves the matrix equation, will balance the chemical equation.
The set of solutions to the matrix equation is a linear space called the kernel of the matrix A. For this space to contain nonzero vectors ν, i.e. to have a positive dimension JN, the columns of the composition matrix A must not be linearly independent. The problem of balancing a chemical equation then becomes the problem of determining the JN-dimensional kernel of the composition matrix. It is important to note that only for JN = 1 will there be a unique preferred solution to the balancing problem. For JN > 1 there will be an infinite number of preferred solutions with JN of them linearly independent. If JN = 0, there will be only the unusable trivial solution, the zero vector.
Techniques have been developed to quickly calculate a set of JN independent solutions to the balancing problem, which are superior to the inspection and algebraic method[citation needed] in that they are determinative and yield all solutions to the balancing problem.
- Example
Using the same chemical equation again, write the corresponding matrix equation:
Its solutions are of the following form, where r is any real number:
The choice of r = 1 and a sign-flip of the first two rows yields the preferred solution to the balancing problem:
Ionic equations
An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Ionic equations are used for single and double displacement reactions that occur in aqueous solutions.
For example, in the following precipitation reaction:
the full ionic equation is:
or, with all physical states included:
In this reaction, the Ca2+ and the NO3− ions remain in solution and are not part of the reaction. That is, these ions are identical on both the reactant and product side of the chemical equation. Because such ions do not participate in the reaction, they are called spectator ions. A net ionic equation is the full ionic equation from which the spectator ions have been removed. The net ionic equation of the proceeding reactions is:
or, in reduced balanced form,
In a neutralization or acid/base reaction, the net ionic equation will usually be:
There are a few acid/base reactions that produce a precipitate in addition to the water molecule shown above. An example is the reaction of barium hydroxide with phosphoric acid, which produces not only water but also the insoluble salt . In this reaction, there are no spectator ions, so the net ionic equation is the same as the full ionic equation.
Double displacement reactions that feature a carbonate reacting with an acid have the net ionic equation:
If every ion is a "spectator ion" then there was no reaction, and the net ionic equation is null.
Generally, if zj is the multiple of elementary charge on the j-th molecule, charge neutrality may be written as:
where the νj are the stoichiometric coefficients described above. The zj may be incorporated as an additional row in the aij matrix described above, and a properly balanced ionic equation will then also obey:
History
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Typesetting
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Notes
- Not to be confused with a related quantity called stoichiometric number.
- Not to be confused with yield (chemistry), a quantification of synthesis efficiency.
- The notation ⇄ was proposed in 1884 by the Dutch chemist Jacobus Henricus van 't Hoff. Van 't Hoff called reactions that didn't proceed to completion "limited reactions". He wrote (translation from French):
Now Mr. Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions, driving the one in the example cited to the formation of sea salt [i.e., NaCl] and nitric acid, [and] the other to hydrochloric acid and sodium nitrate. This consideration, which experiment validates, justifies the expression "chemical equilibrium", which is used to characterize the final state of limited reactions. I would propose to translate this expression by the following symbol:
- HCl + NO3 Na ⇄ NO3 H + Cl Na.
I thus replace, in this case, the = sign in the chemical equation by the sign ⇄, which in reality doesn't express just equality but shows also the direction of the reaction. This clearly expresses that a chemical action occurs simultaneously in two opposing directions.
- The notation
was suggested by Hugh Marshall in 1902.
- Triangle (△) was originally the alchemical symbol for fire.
- This expression comes from the Planck equation for the energy of a photon, E = hν. The Greek letter ν ("nu") is sometimes mistakenly replaced with a Latin letter v ("vee").
- A negative stoichiometric coefficient signifies a substance placed on the incorrect side of the chemical equation.
- An equivalent approach is flipping the signs of aij for reactants instead of replacing the stoichiometric coefficients sj with stoichiometric numbers νj.
References
- IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "chemical reaction equation". doi:10.1351/goldbook.C01034
- Crosland, M.P. (1959). "The use of diagrams as chemical 'equations' in the lectures of William Cullen and Joseph Black". Annals of Science. 15 (2): 75–90. doi:10.1080/00033795900200088.
- van 't Hoff, J.H. (1884). Études de Dynamique Chemique [Studies of chemical dynamics] (in French). Amsterdam, Netherlands: Frederik Muller & Co. pp. 4–5.
Or M. Pfaundler a relié ces deux phénomênes ... s'accomplit en même temps dans deux sens opposés.
- Marshall, Hugh (1902). "Suggested Modifications of the Sign of Equality for Use in Chemical Notation". Proceedings of the Royal Society of Edinburgh. 24: 85–87. doi:10.1017/S0370164600007720.
- McLean, Les. "Why can't we write the symbol "-" in chemical equations?". Quora.
- "Why is the minus sign (-) not allowed in reaction equations?". Stack Exchange. 2017-09-20. Answer by Nicolau Saker Neto. Archived from the original on 2021-06-15.
- Thorne, Lawrence R. (2010). "An Innovative Approach to Balancing Chemical-Reaction Equations: A Simplified Matrix-Inversion Technique for Determining the Matrix Null Space". Chem. Educator. 15: 304–308. arXiv:1110.4321.
- Holmes, Dylan (2015). "The null space's insight into chemical balance". Dylan Holmes. Retrieved Oct 10, 2017.
- James E. Brady; Frederick Senese; Neil D. Jespersen (December 14, 2007). Chemistry: matter and its changes. John Wiley & Sons. ISBN 9780470120941. LCCN 2007033355.
This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations October 2023 Learn how and when to remove this message A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas The reactant entities are given on the left hand side and the product entities are on the right hand side with a plus sign between the entities in both the reactants and the products and an arrow that points towards the products to show the direction of the reaction The chemical formulas may be symbolic structural pictorial diagrams or intermixed The coefficients next to the symbols and formulas of entities are the absolute values of the stoichiometric numbers The first chemical equation was diagrammed by Jean Beguin in 1615 StructureA chemical equation see an example below consists of a list of reactants the starting substances on the left hand side an arrow symbol and a list of products substances formed in the chemical reaction on the right hand side Each substance is specified by its chemical formula optionally preceded by a number called stoichiometric coefficient The coefficient specifies how many entities e g molecules of that substance are involved in the reaction on a molecular basis If not written explicitly the coefficient is equal to 1 Multiple substances on any side of the equation are separated from each other by a plus sign As an example the equation for the reaction of hydrochloric acid with sodium can be denoted 2HCl 2Na 2NaCl H2 Given the formulas are fairly simple this equation could be read as two H C L plus two N A yields two N A C L and H two Alternately and in general for equations involving complex chemicals the chemical formulas are read using IUPAC nomenclature which could verbalise this equation as two hydrochloric acid molecules and two sodium atoms react to form two formula units of sodium chloride and a hydrogen gas molecule Reaction types Different variants of the arrow symbol are used to denote the type of a reaction displaystyle rightarrow net forward reaction displaystyle rightleftarrows reaction in both directions displaystyle rightleftharpoons equilibrium displaystyle stoichiometric relation displaystyle leftrightarrow resonance not a reaction State of matter To indicate physical state of a chemical a symbol in parentheses may be appended to its formula s for a solid l for a liquid g for a gas and aq for an aqueous solution This is especially done when one wishes to emphasize the states or changes thereof For example the reaction of aqueous hydrochloric acid with solid metallic sodium to form aqueous sodium chloride and hydrogen gas would be written like this 2HCl aq 2Na s 2NaCl aq H2 g That reaction would have different thermodynamic and kinetic properties if gaseous hydrogen chloride were to replace the hydrochloric acid as a reactant 2HCl g 2Na s 2NaCl s H2 g Alternately an arrow without parentheses is used in some cases to indicate formation of a gas or precipitate This is especially useful if only one such species is formed Here is an example indicating that hydrogen gas is formed 2HCl 2Na 2 NaCl H2 Catalysis and other conditions The Baker Venkataraman rearrangement needs a base as a catalyst If the reaction requires energy it is indicated above the arrow A capital Greek letter delta D or a triangle is put on the reaction arrow to show that energy in the form of heat is added to the reaction The expression hn is used as a symbol for the addition of energy in the form of light Other symbols are used for other specific types of energy or radiation Similarly if a reaction requires a certain medium with certain specific characteristics then the name of the acid or base that is used as a medium may be placed on top of the arrow If no specific acid or base is required another way of denoting the use of an acidic or basic medium is to write H or OH or even acid or base on top of the arrow Specific conditions of the temperature and pressure as well as the presence of catalysts may be indicated in the same way Notation variants This illustration of a mechanism for acid catalyzed hydrolysis of an amide puts some reactants and products above the arrows and to their branches to allow chaining of the chemical equations The standard notation for chemical equations only permits all reactants on one side all products on the other and all stoichiometric coefficients positive For example the usual form of the equation for dehydration of methanol to dimethylether is 2 CH3OH CH3OCH3 H2O Sometimes an extension is used where some substances with their stoichiometric coefficients are moved above or below the arrow preceded by a plus sign or nothing for a reactant and by a minus sign for a product Then the same equation can look like this 2CH3OH H2OCH3OCH3 displaystyle ce 2CH3OH gt overset ce H2O CH3OCH3 Such notation serves to hide less important substances from the sides of the equation to make the type of reaction at hand more obvious and to facilitate chaining of chemical equations This is very useful in illustrating multi step reaction mechanisms Note that the substances above or below the arrows are not catalysts in this case because they are consumed or produced in the reaction like ordinary reactants or products Another extension used in reaction mechanisms moves some substances to branches of the arrow Both extensions are used in the example illustration of a mechanism Use of negative stoichiometric coefficients at either side of the equation like in the example below is not widely adopted and is often discouraged 2CH3OH H2O CH3OCH3 displaystyle ce 2 CH3OH H2O gt CH3OCH3 Balancing chemical equationsBecause no nuclear reactions take place in a chemical reaction the chemical elements pass through the reaction unchanged Thus each side of the chemical equation must represent the same number of atoms of any particular element or nuclide if different isotopes are taken into account The same holds for the total electric charge as stated by the charge conservation law An equation adhering to these requirements is said to be balanced A chemical equation is balanced by assigning suitable values to the stoichiometric coefficients Simple equations can be balanced by inspection that is by trial and error Another technique involves solving a system of linear equations Balanced equations are usually written with smallest natural number coefficients Yet sometimes it may be advantageous to accept a fractional coefficient if it simplifies the other coefficients The introductory example can thus be rewritten as HCl Na NaCl 12H2 displaystyle ce HCl Na gt NaCl 1 2 H2 In some circumstances the fractional coefficients are even inevitable For example the reaction corresponding to the standard enthalpy of formation must be written such that one molecule of a single product is formed This will often require that some reactant coefficients be fractional as is the case with the formation of lithium fluoride Li s 12F2 g LiF s displaystyle ce Li s 1 2F2 g gt LiF s Inspection method As seen from the equation CH4 2 O2 CO2 2 H2 O a coefficient of 2 must be placed before the oxygen gas on the reactants side and before the water on the products side in order for as per the law of conservation of mass the quantity of each element does not change during the reactionP4O10 6 H2O 4 H3PO4 This chemical equation is being balanced by first multiplying H3PO4 by four to match the number of P atoms and then multiplying H2O by six to match the numbers of H and O atoms The method of inspection can be outlined as setting the most complex substance s stoichiometric coefficient to 1 and assigning values to other coefficients step by step such that both sides of the equation end up with the same number of atoms for each element If any fractional coefficients arise during this process the presence of fractions may be eliminated at any time by multiplying all coefficients by their lowest common denominator Example Balancing of the chemical equation for the complete combustion of methane CH4 O2 CO2 H2O displaystyle ce mathord CH4 mathord O2 gt mathord CO2 mathord H2O dd dd is achieved as follows A coefficient of 1 is placed in front of the most complex formula CH4 1CH4 O2 CO2 H2O displaystyle ce 1 CH4 mathord O2 gt mathord CO2 mathord H2O The left hand side has 1 carbon atom so 1 molecule of CO2 will balance it The left hand side also has 4 hydrogen atoms which will be balanced by 2 molecules of H2O 1CH4 O2 1CO2 2H2O displaystyle ce 1 CH4 mathord O2 gt 1 CO2 2H2O Balancing the 4 oxygen atoms of the right hand side by 2 molecules of O2 yields the equation 1CH4 2O2 1CO2 2H2O displaystyle ce 1 CH4 2 O2 gt 1 CO2 2 H2O The coefficients equal to 1 are omitted as they do not need to be specified explicitly CH4 2O2 CO2 2H2O displaystyle ce CH4 2 O2 gt CO2 2 H2O It is wise to check that the final equation is balanced i e that for each element there is the same number of atoms on the left and right hand side 1 carbon 4 hydrogen and 4 oxygen System of linear equations For each chemical element or nuclide or unchanged moiety or charge i its conservation requirement can be expressed by the mathematical equation j reactantsaijsj j productsaijsj displaystyle sum j in text reactants a ij s j sum j in text products a ij s j where aij is the number of atoms of element i in a molecule of substance j per formula in the chemical equation and sj is the stoichiometric coefficient for the substance j This results in a homogeneous system of linear equations which are readily solved using mathematical methods Such system always has the all zeros trivial solution which we are not interested in but if there are any additional solutions there will be infinite number of them Any non trivial solution will balance the chemical equation A preferred solution is one with whole number mostly positive stoichiometric coefficients sj with greatest common divisor equal to one Example Let us assign variables to stoichiometric coefficients of the chemical equation from the previous section and write the corresponding linear equations s1CH4 s2O2 s3CO2 s4H2O displaystyle ce mathit s 1 CH4 mathit s 2 O2 gt mathit s 3 CO2 mathit s 4 H2O C s1 s3H 4s1 2s4O 2s2 2s3 s4 displaystyle quad begin aligned text C amp amp s 1 amp s 3 text H amp amp 4s 1 amp 2s 4 text O amp amp 2s 2 amp 2s 3 s 4 end aligned All solutions to this system of linear equations are of the following form where r is any real number s1 rs2 2rs3 rs4 2r displaystyle begin aligned s 1 amp r s 2 amp 2r s 3 amp r s 4 amp 2r end aligned The choice of r 1 yields the preferred solution s1 1s2 2s3 1s4 2 displaystyle begin aligned s 1 amp 1 s 2 amp 2 s 3 amp 1 s 4 amp 2 end aligned which corresponds to the balanced chemical equation CH4 2O2 CO2 2H2O displaystyle ce CH4 2 O2 gt CO2 2 H2O Matrix method The system of linear equations introduced in the previous section can also be written using an efficient matrix formalism First to unify the reactant and product stoichiometric coefficients sj let us introduce the quantity nj sjfor a reactant sjfor a product displaystyle nu j begin cases s j amp text for a reactant s j amp text for a product end cases called stoichiometric number which simplifies the linear equations to j 1Jaijnj 0 displaystyle sum j 1 J a ij nu j 0 where J is the total number of reactant and product substances formulas in the chemical equation Placement of the values aij at row i and column j of the composition matrix A a1 1a1 2 a1 Ja2 1a2 2 a2 J displaystyle begin bmatrix a 1 1 amp a 1 2 amp cdots amp a 1 J a 2 1 amp a 2 2 amp cdots amp a 2 J vdots amp vdots amp ddots amp vdots end bmatrix and arrangement of the stoichiometric numbers into the stoichiometric vector n n1n2 nJ displaystyle begin bmatrix nu 1 nu 2 vdots nu J end bmatrix allows the system of equations to be expressed as a single matrix equation An 0 Like previously any nonzero stoichiometric vector n which solves the matrix equation will balance the chemical equation The set of solutions to the matrix equation is a linear space called the kernel of the matrix A For this space to contain nonzero vectors n i e to have a positive dimension JN the columns of the composition matrix A must not be linearly independent The problem of balancing a chemical equation then becomes the problem of determining the JN dimensional kernel of the composition matrix It is important to note that only for JN 1 will there be a unique preferred solution to the balancing problem For JN gt 1 there will be an infinite number of preferred solutions with JN of them linearly independent If JN 0 there will be only the unusable trivial solution the zero vector Techniques have been developed to quickly calculate a set of JN independent solutions to the balancing problem which are superior to the inspection and algebraic method citation needed in that they are determinative and yield all solutions to the balancing problem Example Using the same chemical equation again write the corresponding matrix equation s1CH4 s2O2 s3CO2 s4H2O displaystyle ce mathit s 1 CH4 mathit s 2 O2 gt mathit s 3 CO2 mathit s 4 H2O C H O 101040020221 n1n2n3n4 0 displaystyle begin matrix text C text H text O end matrix quad begin bmatrix 1 amp 0 amp 1 amp 0 4 amp 0 amp 0 amp 2 0 amp 2 amp 2 amp 1 end bmatrix begin bmatrix nu 1 nu 2 nu 3 nu 4 end bmatrix mathbf 0 Its solutions are of the following form where r is any real number n1n2n3n4 s1 s2s3s4 r 1 212 displaystyle begin bmatrix nu 1 nu 2 nu 3 nu 4 end bmatrix begin bmatrix s 1 s 2 s 3 s 4 end bmatrix r begin bmatrix 1 2 1 2 end bmatrix The choice of r 1 and a sign flip of the first two rows yields the preferred solution to the balancing problem n1 n2n3n4 s1s2s3s4 1212 displaystyle begin bmatrix nu 1 nu 2 nu 3 nu 4 end bmatrix begin bmatrix s 1 s 2 s 3 s 4 end bmatrix begin bmatrix 1 2 1 2 end bmatrix Ionic equationsAn ionic equation is a chemical equation in which electrolytes are written as dissociated ions Ionic equations are used for single and double displacement reactions that occur in aqueous solutions For example in the following precipitation reaction CaCl2 2AgNO3 Ca NO3 2 2AgCl displaystyle ce CaCl2 2AgNO3 gt Ca NO3 2 2 AgCl v the full ionic equation is Ca2 2Cl 2Ag 2NO3 Ca2 2NO3 2AgCl displaystyle ce Ca 2 2Cl 2Ag 2NO3 gt Ca 2 2NO3 2AgCl v or with all physical states included Ca2 aq 2Cl aq 2Ag aq 2NO3 aq Ca2 aq 2NO3 aq 2AgCl displaystyle ce Ca 2 aq 2Cl aq 2Ag aq 2NO3 aq gt Ca 2 aq 2NO3 aq 2AgCl v In this reaction the Ca2 and the NO3 ions remain in solution and are not part of the reaction That is these ions are identical on both the reactant and product side of the chemical equation Because such ions do not participate in the reaction they are called spectator ions A net ionic equation is the full ionic equation from which the spectator ions have been removed The net ionic equation of the proceeding reactions is 2Cl 2Ag 2AgCl displaystyle ce 2Cl 2Ag gt 2AgCl v or in reduced balanced form Ag Cl AgCl displaystyle ce Ag Cl gt AgCl v In a neutralization or acid base reaction the net ionic equation will usually be H aq OH aq H2O l displaystyle ce H aq OH aq gt H2O l There are a few acid base reactions that produce a precipitate in addition to the water molecule shown above An example is the reaction of barium hydroxide with phosphoric acid which produces not only water but also the insoluble salt In this reaction there are no spectator ions so the net ionic equation is the same as the full ionic equation 3Ba OH 2 2H3PO4 6H2O Ba3 PO4 2 displaystyle ce 3Ba OH 2 2H3PO4 gt 6H2O Ba3 PO4 2 v 3Ba2 6OH 6H 2PO43 phosphate 6H2O Ba3 PO4 2 barium phosphate displaystyle ce 3Ba 2 6OH 6H underbrace ce 2PO4 3 ce phosphate ce gt 6H2O underbrace Ba3 PO4 2 v barium phosphate Double displacement reactions that feature a carbonate reacting with an acid have the net ionic equation 2H CO32 carbonate H2O CO2 displaystyle ce 2H underbrace ce CO3 2 ce carbonate ce gt H2O CO2 If every ion is a spectator ion then there was no reaction and the net ionic equation is null Generally if zj is the multiple of elementary charge on the j th molecule charge neutrality may be written as j 1Jzjnj 0 displaystyle sum j 1 J z j nu j 0 where the nj are the stoichiometric coefficients described above The zj may be incorporated as an additional row in the aij matrix described above and a properly balanced ionic equation will then also obey j 1Jaijnj 0 displaystyle sum j 1 J a ij nu j 0 HistoryThis section is empty You can help by adding to it June 2022 TypesettingThis section is empty You can help by adding to it June 2022 NotesNot to be confused with a related quantity called stoichiometric number Not to be confused with yield chemistry a quantification of synthesis efficiency The notation was proposed in 1884 by the Dutch chemist Jacobus Henricus van t Hoff Van t Hoff called reactions that didn t proceed to completion limited reactions He wrote translation from French Now Mr Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions driving the one in the example cited to the formation of sea salt i e NaCl and nitric acid and the other to hydrochloric acid and sodium nitrate This consideration which experiment validates justifies the expression chemical equilibrium which is used to characterize the final state of limited reactions I would propose to translate this expression by the following symbol HCl NO3 Na NO3 H Cl Na I thus replace in this case the sign in the chemical equation by the sign which in reality doesn t express just equality but shows also the direction of the reaction This clearly expresses that a chemical action occurs simultaneously in two opposing directions The notation displaystyle ce lt gt was suggested by Hugh Marshall in 1902 Triangle was originally the alchemical symbol for fire This expression comes from the Planck equation for the energy of a photon E hn The Greek letter n nu is sometimes mistakenly replaced with a Latin letter v vee A negative stoichiometric coefficient signifies a substance placed on the incorrect side of the chemical equation An equivalent approach is flipping the signs of aij for reactants instead of replacing the stoichiometric coefficients sj with stoichiometric numbers nj ReferencesIUPAC Compendium of Chemical Terminology 2nd ed the Gold Book 1997 Online corrected version 2006 chemical reaction equation doi 10 1351 goldbook C01034 Crosland M P 1959 The use of diagrams as chemical equations in the lectures of William Cullen and Joseph Black Annals of Science 15 2 75 90 doi 10 1080 00033795900200088 van t Hoff J H 1884 Etudes de Dynamique Chemique Studies of chemical dynamics in French Amsterdam Netherlands Frederik Muller amp Co pp 4 5 Or M Pfaundler a relie ces deux phenomenes s accomplit en meme temps dans deux sens opposes Marshall Hugh 1902 Suggested Modifications of the Sign of Equality for Use in Chemical Notation Proceedings of the Royal Society of Edinburgh 24 85 87 doi 10 1017 S0370164600007720 McLean Les Why can t we write the symbol in chemical equations Quora Why is the minus sign not allowed in reaction equations Stack Exchange 2017 09 20 Answer by Nicolau Saker Neto Archived from the original on 2021 06 15 Thorne Lawrence R 2010 An Innovative Approach to Balancing Chemical Reaction Equations A Simplified Matrix Inversion Technique for Determining the Matrix Null Space Chem Educator 15 304 308 arXiv 1110 4321 Holmes Dylan 2015 The null space s insight into chemical balance Dylan Holmes Retrieved Oct 10 2017 James E Brady Frederick Senese Neil D Jespersen December 14 2007 Chemistry matter and its changes John Wiley amp Sons ISBN 9780470120941 LCCN 2007033355