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Stoichiometric Chemical Balancing and Infinitely Many Solutions

During your life you have witnessed numerous chemical reactions. How would you describe them to someone else? How coidd you obtain quantitative information about the reaction? Chemists use stoichiometric chemical equations[1] to answer these questions.

Bv definition, a chemical equation is a written representation of a chemical reaction, showing the reactants and products, their physical states, and the direction in which the reaction proceeds. In addition, many chemical equations designate the conditions necessary (such as high temperature) for the reaction to occur. A chemical equation provides stoichiometric information about a chemical reaction, only if the equation is balanced.

For a chemical equation to be balanced, the same number of each kind of atom must be present on both sides of the chemical equation. The French chemist Antoine-Laurent de Lavoisier[2] (1743-1794) introduced the law of conservation of matter during the latter half of the eighteenth century. The conservation law states that matter can neither be created nor destroyed, de Lavoisier’s principles of naming chemical substances are still used today.

John Dalton[3] (1766-1844), developer of the first useful atomic theory of matter in the early 1800s, was the first to associate the ancient idea of atoms with stoichiometry. Dalton concluded, while studying meteorology, that evaporated water exists in air as an independent gas. Solid bodies cannot occupy the same space at the same time, but obviously water and air could. If the water and air were composed of discrete particles, evaporation might be viewed as mixing their separate particles. He performed a series of experiments on mixtures of gases to determine what effect properties of the individual gases had on the properties of the mixture as a whole. While trying to explain the results of those experiments, Dalton hypothesized that the sizes of the part icles making up different gases must be different . Dalton wrote

it became an object to determine the relative sizes and weights, together with the relative numbers of atoms entering into such combinations... Thus a train of investigation was laid for determining the number and weight of all chemical elementary particles which enter into any sort of combination one with another.[4]

According to Dalton’s Atomic Theory of Matter of 1803, all substances are composed of atoms. During a chemical reaction atoms may be combined, separated, or rearranged, but not created or destroyed. The postulates include:

  • 1. All matter is composed of atoms, indivisible and indestructible objects, which are the ultimate chemical particles.
  • 2. All atoms of a given element are identical, both in mass and in properties. Atoms of different elements have different masses and different properties.
  • 3. Compounds are formed by combination of two or more different kinds of atoms. Atoms combine in the ratio of small whole numbers.
  • 4. Atoms are the units of chemical change. A chemical reaction involves only the combination, separation, or rearrangement of atoms.

Let’s examine the meaning of chemical equations and compare them to mathematical equations to gain some insights. A chemical equation identifies the starting and finishing chemicals as reactants and products: reactants —> products.

Chemical Equations vs. Mathematical Equations

We usually think of an equat ion like x + 2x = 3x as purely mathematical, even if x represents a physical quantity like distance or mass. A chemical equation may look like a mathematical equation, but it describes experimental observations: the quantities and kinds of reactants and products for a particular chemical reaction. Reactants appear on the left side of a chemical equation; products on the right. For example, see Table 5.4. The products, which are the result of combining the reactants, are known from experimental observations - they are not derived mathematically.

TABLE 5.4: Chemical Equation vs. Mathematical Equation

Chemical Equation

Mathematical Equation

Combustion of Propane C3H[5] + 5O2 —> ЗСО2 + 4H2O

Linear Equation x = 2x + 3

Left: reactants; right: products

Balanced when it reflects the conservation of matter

No standard left/right order

Solved when all values that give a true statement are found

In fact, combining the same reactants at different concentrations or temperatures can often produce different products from the same reactants. First-year chemistry students cannot predict these effects, and are generally not asked to predict them. On the other hand, a chemical equation is similar to a mathematical equation in that there are certain restrictions on what may appear on the left and right sides. These mathematical rules represent the effects of conservation of matter on the reaction—conservation of matter says that no atoms are destroyed or created during a chemical reaction.

How a Chemist Approaches Balancing an Equation

Many chemical equations, in the view of the chemists, can be balanced by inspection, that is, by the process of “trial and error.” The objectives of a chemist are:

  • • Recognize a balanced equation.
  • • Recognize an unbalanced equation.
  • • Balance by inspecting chemical equations with given reactants and products.

According to chemistry textbooks, the step-wise procedure to balance equations is:

Step 1. Determine what reaction is occurring: know the reactants, the products, and the physical states.

Step 2. Write the unbalanced equation that summarizes the reaction described in Step 1.

Step 3. Balance the equation by inspection, starting with the most complicated molecules. Do not change the identities of any reactant or product.

Thus, balancing the equation is done by inspection, a “trial and error” process that some students catch on to quickly, but others struggle with in frustration.

Balancing Equations with Systems of Equations

Balancing chemical equations can be an application of solving a system of linear equations. Placing variables as the multipliers for each compound and making equations for each type of atom in the reaction results in a system of linear equations. This system is usually under-determined, there are more variables than equations. An under-determined system has infinitely many solutions. Our goal is to provide a procedure to find an integer solution from among the infinitely many solutions. This method is best grasped through an example which also lays the foundation for balancing more complicated oxidation-reduction equations.

Example 5.7. A Sulfur Dioxide Reaction.

Balance the chemical equation

Step 1. Introduce multipliers for each compound. There are three compounds, so we identify three multipliers: {x, X2, а,'з}. The reaction equation becomes

Step 2. Identify all elements: S (sulfur) and О (oxygen). Set up an equation for each element involving the amounts (multipliers) for that chemical on the reactant side equaling the amount on the product side.

Step 3. Create a homogeneous system; that is, put all variables on the left side of the equation and zero on the right.

Step 4. Write the system in matrix notation Ax = b where A is the coefficient matrix and b is the column vector of zeros. Then form the augmented matrix for the system.

Step 5. Apply Gaussian elimination (row reduction) to the augmented matrix [A | b] to obtain [// | c], where H is in reduced row echelon form.

Read the system’s solution: x — Ж3/6 = 0 and ж 2 — а?з = 0.

Balancing the chemical equation means finding the smallest whole numbers xi, X2, and Ж3 solving the system. Since there are more equations than unknowns, there is an infinite number of solut ions. Note that Ж3 is part of every equation, so Ж3 can equal anything. The fraction 1/6 is a coefficient of Ж3 in one of the equations, so choose ж3 to be the smallest whole number that eliminates the fraction; i.e., choose Ж3 = 6. Then x = 1 and Ж2 = 6.

The balanced reaction equation is

We are presented with the following more complicated unbalanced equation that would be difficult to solve by inspection.

Example 5.8. Ethylenediamine Mixed with Dinitrogen Tetroxide.

Balance the reaction

Step 1. Introduce five multipliers x through Ж5, one for each of the 5 terms. Step 2. List the equation for each element.

Step 3. Write the augmented matrix for the homogeneous system.

Step 4. Maple’s ReducedRowEchelonForm yields

Step 5. The least common multiple of the denominators for x^s coefficients is 4; choose гс5 = 4. Then aq = 1. ж2 = 2, x3 = 3, and x4 = 2.

The balanced equation is

Check the result—count the elements in the equations to make sure each balances.


In Exercises 1 to 4 balance the basic chemical reactions.

1. Copper plus silver nitrate displacement/redux reaction:

  • (Elements: copper (Cu), silver (Ag), nitrogen (N) and oxygen (O).)
  • 2. Zinc and hydrochloric acid replacement reaction:

  • (Elements: zinc (Zn), hydrogen (H), and chlorine (Cl).)
  • 3. Ferrous oxide to ferric oxide reaction:

  • (Elements: iron (Fe) and oxygen (0).)
  • 4. Calcium hydroxide and phosphoric acid neutralization reaction:

(Elements: calcium (Ca), oxygen (0), hydrogen (H), and phosphorus (P).)


Project 1. Explain why we need to use the smallest integer solution for the balanced chemical reactions. Provide examples.

Project 2. The reduction of kerosene takes place in three steps Balance the three reactions simultaneously.

References and Further Reading

[BKWK2009] Robert T. Balmer, William D. Keat, George Wise, Philip Kosky, Exploring Engineering: an Introduction to Engineering and Design, Elsevier Science & Tech., 2009.

[FVC2007a] William P. Fox, K. Varanzo, and .1. Croteau, “Mathematical Modeling of Chemical Stoichiometry”, PRIMUS, XVII(4), 301-315, 2007.

[FVC2007b] William P. Fox, K. Varanzo, and .1. Croteau, “Oxidation- Reduction Chemical Equation Balancing with MAPLE”, Computers in Education Journal (COED), VOL XII (2), April-June 2007, pages 50-57.

[LLM2016] D. Lay, S. Lay, amd .1. McDonald, Linear Algebra and its Applications, Pearson Education, London, U.K., 2016.

[MB1985] Ronald E. Miller and Peter D. Blair, Input-Output Analysis: Foundations and Extensions, Prentice Hall, 1985.

[Tro2016] Nivaldo .1. Tro, Chemistry: A Molecular Approach, 4th ed, Pearson, 2016.


  • [1] See LibreTexts’ Chemistry Library “Stoichiometry and Balancing Equations”.
  • [2] See
  • [3] See
  • [4] The quote is from Dalton’s notes for a lecture to the Royal Institution in London in1810.
  • [5] Write the unbalanced equation when given compound names for reactantsand products.
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