Type an equation without coefficients —Fe2O3 + CO -> Fe + CO2, KMnO4 + HCl -> KCl + MnCl2 + Cl2 + H2O or MnO4^- + Fe^2+ + H^+ -> Mn^2+ + Fe^3+ + H2O— and get the smallest integer coefficients. Choose the method —trial and error, algebraic, matrix or all— and download the detailed PDF solution. You see trial and error (as done by hand, with the tries), the algebraic method (one unknown per species and one equation per element) and the matrix method (composition matrix and Gauss-Jordan), with exact fractions and the final check.
In a chemical reaction atoms are neither created nor destroyed: they are only rearranged. This is the law of conservation of mass, stated by Lavoisier. A chemical equation is therefore balanced when each element appears the same number of times in the reactants and in the products and, if there are ions, when the total charge is also the same on both sides. This guide explains the three methods the tool uses and works complete examples.
Balancing only changes the coefficients, the numbers in front of each formula, which say how many units of each species take part. The subscripts are never touched: changing H2O into H2O2 balances nothing, it turns water into another substance (hydrogen peroxide). The result is given with the smallest possible whole-number coefficients; a coefficient of 1 is not written.
This is the method done by hand in class. The coefficient of the most complex species is set to 1 and the elements are balanced in an order that avoids undoing earlier work: first the metals, then the nonmetals other than H and O, then hydrogen and finally oxygen, which usually appears in many species. If a fraction appears, the whole equation is multiplied by the denominator. It works well for simple equations, but complicated redox reactions can take many attempts.
Each coefficient gets an unknown and one equation is written per element (atoms on the left = atoms on the right). Example, the combustion of propane:
a C3H8 + b O2 → c CO2 + d H2O
There are four unknowns and three equations, so one is fixed: a = 1. Then c = 3, d = 4 and 2b = 6 + 4, that is, b = 5. Result: C3H8 + 5 O2 → 3 CO2 + 4 H2O. Check: 3 C, 8 H and 10 O on each side.
This is the algebraic method written with matrices, the most systematic one and the one used for large equations. Each row of the composition matrix is an element and each column a species; products get a negative sign. For propane (columns C3H8, O2, CO2, H2O):
C: [3 0 −1 0] H: [8 0 0 −2] O: [0 2 −2 −1]
The coefficients are a vector x with A·x = 0 (the null space of the matrix). The matrix is taken to reduced row echelon form by Gauss-Jordan elimination and the solution is read off, then scaled to whole numbers: x = (1, 5, 3, 4). If the null space has dimension greater than 1, the "equation" actually mixes several independent reactions and has no single balance; if it only has the zero solution, the equation is set up wrong (a species is missing or extra). The tool warns in both cases.
When there are ions, charge must be conserved as well as atoms. Example, the oxidation of iron(II) by permanganate in acid solution:
MnO4− + 5 Fe2+ + 8 H+ → Mn2+ + 5 Fe3+ + 4 H2O
Manganese goes from +7 to +2 (gains 5 electrons) and each iron from +2 to +3 (loses 1), so 5 Fe2+ are needed per MnO4−. Charge check: on the left −1 + 10 + 8 = +17; on the right +2 + 15 = +17. In the algebraic and matrix methods the charge is simply one more row of the matrix, with the charge of each species.
These methods appear in general chemistry textbooks, for example: R. Chang and K. A. Goldsby, Chemistry (McGraw-Hill); T. L. Brown, H. E. LeMay, B. E. Bursten et al., Chemistry: The Central Science (Pearson). The matrix method is a direct application of linear algebra (the null space of a matrix). Simuveliq is not affiliated with these authors or publishers.
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