# Chemical equation balancer

This online calculator balances equations of chemical reactions using algebraic method.

There are several methods of balancing chemical equations:

1. Inspection method, or "hit & trial" method
2. Algebraic method
3. Method proposed by Arcesio Garcia
4. Oxidation number change method
5. Ion-electron method, or half-reaction method

The last two are used for redox reactions.

This chemical equation balancer uses the algebraic method – which is usually quite complex for manual calculations, however, it fits the computer program perfectly.

The algebraic method is based on the Law of Conservation of Mass – that matter can neither be created nor destroyed. Therefore, the number of each type of atom on each side of a chemical equation must be the same. Balancing chemical equations is the process of ensuring the conservation of matter. So, you just need to create a set of algebraic equations expressing the number of atoms of each element involved in the reaction and solve it. Therefore this method could be used for any type of chemical reaction (including redox reactions).

Let me illustrate this method by example.

Consider the reaction:
$FeCl_2+Na_3PO_4=Fe_3(PO_4)_2+NaCl$

We start by introducing unknown coefficients:
$x_1FeCl_2+x_2Na_3PO_4=x_3Fe_3(PO_4)_2+x_4NaCl$

Then we write the balance equations for each element in terms of the unknowns:
For Fe: $x_1*1=x_3*3$
For Cl: $x_1*2=x_4*1$
For Na: $x_2*3=x_4*1$
For P: $x_2*1=x_3*2$
For O: $x_2*4=x_3*8$

They will form a system of linear equations:
$\begin{cases}x_1-3x_3=0; \\2x_1-x_4=0;\\3x_2-x_4=0;\\x_2-2x_3=0;\\4x_2-8x_3=0;\end{cases}$

Here we have five equations for four unknowns, however, the last one is dependent on the fourth, so it can be omitted.

Now we can rewrite this system in matrix form:
$\begin{array}{|cccc|c|} 1 & 0 & -3 & 0 & 0 \\ 2 & 0 & 0 & -1 & 0 \\ 0 & 3 & 0 & -1 & 0 \\ 0 & 1 & -2 & 0 & 0 \\ \end{array}$

This system could be solved by using the Gaussian elimination method. Of course, you could not expect that the number of unknowns will always be equal to the number of equations. However, the Gaussian elimination method actually could find a solution for any number of equations and unknowns. I have created a special calculator that implements the Gaussian elimination method – The General Solution of a System of Linear Equations using Gaussian elimination – in the form suitable for chemical reactions. In short, it just keeps all fractions, and gets to a whole integers solution at the end.

Therefore, the calculator below simply parses the chemical reaction, creates a system of linear equations and feeds it to the above-mentioned Gaussian elimination calculator. The returned solution is then used to display the balanced equation.

Note: Always use the upper case for the first character in the element name and the lower case for the second character, as in the periodic table. Compare: Co – cobalt and CO – carbon monoxide. Thus, Na3PO4 — correct form, na3po4 — incorrect form.

If you're unable to balance a chemical reaction using this chemical reaction balancer, there's a good chance that you've made an error in the reaction. In such cases, you can search for the correct reaction using The Chemical Reaction Search Calculator.

#### Chemical equation balancer

Balanced equation

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PLANETCALC, Chemical equation balancer