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# Chemical equation balance

This online calculator balances equations of chemical reactions.

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This online calculator balances equations of chemical reactions. 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 calculator uses algebraic method - it is usually quite complex for manual calculations, however, it perfectly fits for computer program.

Algebraic method is based on Law of Conservation of Mass: Matter cannot 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 Gaussian elimination method. Of course you could not expect that number of unknowns will always be equal to the number of equations. However, the Gaussian elimination method actually could find solution for any number of equations and unknowns. I have created special calculator which implements Gaussian elimination method /6200/ in the form suitable for chemical reactions. In short, it just keeps all fractions, and gets whole integers solution at the end.

Therefore, calculator below simply parses chemical reaction, creates system of linear equations and feeds it to the abovementioned Gaussian elimination calculator. Returned solution is then used to display 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 periodic table. Compare: Co - cobalt and CO - carbon monoxide. Thus, Na3PO4 — correct form, na3po4 — incorrect form.

### Chemical equation balance

Balanced equation

Matrix equation

Matrix solution
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