How to balance chemical equations using algebraic method?

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How to balance chemical equations using algebraic method?

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Answer 1 · 2017-01-18T20:53:25

Here's how you do it.

Explanation:

Assume you must balance the equation

${CaCO}_{3} + H_{3} {PO}_{4} \to {Ca}_{3} {({PO}_{4})}_{2} + H_{2} {CO}_{3}$ $"CaCO"_3 + "H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "H"_2"CO"_3$

You first write the equation using letter variables for the coefficients:

$a {CaCO}_{3} + b H_{3} {PO}_{4} \to c {Ca}_{3} {({PO}_{4})}_{2} + d H_{2} {CO}_{3}$ $a"CaCO"_3 + b"H"_3"PO"_4 → c"Ca"_3("PO"_4)_2 + d"H"_2"CO"_3$

Then you set up a series of simultaneous equations, one for each element.

$Ca : m l a m m l = 3 c$ $"Ca":color(white)(ml) a color(white)(mml)= 3c$
$C : m l l a m m l l = m m m d$ $"C":color(white)(mll)acolor(white)(mmll) = color(white)(mmm)d$
$O : m 3 a + 4 b = 8 c + 3 d$ $"O": color(white)(m)3a + 4b = 8c +3d$
$H : m m m l l 3 b = m m l l 2 d$ $"H": color(white)(mmmll)3b = color(white)(mmll)2d$
$P : m m m m l b = 2 c$ $"P": color(white)(mmmml)b = 2c$

Now you solve the five simultaneous equations.

It looks almost impossible, but we know that the coefficients must be integers.

Let's set $c = 1$ $c = 1$ .

Then $l a = 3$ $color(white)(l) a = 3$ and

$m m l l d = a = 3$ $color(white)(mmll)d = a = 3$

$m m l l b = 2 c = 2$ $color(white)(mmll)b = 2c = 2$

So $a = 3$ $a = 3$ ; $b = 2$ $b = 2$ ; $c = 1$ $c = 1$ ; $d = 3$ $d = 3$

The balanced equation is

${3CaCO}_{3} + {2H}_{3} {PO}_{4} \to {Ca}_{3} {({PO}_{4})}_{2} + {3H}_{2} {CO}_{3}$ $"3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO"_3$

Here's a useful equation on balancing equations by the algebraic method.

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How to balance chemical equations using algebraic method?

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