Health hazard alert: The following technique is difficult and involves the creative use of algebra. It is recommended only for students who are attempting the balancing equations contest, who have a strong interest in algebra, and lots of time on their hands. It is well beyond the scope of normal balancing of equations and beyond the scope of chemistry 3. Read at your own risk. Your teachers will be happy to help you with these ideas, but cannot, of course, help you to apply them to any of the contest equations.

A simple example: H₂ + O₂ = H₂O

(1) Insert lower case letters as coefficients of the balanced equation.

a H₂ + b O₂ = c H₂O

(2) Write equations, balancing each atom, by multiplying subscripts and coefficients.

for atom H: 2a = 2c, which reduces to “a=c”; for atom O: “2b = c”

(3) Use algebra to relate all coefficients in a single line. “a = c = 2b”

(4) Select a number, usually 1, for the coefficient with the largest multiplier (b).

(5) Using simple algebra, if b = 1, then a = 2 and c = 2.

(6) Write the balanced equation by substituting coefficients for a, b, and c.

2H₂ + O₂ = 2H₂O

A slightly more complicated example: Al(OH)₃ + H₂SO₄ = Al₂(SO₄)₃ + H₂O

(1) a Al(OH)₃ + b H₂SO₄ = c Al₂(SO₄)₃ + d H₂O

(2) for Al: “a = 2c”; for O: “3a + 4b = 12c + d”; for H: “3a + 2b = 2d”; for S: “b = 3c”

(3) Combine the expressions for Al and S first, since they are the simplest.

But “a = 2c” and “b = 3c” are not that easy to combine. The term in common is c. How can I get equal numbers of c? Aim for 6c. Thus: 3a = 6c and 2b = 6c. So “3a = 2b = 6c”

The expression for H includes a new term, d. Let’s solve for this in terms of any one of the other variables. for H: “2d = 3a + 2b”.

(You can also use the expression for oxygen, however this is a bit more complicated so it will take longer to get to the same place. You might wish to try it.)

We already found that “2b = 3a = 6c”, so we substitute using the relationship “2b = 3a”.

for H: 2d = 3a + 2b

2d = 3a + 3a

2d = 6a or “d = 3a”

How can we relate our previous equation, “3a=2b=6c” with our new equation “d=3a”?

Since “3a” is in the first equation, simply add “=d” “3a = 2b = 6c = d”

(Note: This method is much rougher than it seems. Just what the coefficients are, is not important, so long as they come out in the correct ratios. You might have gotten “6a=4b=12c=2d” or “30a=20b=60c=10d” as examples. They will all reduce to the correctly balanced equation.)

(4) Set c = 1

(5) Solve for a: 3a = 6, so a = 2.

Solve for b: 2b = 6, so b = 3

Solve for d: d = 6

(6) 2 Al(OH)₃ + 3 H₂SO₄ = 1 Al₂(SO₄)₃ + 6 H₂O

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