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### Balancing Chemical Equations

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Date: 09/21/2000 at 20:02:17
From: Dan Kneezel
Subject: Balancing chemical equations

I know this question has its roots in chemistry, but I was highly
intrigued and figured I'd pass it along and see what the pros could
do with it.

Is it possible to create a finitely terminating algorithm to
balance a chemical equation, i.e., find the coefficients to balance
this example problem: _C8H8 + _O2 --> _H2O + _CO2?

I know that both the introductory and AP level classes that I've
taken suggest solely the trial and error method, but it seems highly
likely that a more formal approach is both possible and time
efficient.

I thought about the question for a little while, though not with much
concentration, I don't have time to.  I suppose it could be solved
with matrices and/or simultaneous equations.

If there is a method for balancing the equations mathematically,
please pass the information along. My math background should be
extensive enough that I will at the very least understand the gist of
what is being said. If it is not possible to create a finitely

Dan Kneezel
```

```
Date: 09/24/2000 at 10:18:35
From: Doctor Ian
Subject: Re: Balancing chemical equations

Hi Dan,

You're right, simultaneous equations can be used to balance an
equation.

_C8H8 + _O2 --> _H2O + _CO2

we can assign unknowns (variables) to the blanks,

uC8H8 + vO2 --> xH2O + yCO2

which leads directly to a system of equations:

C:  8u = y

H:  8u = 2x

O:  2v = x + 2y

One thing you can say for sure is this: The solution won't be unique.
(For example, you can always multiply all the unknowns by any integer,
and get another 'solution'.)  So you'll always have more unknowns than
equations.

One way to proceed is by eliminating variables.  For example, we can
substitute 8u for y wherever y occurs, giving us

8u = 2x

2v = x + 2(8u)

= x + 16u

Then we can substitute 4u for x wherever x occurs, giving us

2v = 4u + 16u

= 20 u

v = 10u

This delineates the space of solutions. Once you choose a value for v,
the other coefficients are determined.

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Physics/Chemistry

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