Balancing Chemical Equations

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 
terminating algorithm, please explain why.

Thank you in advance,
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.

To take your example, 

  _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. 

I hope this helps. Write back if you have more questions, about this 
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/