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### Linear Equations: Chemical Formulas

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Date: 10/31/97 at 06:49:10
From: meliebe
Subject: Chemical equation

Dear sirs,

We have difficulties in in finding the numbers to enable the following
chemical formula: H2O2 + O3 = H2O + O2

Please be so kind as to advise us of the answer, if it's possible.

Thanks,
Noa Lieber
```

```
Date: 10/31/97 at 09:24:26
From: Doctor Rob
Subject: Re: Chemical equation

Good question!  These kinds of formulas lead to simultaneous systems
of linear equations.

Let the numbers sought be a, b, c, and d, so that the chemical
reaction is given by:

a H2O2 + b O3 --> c H2O + d O2.

Count the hydrogen and oxygen atoms on each side of the reaction.
They must be equal for the reaction to balance:

H:  2*a + 0*b = 2*c + 0*d,
O:  2*a + 3*b = 1*c + 2*d.

This tells us that

c = a,
a = 2*d - 3*b.

Now probably you want a, b, c, and d all to be positive integers (or
maybe you only insist that they be nonnegative). Aside from that,
there are no restrictions on them. It looks to me as though you could
pick any positive integer value for b, any positive integer value for
d such that d > 3*b/2, compute a = c = 2*d - 3*b > 0 from the above,
and you would get a chemical reaction that balances. For example,
b = 1, d = 2 would do, and a = c = 1. On the other hand, b = 3, d = 7
also works, whence a = c = 5.

Probably you are not interested in the case where all of a, b, c,
and d have a common divisor D > 1, since you can balance the equation
with the smaller numbers a/D, b/D, c/D, and d/D in that case. To avoid
this situation, just pick your b and d so as not to have a common
divisor bigger than 1 (i.e., be relatively prime).

This situation is a bit unusual, since usually you will not have two
values (b and d above) which you can pick freely, but only one. The
restriction that the numbers be integers with no common divisor D > 1
will then force a single choice of that free value, and all of the
others will be uniquely determined.

Example:  a H2S04 + b NaOH --> c NaHSO4 + d H20

H:  2*a + 1*b = 1*c + 2*d,
S:  1*a + 0*b = 1*c + 0*d,
O:  4*a + 1*b = 4*c + 1*d,
Na: 0*a + 1*b = 1*c + 0*d,

or, rewriting them,

2*a +   b -   c - 2*d = 0,
a       -   c       = 0,
4*a +   b - 4*c -   d = 0,
b -   c       = 0.

These are equivalent to a = b = c = d. The common divisor condition
forces us to pick a = b = c = d = 1. Sure enough, it checks!

Example:  a Na + b H20 -> c NaOH + d H2

Na:  a = c
H:   2*b = c + 2*d
O:   b = c

These are equivalent to a = b = c = 2*d. The common divisor condition
forces us to pick a = b = c = 2, d = 1. This one checks, too!

If the system of equations has no solutions, then the chemical
reaction cannot balance, and there must either be an additional
ingredient on the left, or an additional product on the right.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

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