Balancing Abstract Chemical Equations with One Kind of Atom

a

5

b

7

c

19

Diophantine equation

Frobenius number

solution

x

A

5

+ y

A

7

= z

A

19

5x+7y19z

23

This Demonstration solves chemical-like equations with one kind of atom, represented by the letter

A

. An expression like

A

n

can be thought of as a "molecule" of

n

atoms of

A

. For the equation to be balanced, the counts of the atoms on both sides must be the same. Balancing the equation

x

A

a

+y

A

b

=z

A

c

is equivalent to solving the Diophantine equation

ax+by=cz

, where parameters

a

,

b

, and

c

are positive integers, and the solution should be in non-negative integers

x

,

y

and positive

z

.

The Diophantine equation

ax+by=d

, where

a

,

b

are positive and with a solution in non-negative integers, is a Frobenius equation. The largest

d

for which the equation has no solution is called the Frobenius number. So if

f

is the Frobenius number of the equation, then the Frobenius equation has solutions for all

d>f

. The problem of balancing the chemical equation is reduced to solving the Frobenius equations

ax+by=cz

for

z=1,2,…,k

, where

k

is the smallest number for which

f<ck

and

cz

is as small as possible.