# Balancing Abstract Chemical Equations with One Kind of Atom

Balancing Abstract Chemical Equations with One Kind of Atom

This Demonstration solves chemical-like equations with one kind of atom, represented by the letter . An expression like can be thought of as a "molecule" of atoms of . For the equation to be balanced, the counts of the atoms on both sides must be the same. Balancing the equation is equivalent to solving the Diophantine equation , where parameters , , and are positive integers, and the solution should be in non-negative integers , and positive .

A

A

n

n

A

x+y=z

A

a

A

b

A

c

ax+by=cz

a

b

c

x

y

z

The Diophantine equation , where , are positive and with a solution in non-negative integers, is a Frobenius equation. The largest for which the equation has no solution is called the Frobenius number. So if is the Frobenius number of the equation, then the Frobenius equation has solutions for all . The problem of balancing the chemical equation is reduced to solving the Frobenius equations for , where is the smallest number for which and is as small as possible.

ax+by=d

a

b

d

f

d>f

ax+by=cz

z=1,2,…,k

k

f<ck

cz