How Do I Balance Thee? ... Let Me Count the Ways!

I'm right! You're wrong! These are the comments that you may hear when your students are given the following equation to balance:

KClO₃ + HCl -> KCl + H₂O + Cl₂ + ClO₂

The source of this reaction was M. McMillan in A Demonstration-A-Day (Calhoun Scientific: Battle Creek, MI, 1992). The author suggests that this would be a good equation for students to try to balance by trial and error because it has two different sets of coefficients that are not multiples of each other. I gave the equation as an extra-credit problem to my chemistry students and, to my amazement, they turned in ten different sets of coefficients which correctly balanced the equation. Wondering if my students had found all of the possibilities, I balanced the equation myself and continued to get new combinations .

The following are some relationships that will help in balancing this equation by trial and error:

·The coefficient for KClO₃ will be the same as the coefficient for the KCl because there is only one atom of potassium in each compound on both sides of the equation.

·The coefficient of ClO₂ will always be an even number. This is due to the even coefficient of HCl, and an even number of chlorine atoms in Cl₂ despite its coefficient.

·After setting the coefficient of ClO₂, the remaining oxygen atoms are balanced with the coefficient for H₂O.

·The coefficient of HCl will be twice the coefficient of H₂O because there are two atoms of hydrogen in H₂O and only one in HCl. The hydrogen atoms now balance.

·Finally, the coefficient for Cl₂ is established by subtracting the coefficient of ClO₂ from that of HCl and dividing the result by two.

Trying to detect some pattern among the various combinations, I systematically ordered the coefficients as shown in Table 1. Table 1 could be expanded using a spreadsheet program to give additional combinations.

Given Table 1 and using some deductive reasoning, students could be challenged to determine the mathematical relationships among the coefficients. If x = the coefficient of KClO₃, and y = the coefficient of ClO₂, then the coefficients are:

KCl = x

HCl = 2 (3x - 2y)

H₂O = 3x - 2y

Cl₂ = [2 (3x - 2y) - y] / 2

Students need to realize that there is another factor, the transfer of electrons, that needs to be balanced. In order to determine the correct balanced equation, students can be introduced to the balancing of redox equations. The solution to the equation that negates all the others is as follows:

2 KClO₃ + 4 HCl -> 2 KCl + 2 H₂O + Cl₂ + 2ClO₂

Are there any other chemical equations that provide so many sets of coefficients?