CHEMICAL EQUATIONS COEFFICIENTS ### Chemistry or Mathematics?

Finding coefficients for chemical reactions is often complicated, especially in the case of redox processes. The classical approach uses oxidation numbers and many chemists cannot imagine any way to find the solution other than balancing half-reactions for the process. In some cases however it is hard to use this method because oxidation numbers cannot be unequivocally attributed. Luckily it turns out that oxidation numbers and half-reaction are unnecessary! Another method exists that uses algebra to find the solution. Mathematics provides a general way to find reaction coefficients. In fact it provides better solutions than chemistry itself! Such an idea might seem a bit strange for a chemist. For me it was a real shock when I realized the power of mathematics. That is why the calculator was created - to prove that this algebraic technique works. I believe that students will find it very useful in determining the correct coefficients.

To see how the program finds the solution let`s start from a simple example.
All the examples can be cut and pasted into the calculator

CO + H2O --> CO2 + H2

Use the calculator with the explanation box checked to see the output. With mathematics in mind you can write such a simple equation with variables a, b, c denoting the coefficients to be found:

a CO + b H2O + c CO2 = H2

All you have to do is balance the quantity for each element on the left and the right side of the equation. You obtain a system of equations:

for C: 1a + 0b + 1c = 0
for O: 1a + 1b + 2c = 0
for H: 0a + 2b + 0c = 2

To solve this system is to find the value of variables a, b, c which are the desired coefficients. What would happen if we mistakenly mixed it up a bit?

CO2 + H2O --> CO + H2

If one did not notice the mistake one could not find the solution by means of the classical approach. Try by yourself. But the calculator still works! Following the methodology described you will find that the value of some variables will be negative so they should be placed on the opposite side of the equation. Algebra decides what will be the product and what will be the substrate! If you are an experienced chemist you will easily find the mistake in the mixed equation thanks to your chemical knowledge because the example is simple. And what about the one below? Is it right or wrong?:

P2I4 + H3PO4 -> P4 + H2O + PH4I

Try to solve it manually. Unexpectedly hard, isn`t it. Use the calculator then. But I can assure you that it really does not need any chemical analysis to find the true shape of the process. Imagine that such compounds exist and react as follows:

DogCat + Flea2Cat --> DogCat2 + Flea2

What would happen if we mistakenly mixed it up a bit again?

DogCat2 + Flea2Cat --> DogCat + Flea2

One could never find the coefficients for this equation using the classical approach. You cannot see where the error is at first glance. But try this in the calculator. It still works! I do hope this proves that chemical knowledge is not so important as mathematics when finding coefficients. Don`t get me wrong. I do not want to underestimate chemistry. The calculator is not supposed to predict what the products will be. Chemistry is really needed here. You must define all the reagents. The calculator can only analyze the given molecules and their elements by means of simple algebraic approach. Amazing that this mathematical procedure is able to group substrates apart from products. Furthermore, it can find the coefficients for only just hypothetically possible reactions. It does not care if the reaction is probable from the chemical point of view, for example:

KMnO4 --> K3MnO5 + Mn3O7 + O3

But if the reaction is found to be impossible for mathematical reasons one can be sure that it cannot occur in a test tube:

H3PO4 --> H2O + P2O3

To really test the calculator try this monster. I hope you never see one like this during your exams:

[Cr(N2H4CO)6]4[Cr(CN)6]3 + KMnO4 + H2SO4 -> K2Cr2O7 + MnSO4 + CO2 + KNO3 + K2SO4 + H2O

Some references:

J.Chem.Educ. 62, 507 (1985); 72, 894 (1995) [finding coefficient with algebraic method]
J.Chem.Educ. 73, 507 (1996); 74, 1367, 1368 (1997) [examples of equations]
J.Chem.Educ. 54, 704 (1977); 59, 728 (1982); 69, 279 (1992); 71, 295 (1994); 74; 1369 (1997) [use of matrix calculus] 