Here’ s a cool application of linear equations I first encountered about 20 years ago working with chemistry colleague Penney Sconzo at my former school in Atlanta, GA. Many students struggle early in their first chemistry classes with balancing equations. Thinking about these as generalized systems of linear equations gives a universal approach to balancing chemical equations, including ionic equations.
This idea makes a brilliant connection if you teach algebra 2 students concurrently enrolled in chemistry, or vice versa.
FROM CHEMISTRY TO ALGEBRA
Consider burning ethanol. The chemical combination of ethanol and oxygen, creating carbon dioxide and water:
But what if you didn’t know that 1 molecule of ethanol combined with 3 molecules of oxygen gas to create 2 molecules of carbon dioxide and 3 molecules of water? This specific set coefficients (or multiples of the set) exist for this reaction because of the Law of Conservation of Matter. While elements may rearrange in a chemical reaction, they do not become something else. So how do you determine the unknown coefficients of a generic chemical reaction?
Using the ethanol example, assume you started with
for some unknown values of w, x, y, and z. Conservation of Matter guarantees that the amount of carbon, hydrogen, and oxygen are the same before and after the reaction. Tallying the amount of each element on each side of the equation gives three linear equations:
where the coefficients come from the subscripts within the compound notations. As one example, the carbon subscript in ethanol ( ) is 2, indicating two carbon atoms in each ethanol molecule. There must have been 2w carbon atoms in the w ethanol molecules.
This system of 3 equations in 4 variables won’t have a unique solution, but let’s see what my Nspire CAS says. (NOTE: On the TI-Nspire, you can solve for any one of the four variables. Because the presence of more variables than equations makes the solution non-unique, some results may appear cleaner than others. For me, w was more complicated than z, so I chose to use the z solution.)
All three equations have y in the numerator and denominators of 2. The presence of the y indicates the expected non-unique solution. But it also gives me the freedom to select any convenient value of y I want to use. I’ll pick to simplify the fractions. Plugging in gives me values for the other coefficients.
Substituting these into (2) above gives the original equation (1).
Traditionally, chemists write these equations with the lowest possible natural number coefficients, but thinking of them as systems of linear equations makes another reality obvious. If 1 molecule of ethanol combines with 3 molecules of hydrogen gas to make 2 molecules of carbon dioxide and 3 molecules of water, surely 10 molecule of ethanol combines with 30 molecules of hydrogen gas to make 20 molecules of carbon dioxide and 30 molecules of water (the result of substituting instead of the used above).
You could even let to get , , and . Shifting units, this could mean a half-mole of ethanol and 1.5 moles of hydrogen make a mole of carbon dioxide and 1.5 moles of water. The point is, the ratios are constant. A good lesson.
ANOTHER QUICK EXAMPLE:
Now let’s try a harder one to balance: Reacting carbon monoxide and hydrogen gas to create octane and water.
Setting up equations for each element gives
I could simplify the hydrogen equation, but that’s not required. Solving this system of equations gives
Nice. No fractions this time. Using gives , , and , or
EXTENSIONS TO IONIC EQUATIONS:
Now let’s balance an ionic equation with unknown coefficients a, b, c, d, e, and f:
In addition to writing equations for barium, oxygen, hydrogen, and phosphorus, Conservation of Charge allows me to write one more equation to reflect the balancing of charge in the reaction.
Solving the system gives
Now that’s a curious result. I’ll deal with the zeros in a moment. Letting gives and , indicating that 3 molecules of ionic barium combine with 2 molecules of ionic phosphate to create a single uncharged molecule of barium phosphate precipitate.
The zeros here indicate the presence of “spectator ions”. Basically, the hydroxide and hydrogen ions on the left are in equal measure to the liquid water molecule on the right. Since they are in equal measure, one solution is
You still need to understand chemistry and algebra to interpret the results, but combining algebra (and especially a CAS) makes it much easier to balance chemical equations and ionic chemical equations, particularly those with non-trivial solutions not easily found by inspection.
The minor connection between science (chemistry) and math (algebra) is nice.
As many others have noted, CAS enables you to keep your mind on the problem while avoiding getting lost in the algebra.