AP Chemistry exam: making math easy by using dimensional analysis

The AP Chemistry exam is filled to the brim with calculations. You’ll regularly mash the multiplication and division keys on your calculator when figuring out how much precipitate forms by mixing together two solutions of differing concentrations. On the multiple choice section, you’ll even have to work it out by hand. It can be challenging to keep all of your information well-organized and streamlined.

Cue stage right: dimensional analysis.

What is dimensional analysis?

In essence, dimensional analysis is a technique that allows you to condense a lot of lines of work into significantly fewer steps. The key here is to make use of conversion factors: ratios that connect two ideas. These conversion factors are often presented as “THIS is the same thing as THAT,” but that’s not exactly true. Instead, think of it as a relationship between two variables, along the lines of “For every X number of THIS, I connect it with Y number of THAT.”

When you properly set up the conversion factor, you have a quantity that “acts” like the number 1. Multiplying a bunch of conversion factors in a row is like multiplying by 1 a bunch of times in a row. It doesn’t change your original relationships.

Confused? Let’s get out of the land of abstract math and look at an example to see how to put dimensional analysis into practice.

How does it work?

You own a bakery in France that sells croissants by the dozen. With the money (in euros, €) you make from selling your croissants, you like to buy gyoza in Japan (in yen, ¥) which are purchased in packs of 25. You’re also given the following information:

1 dozen croissants = €15.99
1 25-pack of gyoza = ¥1250
€1.00 = \$1.16
\$1.00 = ¥112.80

After you curse out the people in the international marketplace for not having a convenient euro-to-yen conversion, you put your baking team to work and pump out 1000 dozen croissants. How many 25-packs of dumplings could you buy?

Yes, you can set up four different equations and achieve the correct answer. You can figure out how many euros you make from selling the croissants, convert that into dollars, convert that into yen, and then figure out how many packs of gyoza you could purchase. But setting up four separate equations can lead to some disorganized work on your paper. You could pick up the wrong piece of information or multiply when you mean to divide.

By using dimensional analysis, you can clean up your work and get a much clearer picture of what’s going on. This is where our conversion factors come into play.

Start with the original amount you made: the 1000 dozen croissants. Your first conversion factor says that 1 dozen croissants will be sold for €15.99. You can multiply your 1000 dozen croissants by €15.99/1 dozen croissants. The “dozen croissants” unit associated with the 1000 cancels out with the “dozen croissants” unit in the denominator of your conversion factor. Furthermore, you’re multiplying by something that acts like 1, so you’re not altering any quantities. In doing so, you convert your croissants into cold hard cash (in euros). You continue setting up your other conversion factors in the same manner, getting something that looks like this:

If you use the method where you set up multiple individual equations, you can very well end up with the same answer of 1673.8 25-packs of dumplings. However, using dimensional analysis clearly labels all your units and how they cancel out with one another until you’re left with the unit you actually want (the 25-pack of dumplings). You greatly reduce the chance of mistaking a multiplication for a division or inserting the wrong calculation into the next step.

Here's another example.

To see how dimensional analysis pans out in a chemistry problem, let’s move from baked goods to cars.

The engine in your car takes octane fuel (C8H18) and combusts it. You don’t know how much octane you used up on a recent drive, but you did observe that you produced 100 grams of water in the end. How many grams of CO2 did you produce?

The AP Chemistry exam won’t often present you conversion factors explicitly, but you can calculate them for yourself by figuring out the balanced reaction, manipulating formulas, and calculating molar masses on your own.

Balanced Reaction: 2 C8H18 + 25 O2 -> 16 CO2 + 18 H2O
Molecular Weight of H2O = 18 grams per mole
Molecular Weight of CO2 = 44 grams per mole

Though dimensional analysis works best with classic stoichiometry problems, you can adjust it as you need to fit your situation. Are you working with solutions? Treat your molarity as a moles per liter conversion factor. How about measuring how much heat energy is created in a reaction? Throw it into a conversion factor that incorporates one of your products or reactants.

It doesn’t matter whether you’re handling intricacies of international culinary business practices or estimating how much carbon dioxide you’re putting in the air based on the mass of residual water left behind. Dimensional analysis will always provide a clear and clean path to getting you to what a question is specifically asking for.