The Way You SHOULD Be Solving Stoichiometry Problems, Part 2: Using Dimensional Analysis

In the first blog post of this series, I showed you a method to solve stoichiometry problems that helps you keep organized and avoid getting confused.  Most of my clients like this approach.

However, one of my clients told me, “I think your method makes sense, but my teacher won’t let me do it this way.”

I am not going to address the question of why any technique that consistently gives the right answer should even be a problem for a teacher.  Instead, I will point out that it is thankfully not too much trouble to use the process I showed and report the answer in a way that would be acceptable to most teachers.

Most Chemistry teachers use a method called dimensional analysis, which looks like a series of multiplied fractions.  Each fraction in the series is just a way of expressing one of the arrow conversions that we looked at in Part 1 of this series.  So we can think of these fractions as just a way of reporting all the conversions that we used to get from the initial quantity given in the problem all the way to the final answer.

Let’s redo one of the stoichiometry problems we worked on in the last post, only this time reporting the answer using dimensional analysis.  As a reminder, the first problem we looked at last time was:

Fe₂O₃ (s) + 2 Al (s) → 2 Fe (s) + Al₂O₃ (s)

The above equation refers to the thermite reaction, which is used for certain kinds of welding.  If 52.7 grams of iron (III) oxide reacts with an excess of aluminum, what mass of iron can be produced?

And this was the solution from the last post:

The dimensional analysis method involves writing an equation.  When I report my answer using this technique, I want to start by writing the quantity I was initially given on the left side of the equation.  In this case, the given quantity is 52.7 grams of iron (III) oxide.  I want to be sure to write this quantity, and ALL quantities in the problem, with both a UNIT and a SUBSTANCE.  So I want to express it as “52.7 g Fe₂O₃” and not just “52.7” or “52.7 g”.  This is very important for the process not to cause confusion later!

On the right side of the equation, I also want to write the UNIT and SUBSTANCE of the result.  In this case, the problem is asking for the mass of iron produced, so I will write “___ g Fe” on the right side.  (I can’t put a number since I have not solved the problem yet!)  I personally find this very helpful, as it reminds me of what my final goal is.  So my solution so far looks like:

I will be multiplying the number on the left by a series of fractions.  Each fraction will correspond to exactly one arrow I had in my table.  Since my table has three arrows, I will need to multiply the left side by three fractions:

Now I will fill in the fractions.

I know that the first fraction corresponds to the first arrow in my table.  That is the purple one that goes from grams of iron (III) oxide to moles of iron (III) oxide.  I know that the conversion between these two quantities is the molar mass of Fe₂O₃, which is 159.70 g / mol.  I need to express this conversion as a fraction with both a unit and a substance, in both the numerator and denominator of the fraction, like this:   

When I place this fraction after the given on the left side, I have a choice of how I want to orient that fraction: I can either place it so that the 159.70 g Fe₂O₃ is on the top of the fraction and 1 mol Fe₂O₃ is in the bottom, or I can flip it so that the 159.70 g Fe₂O₃ goes to the bottom of the fraction, and the 1 mol Fe₂O₃ is on top.  The number 159.70 always has to be on the same side as the grams unit and not the moles unit, since that is how the molar mass is defined.  But how do I decide which orientation is better?      

What I want to do is to treat the units and substances in the problem as if they were algebra variables, and orient my fraction so that the units in the factor to the left get algebraically cancelled out.  Since the factor immediately to the left has units of g Fe₂O₃, I want to orient my fraction so that the 159.70 g Fe₂O₃ is at the bottom of the fraction.  I can then act as if “g Fe₂O₃” is a variable name and cancel it out from the factor to the left:

Note that if I had the conversion fraction in the other orientation, with the 159.70 g Fe₂O₃ on top, then I would not have been able to cancel out the units in this way.

My next conversion is the green arrow in the table, and I know that this is where I went from moles of iron (III) oxide to moles of iron.  From the reaction equation, I also know that 2 moles of iron are produced from each mole of iron(III) oxide that is consumed.  Since I want to cancel out the “mol Fe₂O₃” unit from the previous fraction, I am going to orient my next fraction like this:

so that I can cancel the “mol Fe₂O₃” as shown above.

Finally, I want to create a fraction for the final purple arrow, which went from moles of iron to grams of iron in the table.  This is the conversion using the molar mass of iron, which this time I express with the 55.85 g Fe on top, so that moles of Fe will cancel from the previous fraction:

I note that the only units left over on the left side match the units on the right side exactly, so I have confidence that I set up the multiplication correctly.  This is why I listed the unit and substance of the answer even before I started writing any fractions — it is a great sanity check to make sure that my setup is what I wanted it to be.

I can now multiply out all the numbers to arrive at the solution:

which is exactly what I got last time.  This is the format that most Chemistry teachers are expecting, so if you report your solution this way, your teacher should accept your answer.

I hope you find my post helpful!  If you would like more help in Chemistry, I would love to work with you on 1:1 tutoring.  Please see my website at https://andrewjeungtutoring.com .