Dimensional Analysis Introduction Conversion Factors and Their Reciprocals Use of Conversion Factors: Some Examples Developing the Units Path Summary and Helpful Hints
Example 1: Suppose you want to convert $100 in U.S. dollars to Canadian dollars. You know that the exchange rate is currently $1.50 Cdn. for every $1.00 U.S. How much Canadian cash will you get?
Solution. You probably solved this in your head: you'll expect to receive $150 Canadian for your $100 U.S. But think carefully! This isn't intuitively obvious, you had to learn how to do it. The steps you probably followed, so fast you didn't realize, were these:
1. You recognized that you were starting with $100 U.S. (the starting quantity)
2. You got a conversion factor from the English sentence that tells you the exchange rate: the conversion factor is3. You then multiplied the starting quantity by the conversion factor to get your answer (the desired quantity):
This gives us a method for solving many problems that aren't so obvious as converting U.S. to Canadian dollars.

Dimensional analysis is sometimes called the factorlabel
method or the factorunit method. That is because it
involves two quantities: a mathematical proportionality that is
thefactor and an expression of the units, or label.
Every conversion factor has a reciprocal that can be used for the opposite conversion. To convert $U.S. to Canadian dollars, we used the factor
However, if we wanted to convert Canadian dollars to U.S. dollars, we would use the reciprocal of the above factor, or
So, if we had $500 Cdn., we could determine how much U.S. money we could get in exchange:
Below are some other conversion factors you will have seen in your studies of chemistry.














Example 1. (a) Write the reciprocal for the following conversion factor: 1 mol NaOH / 40.0 g NaOH. (b) Which of these (the original factor or its reciprocal) do we use if we want to change the mass of NaOH (49.7 g) to moles?
Answer: (a) 40.0 g NaOH / mol NaOH
(b) the original factor: (49.7 g NaOH) x (1 mol NaOH / 40.0 g NaOH) gives units of moles NaOH
Example 2. (a) Write the reciprocal for the following conversion factor: 3 atoms S / molecule P_{4}S_{3}. (b) Which of these (the original factor or its reciprocal) do we use if we want to calculate how many molecules of P_{4}S_{3} can be made from a 1200 S atoms?
Answer: (a) 1 molecule P_{4}S_{3} / 3 atoms S
(b) the reciprocal factor: (1200 S atoms) x (1 molecule P_{4}S_{3} / 3 atoms S) gives units of molecules of P_{4}S_{3}
Example 3. (a) Write the reciprocal of the following conversion factor: 3 mol AgCl / mol CrCl_{3}. (b) Which of these (the original factor or its reciprocal) do we use if we want to calculate how many moles of AgCl can be prepared from 0.1250 moles of CrCl_{3}?
Answer: (a) 1 mol CrCl_{3} / 3 mol AgCl
(b) the original factor: (0.1250 mol CrCl_{3}) x ( 3 mol AgCl / mol CrCl_{3}) has units of moles AgCl
Example 4. (a) Write the conversion factor for "36.3% chocolate chips in Chips Ahoy cookies," then write its reciprocal. Which of these (the original or its reciprocal) do we use to calculate how many cookies have to be processed to yield a 500. g of chocolate chips?
Answer: (a) conversion factor is 36.3 g chocolate chips / 100 g cookies; reciprocal is 100 g cookies / 36.3 g chocolate chips
(b) the reciprocal factor: (500. g chocolate chips) x (100 g cookies / 36.3 g chocolate chips) has units of mass of cookies.
Example 1. You have 45.5 grams of iron, Fe. What mass (in g) of Fe_{2}O_{3} can you make if you react the iron with oxygen? The balanced equation is
4 Fe + 3 O_{2 }> 2 Fe_{2}O_{3}
Analysis: We want to know how we can get from what we are given (45.5 g Fe) to what we WANT (mass of Fe_{2}O_{3}). Now you know about chemistry and balanced equations, so you know that the MOLE is the important unit here, and we'll have to work through moles. If we want to get from mass of Fe to mass of Fe_{2}O_{3}, we'll have to have a bunch of conversion factors since no single factor that we've been given can get us there.
The balanced equation gives us this conversion factor that relates the product to the reactant: 4 mol Fe / 2 mol Fe_{2}O_{3}. We have two other conversion factors, the molar masses, that allow us to go from masses to moles and vice versa. We also have the reciprocals of these factors.
The sequence of operations is this:
THIS SEQUENCE IS IMPORTANT! It may take a while to figure out, and you may have to play with conversion factors to realize just how you'll get from one step to another. But by now you'll know that mass/mole conversions are made through the conversion factor molar mass or its reciprocal. And the balanced equation gives you the connection between the moles of reactants and products.
So: we begin with 45.5 g Fe, and use the above sequence with its conversion factors chosen so that all the units cancel except "g Fe_{2}O_{3}" which is what we wanted to find:
Example 2. You are a chemist in an oreprocessing plant that produces iron metal. You are told that the new shipment of ore coming in contains 56.3% Fe_{2}O_{3}. Your company asks you: how many tonnes of iron can be obtained from 5.00 x 10^{2} tonnes of ore? (1 tonne = 1000 kg).
Analysis. You are given the amount of ore that has to be processed. That will be your starting point for this question. Once again, you'll need to work through moles. You'll also have to turn "56.3% Fe_{2}O_{3}" into a conversion factor:
56.3 % Fe_{2}O_{3} = 56.3 g Fe_{2}O_{3} / 100 g ore The unit path is then
mass of ore > mass of Fe_{2}O_{3} > moles Fe_{2}O_{3} > moles Fe > mass Fe So we begin with the mass of ore (5.00 x 10^{2} tonnes) and proceed in serval steps:
Step 1: mass of ore to mass of Fe_{2}O_{3} in grams
Step 2: mass of Fe_{2}O_{3} to mass of Fe
Step 3: mass of Fe in grams to mass in tonnesStep 1:
Step 2:
Step 3:
Learning this method takes practice, but it helps if you remember the basic process:
HELPFUL HINTS:
This page is
http://chemiris.labs.brocku.ca/~chemweb/courses/chem180/Dimensional_Analysis.html
Created October 10, 2000 by M. F. Richardson
© Brock University, 2000