L6: Dimensional Analysis

Lesson 6: Dimensional Analysis (Sophomore Level Chemistry)

In this lesson the students will be introduced to the concept of dimensional analysis, or factor label.  They will have just completed a section on units and should be familiar with the basics of families of units such as length: meter being the base, and kilometer and centimeter being other members of the family.  I will enter into discussion with the students about the subject including examples and applications.  Important points in the discussion are that we aren’t changing anything, only converting units, that we are always multiply by one, and that we are trying to get away from the old method of “moving the decimal point.”  The discussion will go like this:

Dimensional Analysis is the name given to converting units in math and science.  All we are doing is converting one thing into another.  We aren’t changing anything about the things we are studying.  Math wise, there is some multiplication and division involved.  However, despite the many different numbers that will be used we are always in a sense multiplying by one.  For every one meter, there are one hundred centimeters; for every one kilometer there are 1000 meters, etc.  To convert what we have to what we want we multiply by a conversion factor that is the same as multiplying by one.  That’s how we know we aren’t changing anything.  Quick example:

How many centimeters is 6.34 meters? 

Our conversion factor is 100 centimeters / 1 meter.  We multiply by this conversion factor (which is the same as multiplying by one).  Just like in math, when we have the same thing on the top and bottom of a fraction, it cancels out and we are left with our answer:

6.34 meters *	100 centimeters
	1 meter

      Meters cancel out and we are left with 634 centimeters.

For the most part, this seems just like our old move the decimal point method of converting units.  However, dimensional analysis can be used to make multiple conversions in the same problem.  Not only can we convert up and down the ladder of a base and family units, but eventually we can convert multiple units at once.  Dimensional analysis allows us to make a bridge from what we have to what we want.  So another example:

Lets say we have 5389 millimeters and we want to know how many kilometers that is, but we don’t know the exact conversion factor.  We do know how many millimeters are in a centimeter, how many centimeters in a meter, and how many meters in a kilometer though.  We go through the same steps as before, but now we have to multiply by more than one conversion factor (even though we are still just multiplying by one)

5389 millimeters *	1 centimeter *	1 meter *	1 kilometer
	10 milimeters	100 centimeters	1000 meters

As before, millimeters, centimeters, and meters all cancel out and we are left with 0.005389 km

Now that we’ve got the general idea, lets try some examples:

First try:  how many seconds are in a year?

1 year *	365 days *	24 hours *	60 minutes *	60 seconds
	1 year	1 day	1 hour	1 minutes

= 31536000 seconds in a year

You are having a pizza party with 15 of your friends and you want to know how much pizza to buy.  Each friend will have four slices of pizza and there are twelve slices in a whole pizza.  How many pizzas should you buy?

15 friends *	4 slices *	1 pizza
	1 friend	12 slices

= 5 pizzas

During a science experiment you measure how fast you can run.  You come up with 15 miles per hour.  You go to submit your results but your teacher tells you that you need to use SI units.  You will need to convert to km/hour.  You don’t know how to do the exact conversion, but your teacher tells you there are 2.54 centimeters in an inch.  How fast can you run in km/hour?

Recognize that we aren’t converting time, so you can leave it off til the end (someday we may need it).

15 miles *	5280 feet *	12 inches *	2.54 centimeters *	1 meter *	1 kilometer
	1 mile	1 foot	1 inch	100 centimeters	1000 meters

= 24.1 km/h

In each case, the students will be providing the work with only slight help when needed from me.  I will encourage them to challenge their misconceptions and work through any problems they may have before I help them.  Once this lesson is complete, the students have an assignment of 15 examples of similar conversions.  If they follow the steps we outlined in class, they will have little trouble with it.