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Unit Factor Analysis


Every physical measurement has three parts: a value, a unit, and a precision. The value is the numerical part of the measurement. The unit is the part that comes after the value: grams, or feet, or gallons, for example. The precision indicates the confidence we have in the value. For example, $29.87 is more precise than $30. This handout will outline a method which can be used to solve about 70% of the numerical problems you will encounter in college. If you learn only one thing from this course, it should be unit factor analysis. (I recommend that you learn more than one thing but if you're trying to economize ...) Remember one of the Chemistry Department mottos: Units Are Your Friends!


What follows is alternately known as "unit factor analysis," "dimensional analysis," or, in Dr. Dunn parlance, "the hotdog method." Let's consider some common hotdogs used in unit conversions:
Liter/Gallon( 3.79 L / 1 gal )
Ounce/Pound( 16 oz / 1 lb )
Inch/Foot( 12 in / 1 ft )
Centimeter/Inch( 2.54 cm / 1 inch )
Weight Percent( xxx g something / hg something that contains it )
Water Density( 1.00 kg water / L water )
Gram/Pound( 454 g / 1 lb )
mL/cm3( 1 mL / 1 cm3 )

Each one of these factors comes from an equality. For example 12 inches equals one foot. If this is true then ( 12 in / 1 ft ) must equal one, or unity since the top and bottom of the fraction are equal. All unit factors equal one, which is why they are called unit factors. Dr. Dunn calls them "hotdogs" because the parentheses remind him of a hotdog bun. The powerful thing is that you can always multiply by one without changing the value of the thing you multiplied. You can multiply by one over and over, no problem. Unit Factor Analysis uses this fact as the basis for what it does. You just keep multiplying by one (in the guise of a unit factor) until your answer has the correct units.

Certain prefixes imply unit factors.
So this gives us unit factors like ( 1000 m / km ), ( 1000 mL / L ), ( 100 g / hg ), etc. Don't confuse the unit m (meter) with the prefix m (milli). You can, for example, have the unit mm which is millimeter.

Unit Factor Analysis

Now that we have some hotdogs, we can use them to solve a problem. Here are the steps in unit factor analysis:

  1. Write the units of the answer to the left of an equal sign leaving room for the value to be added later.
  2. Find a hotdog which resembles the units of the answer and write it to the right of the equal sign. You may have to flip it upside down to get the units to agree. This initial hotdog may come from the problem or it may come from a list of unit factors with which you are already familiar.
  3. Compare the units on the right and left. If they agree, you're ready to do the arithmetic. If they don't agree, add another hotdog on the right, attempting to cancel units which don't belong and add units which do belong. Continue adding hotdogs until the units on the left and right agree.
  4. Now you're ready to do the arithmetic. Enter the numbers into your calculator without rounding any of them off. If your calculator has memory, you may want to store intermediate results. Otherwise write down intermediate results using all the digits your calculator gives you When you've arrived at the final result, write another equal sign and copy your answer using all the digits your calculator gives you.
  5. Finally round off your answer to the same number of digits found in the least precise hotdog in the chain.

An Example

Consider a mead recipe which calls for 15 pounds of honey to make 5 gallons of mead. As a unit factor this becomes ( 15 pounds honey / 5 gallons mead ). We wish to make a smaller batch of mead, say 1.75 Liters (in a 2 L bottle). How much honey should we use? We could choose several units for our answer, pounds, ounces, or grams. Since honey is sold by the ounce in grocery stores, we will choose the ounce as the unit of our answer.

Ounces honey = 1.75 Liters mead ( 1 gallon / 3.79 liters )( 15.0 pounds honey / 5.0 gallons mead)( 16 ounces / pound)
= 22.163588 ounces honey
= 22 ounces honey

Now suppose the grocery store only has bottles with 12, 16, and 32 ounces of honey. The closest we can come will be 2*12=24 ounces. If we want to keep the same ratio of honey to mead without wasting our honey, how much mead should we make?

Liters mead = 24 ounces honey ( 1 pound / 16 ounces )( 5 gallons mead / 15 pounds honey)( 3.79 liters / 1 gallon)
= 1.895 liters mead
= 1.9 liters mead

That is, if we want to use 24 ounces of honey, we will have to fill our 2 L bottle almost full to keep to the recipe proportions.

Extending the Menu

As the semester progresses, we will add new hotdogs to the menu. Any time we encounter an equality we can generate a new kind of hotdog. We will learn, for example, that 1 mole of carbon weighs 12 grams, that 1 mole of glucose weighs 180 grams, and that there are 6 moles of carbon in a mole of glucose. Now, at this point you probably don't even know what a mole is. But you cans still work problems like the following:

What is the weight percent of carbon in glucose?

Well, let's start with the unit of the answer.
( g C / hg glucose ) =
Here hg stands for hectagrams which is 100 grams. We need something that has units of grams of carbon on the right hand side. We only know one thing about grams of carbon and that is that 12 g C = 1 mole C. That is, ( 12 g C / 1 mol C ) is a unit factor.
( g C / hg glucose ) = ( 12 g C / mol C )
We need to get rid of mole C and so we use the other hotdog that has moles C in it.
( g C / hg glucose ) = ( 12 g C / mol C )( 6 mol C / 1 mol glucose)
Yes, and now we have a pesky mol of glucose to get rid of and so we use our final piece of information, (180 g glucose/mol glucose).
( g C / hg glucose ) = ( 12 g C / mol C )( 6 mol C / 1 mol glucose)( 1 mol glucose / 180 g glucose )
Finally we need to convert grams to hectagrams: (100 g/hg)
( g C / hg glucose ) = ( 12 g C / mol C )( 6 mol C / 1 mol glucose)( 1 mol glucose / 180 g glucose )( 100 g / hg )
= 40.000 ( g C / hg glucose )
= 40% carbon in glucose

We have just solved a common general chemistry problem and we don't even know what a mole is. That's some kind of powerful method. All you need is the units of the answer and enough hotdogs to get rid of the units you don't like and introduce the units you do like.

Many times you may know the dimensions of a container and you need to know the volume. If the container is rectangular, simply multiply the height, width, and depth. For a cylindrical container, the volume is (3.14)r2h (r=radius). For a spherical container, the volume is 4(3.14)r3/3. Notice that in each case the unit of volume is (length)3. You can also work backwards: if you know the volume of the container and its shape, you can get the dimensions.

Try your hand at these for practice:

Your gas mileage is 32 miles per gallon. Gas costs $1.37 per gallon. A mile is 5280 feet. How many dollars does it cost you to drive 10 kilometers?

An aquarium measures 36"x24"x12". How many gallons of water does it hold?

An aquarium measures 36"x24"x12". How many pounds of water does it hold?

Criteria for Success

This project is passed by quiz alone. When you are ready, I will give you a single problem to work by Unit Factor Analysis. I will expect that you have memorized the table of unit factors above. In addition, I may give you information that can be turned into unit factors. You will work this problem without notes, but you may use a calculator. If you do not get the correct answer, you fail. You may, however, keep taking the test (one per day) until you pass. Of course the problems will be different from day to day.