Factor Labeling, or Dimensional Analysis
Following is a brief explanation of the mathematical procedure known as Factor Labeling, or Dimensional Analysis.

 Factor labeling, or dimensional analysis, is a method that is often used to help with the day-to-day mathematics that are encountered in Chemistry and Physics. There are as many ways to do a math problem as there are people who are doing the problem. Factor labeling is a method that is very direct and allows the student to set up a complex math problem as a series of factors. The solution to the problem will not require any algebraic solutions.

The process is based on the mathematical idea that multiplying a term by one will not change the value of the term. In factor labeling, a factor is selected that is the scientific equivalent of one. That means that the numerator and denominator of a fraction will be the same value, simply expressed in different units. For instance, the following factors are all the scientific equivalent of one.

(1 foot/12 inches)  or (1 inch/2.54 cm)  or
(1 dozen/12 items)  or (1 mole/6x1023 atoms)

 Each of these fractions can be written in the form seen here, or inverted.  Either way, they will have a value that is equivalent to one.  What determines the way the fraction is written?  What determines whether it is written as seen here, or inverted?

Answer:  The numerator and denominator will be determined by the other terms in the problem.  Ultimately, all the units, in all the factors, will need to cancel out.  At the end of a properly set up problem, the only units to survive will be the units desired in the answer.

Question: How many donuts will be present in 8.5 dozen?

    Answer: Clearly, this problem does not need to be done as a factor labeling problem. However, if it were, here is how it would look.



    Notice that the units in the numerator of the first term, dozen, will cancel the units in the denominator of the second term, namely dozen. As a result the only the units that will be left are "donuts". These are the units that are being asked for in the problem. In addition, the relationship between the numerator and the denominator in the second term provide a fraction that is equivalent to one.

Question: How many donuts will be present in 8.5 dozen?

    Answer: Clearly, this problem does not need to be done as a factor labeling problem. However, if it were, here is how it would look.



    Notice that the units in the numerator of the first term, dozen, will cancel the units in the denominator of the second term, namely dozen. As a result the only the units that will be left are "donuts". These are the units that are being asked for in the problem. In addition, the relationship between the numerator and the denominator in the second term provide a fraction that is equivalent to one.

Question: How many atoms are in a 2.5 mole sample of iron?

    Answer: Start with the initial piece of information given in the problem.



    Multiply this term by a fraction that contains atoms of Fe in the numerator and moles of Fe in the denominator.  Be sure that the numerical relationship between the numerator and denominator is valid.



    Notice that the units of moles Fe will cancel. The surviving units are atoms Fe, the desired units. In addition, the term (6x1023 atoms Fe) is equal to (1 mole Fe), so the fraction created with those two terms has a value that is equivalent to one.

Many times it is possible to do more complex problems by using a single factor. While doing a problem with a single factor may sound easier, it can often become the source of errors. By doing a problem with several factors, each relationship that is needed to do the problem is more clearly defined, and the cancellation of units will indicate whether the problem is set up properly, or not.

Question: How many atoms of Copper are present in a 1.2 kilogram sample?

    Answer: Again, start by writing out the initial piece of information that has been provided.


    Since most problems are best done using gram units, the next factor should convert units to grams.



    Now that the problem is in terms of grams, convert the units to moles of Cu. It would be more direct to convert directly to atom, but for demonstration purposes the mole conversion will be done.



    At this point, the original term has been multiplied by two terms, both with values equivalent to one.  And, the only units not to cancel are the moles of Cu. To make the final conversion to atoms of Cu, it is now necessary to multiply by a final factor.



    At this point, the problem is done. The original number, 1.2 kg Cu, has been multiplied by three terms. Each term has a value that is equivalent to one. Also, the denominator of each term cancels the units that appear in the previous numerator and a new set of units is introduced in the numerator of each new term. Here is the same set up used to do the calculations, but the pairs of terms that cancel have been indicated.



    The only units not to cancel are the (atoms Cu), the desired units.

 While factor labeling may initially seem like a difficult way to do simple problems, as the problems in Chemistry or Physics become more complex, factor labeling will remain simple. Meanwhile, less organized approaches to doing the problems will become complex. With the increase in complexity comes a greater chance for making mistakes. It is best to learn factor labeling on the more simple problems, for practice purposes. Later, the technique will pay great rewards.

Questions and comments should be sent to :

kdrews@bcpl.net

Updated on April 6, 2002