The Foundations of Chemistry

THE UNIT FACTOR METHOD

(DIMENSIONAL ANALYSIS)

Many chemical and physical processes can be described by numerical relationships. In fact,

many of the most useful ideas in science must be treated mathematically. In this section,

we review some problem-solving skills.

The units must alwaysaccompany the numeric value of a measurement, whether we

are writing about the quantity, talking about it, or using it in calculations.

Multiplication by unity (by one) does not change the value of an expression. If we rep-

resent “one” in a useful way, we can do many conversions by just “multiplying by one.”

This method of performing calculations is known as dimensional analysis,the factor-

label method,or the unit factor method.Regardless of the name chosen, it is a pow-

erful mathematical tool that is almost foolproof.

Unit factorsmay be constructed from any two terms that describe the same or equiv-

alent “amounts” of whatever we may consider. For example, 1 foot is equal to exactly 12

inches, by definition. We may write an equation to describe this equality:

1 ft
12 in.

Dividing both sides of the equation by 1 ft gives

 or 1 

The factor (fraction) 12 in./1 ft is a unit factor because the numerator and denominator

describe the same distance. Dividing both sides of the original equation by 12 in. gives

1 
1 ft/12 in., a second unit factor that is the reciprocal of the first. The reciprocal of any

unit factor is also a unit factor.Stated differently, division of an amount by the same amount

always yields one!

In the English system we can write many unit factors, such as

,, ,,

The reciprocal of each of these is also a unit factor. Items in retail stores are frequently

priced with unit factors, such as 39¢/lb and $3.98/gal. When all the quantities in a unit

factor come from definitions, the unit is known to an unlimited (infinite) number of sig-

nificant figures. For instance, if you bought eight 1-gallon jugs of something priced at

$3.98/gal, the total cost would be 8$3.98, or $31.84; the merchant would not round

this to $31.80, let alone to $30.

In science, nearly all numbers have units. What does 12 mean? Usually we must supply

appropriate units, such as 12 eggs or 12 people. In the unit factor method, the units guide

us through calculations in a step-by-step process, because all units except those in the de-

sired result cancel.

2000 lb



1 ton

4 qt



1 gal

1 mi



5280 ft

1 yd



36 in.

1 yd



3 ft

12 in.



1 ft

12 in.



1 ft

1 ft



1 ft

1-9

It would be nonsense to say that the
length of a piece of cloth is 4.7. We
must specify units with the number —
4.7 inches, 4.7 feet, or 4.7 meters, for
instance.

26 CHAPTER 1: The Foundations of Chemistry

Unless otherwise indicated, a “ton”
refers to a “short ton,” 2000 lb. There
are also the “long ton,” which is 2240
lb, and the metric tonne, which is
1000 kg.