Download Dimensional Analysis (The Factor Label Method)

Dimensional Analysis (The Factor Label Method)
Most calculations in science involve measured quantities. In such calculations, the units in
which quantities are measured must be treated mathematically just as the numerical parts
of the quantities are. For example, in multiplying 1.2 cm by 2.0 cm, there are two separate
calculations to be carried out.
First, it is necessary to multiply the two numbers: 1.2 X 2.0 = 2.4
Second, it is necessary to multiply the two units: cm X cm = cm2. The complete answer
then, is 1.2 cm X 2.0 cm = 2.4 cm2.
This concept can be applied in the solution of many problems. The application depends on
the use of a "conversion factor". A conversion factor is a fraction in which the
numberator adn the denominator both represent the same measurement. For example,
the fraction
100 cm
1m
is a unit factor since both the numerator and denominator represent the same length (one
meter). The solved examples illustrate the use of unit factors in solving problems by
dimensional analysis.
Table of Conversion Factors
Example_1: Convert 45.3 cm to its equivalent measurement in mm.
Solution:
Select a conversion factor which will convert the unit "cm" to the unit "mm".
The appropriate conversion factor is: 10 mm / 1 cm. Arrange the problem so
that the given measurement, when multiplied by the correct unit factor, will
yield an answer with the proper label:
45.3 cm X 10 mm = 453 mm
1 cm
Example_2: Change a speed of 72.4 miles per hour to its equvalent in meters per second.
In this example, several conversion factors are needed. One to change the
miles into meters and the other to change hours into seconds?
72.4 mi X 1760 yd X 36 inches X 1 meter X 1 hr
1 hr
1 mi
1 yd
39 inches
3600 s
= 32.7 m
s
Example_3: The density of mercury is 13.6 g/mL. What is the mass in kilograms of a 2 L
commercial flask of mercury?
Set up the problem so that the calculation will yield a result with a mass in
grams.
13.6 g X 1000 mL X 2 L X 1 kg = 27.2 kg
1 mL
1L
1000 g