CK-12-Chemistry Intermediate

http://www.ck12.org Chapter 3. Measurements

Dimensional Analysis and Derived Units

Some units are combinations of SI base units. Aderived unitis a unit that results from a mathematical combination
of SI base units. We have already discussed volume and energy as two examples of derived units. Some others are
listed below (Table3.3).

TABLE3.3: Derived SI Units

Quantity Symbol Unit Unit Abbreviation Derivation

Area A square meter m^2 length×width

Volume V cubic meter m^3 length × width ×

height

Density D kilograms per cubic

meter

kg/m^3 mass / volume

Concentration c moles per liter mol/L amount / volume

Speed (velocity) v meters per second m/s length / time

Acceleration a meters per second

per second

m/s^2 speed / time

Force F newton N mass×acceleration

Energy E joule J force×length

Using dimensional analysis with derived units requires special care. When units are squared or cubed, as with area
or volume, the conversion factors themselves must also be squared. Shown below is the conversion factor for cubic
centimeters and cubic meters.

( 1 m

100 cm

) 3

=^1 m

3

106 cm^3 =^1

Because a cube has 3 sides, each side is subject to the conversion of 1 m to 100 cm. Since 100 cubed is equal to
1 million (10^6 ), there are 10^6 cm^3 in 1 m^3. Two convenient volume units are the liter, which is equal to a cubic
decimeter, and the milliliter, which is equal to a cubic centimeter. The conversion factor would be:

(

1 dm

10 cm

) 3

=^1 dm

3

1000 cm^3 =^1

There are thus 1000 cm^3 in 1 dm^3 , which is the same thing as saying there are 1000 mL in 1 L (Figure3.6).

FIGURE 3.6

There are 1000 cm^3 in 1 dm^3. Since 1

cm^3 is equal to 1 mL and 1 dm^3 is equal

to 1 L, we can say that there are 1000 mL

in 1 L.