What we are actually comparing is the mass per unit volume, that is, the density. In order to determine these densities, we might weigh a cubic centimeter of each material. If the styrofoam weighed 0.10 g and the aerogel 0.001 g, we could describe the density of styrofoam as 0.10 g cm–3 and that of an aerogel as 0.001 g cm–3. (Note that the negative exponent in the units cubic centimeters indicates a reciprocal. Thus 1 cm–3 = 1/cm3 and the units for our densities could be written as , g/cm3, or g cm–3. In each case the units are read as grams per cubic centimeter, the per indicating division.) We often abbreviate "cm3" as "cc", and 1 cm3 = 1 mL exactly, by definition.
In general it is not necessary to weigh exactly 1 cm3 of a material in order to determine its density. We simply measure mass and volume and divide volume into mass:
where ρ = density m = mass V = volume
EXAMPLE 1.9 Calculate the density of (a) a piece of aerogel whose mass is 0.0167 g and which, when submerged, increases the water level in a graduated cylinder by 13.9 ml; (b) a styrofoam cylinder of mass 1.3918 g, radius 0.750 cm, and height 5.25 cm.
a) Since the submerged metal displaces its own volume,
b) The volume of the cylinder must be calculated first, using the formula
V = π r2h = 3.142 × (0.750 cm)2 × 5.25 cm = 9.278 718 8 cm3
|Substance||Density/g/cm3 at 20oC|
|styrofoam||0.03 - 0.12|
|[| Sulfur hexafluoride]||0.006164|
|[| Halon 1301]||.0066|
Note that unlike mass or volume, the density of a substance is independent of the size of the sample. Thus density is a property by which one substance can be distinguished from another. A sample of pure aluminum can be trimmed to any desired volume or adjusted to have any mass we choose, but its density will always be 2.70 g/cm3 at 20°C. The densities of some common pure substances are listed in the Table.
Tables and graphs are designed to provide a maximum of information in a minimum of space. When a physical quantity (number × units) is involved, it is wasteful to keep repeating the same units. Therefore it is conventional to use pure numbers in a table or along the axes of a graph. A pure number can be obtained from a quantity if we divide by appropriate units. For example, when divided by the units gram per cubic centimeter, the density of aluminum becomes a pure number 2.70:
Therefore, a column in a table or the axis of a graph is conveniently labeled in the following form:
This indicates the units that must be divided into the quantity to yield the pure number in the table or on the axis. This has been done in the second column of the Table.