One important parameter that characterizes a macroscopic system is its volume V, the amount of space the system occupies. The SI unit of volume is ${\text{m}}^3$. Nonetheless, both ${\text{cm}}^3$ and, to some extent, liters (L) are widely used metric units of volume. In most cases, you must convert these to ${\text{m}}^3$ before doing calculations.

While it is true that $1\text{m}=100\text{cm}$, it is not true that $1{\text{m}}^3=100\text{c}{\text{m}}^3$. FIGURE 14.1 shows that the volume conversion factor is $1{\text{m}}^3={10}^6\text{c}{\text{m}}^3$. A liter is $1000{\text{cm}}^3$, so $1{\text{m}}^3={10}^3\text{L}$. A milliliter (1 mL) is the same as $1{\text{cm}}^3$.

FIGURE 14.1 There are ${10}^6\text{c}{\text{m}}^3$ in $1{\text{m}}^3$.

A system is also characterized by its density. Suppose you have several blocks of copper, each of different size. Each block has a different mass $m$ and a different volume V. Nonetheless, all the blocks are copper, so there should be some quantity that has the same value for all the blocks, telling us, “This is copper, not some other material.” The most important such parameter is the ratio of mass to volume, which we call the mass density $\rho$ (lowercase Greek rho):

Conversely, an object of density $\rho$ has mass $m=\rho V$.

The SI units of mass density are $\text{kg}\text{/}{\text{m}}^3$. Nonetheless, units of $\text{g}\text{/}\text{c}{\text{m}}^3$ are widely used. You need to convert these to SI units before doing most calculations. You must convert both the grams to kilograms and the cubic centimeters to cubic meters. The net result is the conversion factor

The mass density is usually called simply “the density” if there is no danger of confusion. However, we will meet other types of density as we go along, such as charge density and current density, and sometimes it is important to be explicit about which density you are using. TABLE 14.1 provides a short list of mass densities of various fluids. Notice the enormous difference between the densities of gases and liquids. Gases have lower densities because the molecules in gases are farther apart than in liquids.

What does it mean to say that the density of gasoline is $680\text{kg}\text{/}{\text{m}}^3$ or, equivalently, $0.68\text{g}\text{/}\text{c}{\text{m}}^3?$ Density is a mass-to-volume ratio. It is often described as the “mass per unit volume,” but for this to make sense you have to know what is meant by “unit volume.” Regardless of which system of length units you use, a unit volume is one of those units cubed. For example, if you measure lengths in meters, a unit volume is $1{\text{m}}^3$. But $1{\text{cm}}^3$ is a unit volume if you measure lengths in centimeters, and $1{\text{mi}}^3$ is a unit volume if you measure lengths in miles.

Density is the mass of one unit of volume, whatever the units happen to be. To say that the density of gasoline is $680\text{kg}\text{/}{\text{m}}^3$ is to say that the mass of $1{\text{m}}^3$ of gasoline is 680 kg. The mass of $1{\text{cm}}^3$ of gasoline is 0.68 g, so the density of gasoline in those units is $0.68\text{g}\text{/}\text{c}{\text{m}}^3$.

The mass density is independent of the object’s size. Mass and volume are parameters that characterize a specific piece of some substance—say copper—whereas the mass density characterizes the substance itself. All pieces of copper have the same mass density, which differs from the mass density of any other substance.