# Derived SI Units

We can derive many units from the seven SI base units. For example, we can use the base unit of length to define a unit of volume, and the base units of mass and length to define a unit of density.

## Volume

Volume is the measure of the amount of space occupied by an object. The standard SI unit of volume is defined by the base unit of length (see image below). The standard volume is a cubic meter (m3), a cube with an edge length of exactly one meter. To dispense a cubic meter of water, we could build a cubic box with edge lengths of exactly one meter. This box would hold a cubic meter of water or any other substance.

A more commonly used unit of volume is derived from the decimeter (0.1 m, or 10 cm). A cube with edge lengths of exactly one decimeter contains a volume of one cubic decimeter (dm3). A liter (L) is the more common name for the cubic decimeter. One liter is about 1.06 quarts.

A cubic centimeter (cm3) is the volume of a cube with an edge length of exactly one centimeter. The abbreviation cc (for cubic centimeter) is often used by health professionals. A cubic centimeter is also called a milliliter (mL) and is 1/1000 of a liter.

(a) The relative volumes are shown for cubes of 1 m3, 1 dm3 (1 L), and 1 cm3 (1 mL) (not to scale). (b) The diameter of a dime is compared relative to the edge length of a 1-cm3 (1-mL) cube. Image credit: OpenStax, Chemistry

## Density

We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the base units of mass and length.

The density of a substance is the ratio of the mass of a sample of the substance to its volume. The SI unit for density is the kilogram per cubic meter (kg/m3). For many situations, however, this as an inconvenient unit, and we often use grams per cubic centimeter (g/cm3) for the densities of solids and liquids, and grams per liter (g/L) for gases.

Although there are exceptions, most liquids and solids have densities that range from about 0.7 g/cm3 (the density of gasoline) to 19 g/cm3 (the density of gold). The density of air is about 1.2 g/L. The table shows the densities of some common substances.

### Densities of Common Substances

SolidsLiquidsGases (at 25 °C and 1 atm)
ice (at 0 °C) 0.92 g/cm3water 1.0 g/cm3dry air 1.20 g/L
oak (wood) 0.60–0.90 g/cm3ethanol 0.79 g/cm3oxygen 1.31 g/L
iron 7.9 g/cm3acetone 0.79 g/cm3nitrogen 1.14 g/L
copper 9.0 g/cm3glycerin 1.26 g/cm3carbon dioxide 1.80 g/L
lead 11.3 g/cm3olive oil 0.92 g/cm3helium 0.16 g/L
silver 10.5 g/cm3gasoline 0.70–0.77 g/cm3neon 0.83 g/L
gold 19.3 g/cm3mercury 13.6 g/cm3radon 9.1 g/L

While there are many ways to determine the density of an object, perhaps the most straightforward method involves separately finding the mass and volume of the object, and then dividing the mass of the sample by its volume. In the following example, the mass is found directly by weighing, but the volume is found indirectly through length measurements.

$$\text{density} = \cfrac{\text{mass}}{\text{volume}}$$

### Calculation of Density

Gold—in bricks, bars, and coins—has been a form of currency for centuries. In order to swindle people into paying for a brick of gold without actually investing in a brick of gold, people have considered filling the centers of hollow gold bricks with lead to fool buyers into thinking that the entire brick is gold. It does not work: Lead is a dense substance, but its density is not as great as that of gold, 19.3 g/cm3. What is the density of lead if a cube of lead has an edge length of 2.00 cm and a mass of 90.7 g?

#### Solution

The density of a substance can be calculated by dividing its mass by its volume. The volume of a cube is calculated by cubing the edge length.

$$\text{volume of lead cube}$$$$= 2.00 \text{cm} × 2.00 \text{cm} × 2.00 \text{cm}$$$$= 8.00 \text{cm}^3$$

$$\text{density}$$$$= \cfrac{\text{mass}}{\text{volume}}$$$$= \cfrac{90.7 \text{g}}{8.00 \text{cm}^3}$$$$= \cfrac{11.3 \text{g}}{1.00 \text{cm}^3}$$\(= 11.3 \text{g/cm}^3

(We will discuss the reason for rounding to the first decimal place in one of the next few lessons.)