# 1.7: The International System of Units

- Page ID
- 15193

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- Define and calculate density.

People who live in the United States measure weight in pounds, height in feet and inches, and a car’s speed in miles per hour. In contrast, chemistry and other branches of science use the International System of Units (also known as **SI** after *Système Internationale d’Unités*), which was established so that scientists around the world could communicate efficiently with each other. Many countries have also adopted SI units for everyday use as well. The United States is one of the few countries that has not.

## Base SI Units

**Base (or basic) units**, are the fundamental units of SI. There are seven base units, which are listed in Table \(\PageIndex{1}\), Chemistry uses five of the base units: the mole for amount, the kilogram for mass, the meter for length, the second for time, and the kelvin for temperature. The degree Celsius (°C) is also commonly used for temperature. The numerical relationship between kelvins and degrees Celsius is as follows:

\[K = °C + 273 \label{Eq1} \]

Property |
Unit |
Abbreviation |
---|---|---|

length | meter | m |

mass | kilogram | kg |

time | second | s |

amount | mole | mol |

temperature | kelvin | K |

electrical current | ampere | amp |

luminous intensity | candela | cd |

The United States uses the English (sometimes called Imperial) system of units for many quantities. Inches, feet, miles, gallons, pounds, and so forth, are all units connected with the English system of units. There have been many mistakes due to the improper conversion of units between the SI and English systems.

The size of each base unit is defined by international convention. For example, the *kilogram* is defined as the quantity of mass of a special metal cylinder kept in a vault in France (Figure \(\PageIndex{1}\)). The other base units have similar definitions and standards. The sizes of the base units are not always convenient for all measurements. For example, a meter is a rather large unit for describing the width of something as narrow as human hair. Instead of reporting the diameter of hair as 0.00012 m or as 1.2 × 10^{−4} m using scientific notation as discussed in section 1.4, SI also provides a series of **prefixes** that can be attached to the units, creating units that are larger or smaller by powers of 10.

Common prefixes and their multiplicative factors are listed in Table \(\PageIndex{2}\). (Perhaps you have already noticed that the base unit *kilogram* is a combination of a prefix, kilo- meaning 1,000 ×, and a unit of mass, the gram.) Some prefixes create a multiple of the original unit: 1 kilogram equals 1,000 grams, and 1 megameter equals 1,000,000 meters. Other prefixes create a fraction of the original unit. Thus, 1 centimeter equals 1/100 of a meter, 1 millimeter equals 1/1,000 of a meter, 1 microgram equals 1/1,000,000 of a gram, and so forth.

Prefix |
Abbreviation |
Multiplicative Factor |
Multiplicative Factor in Scientific Notation |
---|---|---|---|

giga- | G | 1,000,000,000 × | 10^{9} × |

mega- | M | 1,000,000 × | 10^{6} × |

kilo- | k | 1,000 × | 10^{3} × |

deca- | D | 10 × | 10^{1} × |

deci- | d | 1/10 × | 10^{−1} × |

centi- | c | 1/100 × | 10^{−2} × |

milli- | m | 1/1,000 × | 10^{−3} × |

micro- | µ* | 1/1,000,000 × | 10^{−6} × |

nano- | n | 1/1,000,000,000 × | 10^{−9} × |

*The letter µ is the Greek lowercase letter for m and is called “mu,” which is pronounced “myoo.” |

Both SI units and prefixes have abbreviations, and the combination of a prefix abbreviation with a base unit abbreviation gives the abbreviation for the modified unit. For example, kg is the abbreviation for kilogram. We will be using these abbreviations throughout this book.

The mass of a body is a measure of its inertial property or how much matter it contains. The weight of a body is a measure of the force exerted on it by gravity or the force needed to support it. Gravity on earth gives a body a downward acceleration of about 9.8 m/s^{2}. In common parlance, weight is often used as a synonym for mass in weights and measures. For instance, the verb “to weigh” means “to determine the mass of” or “to have a mass of.” The incorrect use of weight in place of mass should be phased out, and the term mass used when mass is meant. The SI unit of mass is the kilogram (kg). In science and technology, the weight of a body in a particular reference frame is defined as the force that gives the body an acceleration equal to the local acceleration of free fall in that reference frame. Thus, the SI unit of the quantity weight defined in this way (force) is the newton (N).

## Derived SI Units

**Derived units** are combinations of SI base units. Units can be multiplied and divided, just as numbers can be multiplied and divided. For example, the area of a square having a side of 2 cm is 2 cm × 2 cm, or 4 cm^{2} (read as “four centimeters squared” or “four square centimeters”). Notice that we have squared a length unit, the centimeter, to get a derived unit for area, the square centimeter.

Volume is an important quantity that uses a derived unit. **Volume** is the amount of space that a given substance occupies and is defined geometrically as length × width × height. Each distance can be expressed using the meter unit, so volume has the derived unit m × m × m, or m^{3} (read as “meters cubed” or “cubic meters”). A cubic meter is a rather large volume, so scientists typically express volumes in terms of 1/1,000 of a cubic meter. This unit has its own name—the liter (L). A liter is a little larger than 1 US quart in volume. Below are approximate equivalents for some of the units used in chemistry.

Approximate Equivalents to Some SI Units

- 1 m ≈ 39.36 in. ≈ 3.28 ft ≈ 1.09 yd
- 1 in. ≈ 2.54 cm
- 1 km ≈ 0.62 mi
- 1 kg ≈ 2.20 lb
- 1 lb ≈ 454 g
- 1 L ≈ 1.06 qt
- 1 qt ≈ 0.946 L

Figure \(\PageIndex{2}\): The relative lengths of 1 m, 1 yd, 1 cm, and 1 in. are shown (not actual size), as well as comparisons of 2.54 cm and 1 in., and of 1 m and 1.094 yd. (CC BY 4.0; OpenStax)

As shown in Figure \(\PageIndex{3}\), a liter is also 1,000 cm^{3}. By definition, there are 1,000 mL in 1 L, so 1 milliliter and 1 cubic centimeter represent the same volume.

\[1\; mL = 1\; cm^3 \label{Eq2} \]

Figure \(\PageIndex{3}\): Units of Volume. (a) The relative volumes are shown for cubes of 1 m^{3}, 1 dm

^{3}(1 L), and 1 cm

^{3}(1 mL) (not to scale). A liter (L) is defined as a cube 1 dm (1/10th of a meter) on a side. A milliliter (mL), 1/1,000th of a liter, is equal to 1 cubic centimeter. (b) The diameter of a dime is compared relative to the edge length of a 1-cm

^{3}(1-mL) cube. (CC BY 4.0; OpenStax)

Give the abbreviation for each unit and define the abbreviation in terms of the base unit.

- kiloliter
- microsecond
- decimeter
- nanogram

**Answer a**-
The abbreviation for a kiloliter is kL. Because kilo means “1,000 ×,” 1 kL equals 1,000 L.

**Answer b**-
The abbreviation for microsecond is µs. Micro implies 1/1,000,000th of a unit, so 1 µs equals 0.000001 s.

**Answer c**-
The abbreviation for decimeter is dm. Deci means 1/10th, so 1 dm equals 0.1 m.

**Answer d**-
The abbreviation for nanogram is ng and equals 0.000000001 g.

Give the abbreviation for each unit and define the abbreviation in terms of the base unit.

- kilometer
- milligram
- nanosecond
- centiliter

**Answer a**- km (1,000 m)
**Answer b**- mg (0.001 g)
**Answer c**- ns (0.000000001 s)
**Answer d**- cL (0.01L)

**Energy**, another important quantity in chemistry, is the ability to perform work, such as moving a box of books from one side of a room to the other side. It has a derived unit of kg•m^{2}/s^{2}. (The dot between the kg and m units implies the units are multiplied together.) Because this combination is cumbersome, this collection of units is redefined as a **joule** (J). An older unit of energy, but likely more familiar to you, the calorie (cal), is also widely used. There are 4.184 J in 1 cal. Energy changes occur during all chemical processes and will be discussed in a later chapter.

The food in our diet provides the energy our bodies need to function properly. The energy contained in food could be expressed in joules or calories, which are the conventional units for energy, but the food industry prefers to use the kilocalorie and refers to it as the Calorie (with a capital C). The average daily energy requirement of an adult is about 2,000–2,500 Calories, which is 2,000,000–2,500,000 calories (with a lowercase c).

If we expend the same amount of energy that our food provides, our body weight remains stable. If we ingest more Calories from food than we expend, however, our bodies store the extra energy in high-energy-density compounds, such as fat, and we gain weight. On the other hand, if we expend more energy than we ingest, we lose weight. Other factors affect our weight as well—genetic, metabolic, behavioral, environmental, cultural factors—but dietary habits are among the most important.

In 2008 the US Centers for Disease Control and Prevention issued a report stating that 73% of Americans were either overweight or obese. More alarmingly, the report also noted that 19% of children aged 6–11 and 18% of adolescents aged 12–19 were overweight—numbers that had tripled over the preceding two decades. Two major reasons for this increase are excessive calorie consumption (especially in the form of high-fat foods) and reduced physical activity. Partly because of that report, many restaurants and food companies are working to reduce the amounts of fat in foods and provide consumers with more healthy food options.

**Density** is defined as the mass of an object divided by its volume; it describes the amount of matter contained in a given amount of space.

\[\mathrm{density=\dfrac{mass}{volume}}\label{Eq3} \]

Thus, the units of density are the units of mass divided by the units of volume: g/cm^{3} or g/mL (for solids and liquids), g/L (for gases), kg/m^{3}, and so forth. For example, the density of water is about 1.00 g/cm^{3}, while the density of mercury is 13.6 g/mL. (Remember that 1 mL equals 1 cm^{3}.) Mercury is over 13 times as dense as water, meaning that it contains over 13 times the amount of matter in the same amount of space. The density of air at room temperature is about 1.3 g/L. Table 1.6.3 shows the densities of some common substances.

Solids |
Liquids |
Gases (at 25 °C and 1 atm) |
---|---|---|

ice (at 0 °C) 0.92 g/cm^{3} |
water 1.0 g/cm^{3} |
dry air 1.20 g/L |

oak (wood) 0.60–0.90 g/cm^{3} |
ethanol 0.79 g/cm^{3} |
oxygen 1.31 g/L |

iron 7.9 g/cm^{3} |
acetone 0.79 g/cm^{3} |
nitrogen 1.14 g/L |

copper 9.0 g/cm^{3} |
glycerin 1.26 g/cm^{3} |
carbon dioxide 1.80 g/L |

lead 11.3 g/cm^{3} |
olive oil 0.92 g/cm^{3} |
helium 0.16 g/L |

silver 10.5 g/cm^{3} |
gasoline 0.70–0.77 g/cm^{3} |
neon 0.83 g/L |

gold 19.3 g/cm^{3} |
mercury 13.6 g/cm^{3} |
radon 9.1 g/L |

What is the density of a section of bone if a 25.3 cm^{3} sample has a mass of 27.8 g?

###### Solution

Because density is defined as the mass of an object divided by its volume, we can set up the following relationship:

\[ \begin{align*} \mathrm{density} &=\dfrac{mass}{volume} \\[4pt] &= \dfrac{27.8\:g}{25.3\:cm^3} \\[4pt] &=1.10\:g/cm^3 \end{align*} \nonumber \]

Note that we have limited our final answer to three significant figures.

What is the density of oxygen gas if a 15.0 L sample has a mass of 21.7 g?

**Answer**- 1.45 g/L

Density can be used to convert between the mass and the volume of a substance. This will be discussed in the next section.

## Concept Review Exercises

- What is the difference between a base unit and a derived unit? Give two examples of each type of unit.

2. Do units follow the same mathematical rules as numbers do? Give an example to support your answer.

3. What is density?

## Answers

- Base units are the seven fundamental units of SI; derived units are constructed by making combinations of the base units; Two examples of base units: kilograms and meters (answers will vary); Two examples of derived units: grams per milliliter and joules (answers will vary).

2. yes; \(\mathrm{mL\times\dfrac{g}{mL}=g}\) (answers will vary)

3. Density is defined as the mass of an object divided by its volume

## Key Takeaways

- Recognize the SI base units and derived units.
- Combining prefixes with base units creates new units of larger or smaller sizes.