1.4 Measurements
Learning Objectives
By the end of this section, you will be able to:
- Explain the process of measurement
- Identify the three basic parts of a quantity
- Describe the properties and units of length, mass, temperature, volume, density, and time
- Perform basic unit calculations and conversions in the metric system
Measurements provide much of the information that informs the hypotheses, theories, and laws describing the behavior of matter and energy in both the macroscopic and microscopic domains of chemistry. Every measurement provides three kinds of information: the size or magnitude of the measurement (a number); a standard of comparison for the measurement (a unit); and an indication of the uncertainty of the measurement. While the number and unit are explicitly represented when a quantity is written, the uncertainty is an aspect of the measurement result that is more implicitly represented.
The number in the measurement can be represented in different ways, including decimal form and scientific notation. (Scientific notation is also known as exponential notation; a review of this topic can be found in Appendix B.) For example, the maximum takeoff weight of a Boeing 777-200ER airliner is 298,000 kg, which can also be written as 2.98 × 105 kg. The mass of the average mosquito is about 0.0000025 kg, which can be written as 2.5 × 10−6 kg.
Units, such as liters and centimeters, are standards of comparison for measurements. A 2-L bottle of a soft drink contains a volume of beverage that is twice that of the accepted volume of 1 L. Without units, a number can be meaningless, confusing, or possibly life threatening. Suppose a doctor prescribes phenobarbital to control a patient’s seizures and states a dosage of “100” without specifying units. Not only will this be confusing to the medical professional giving the dose, but the consequences can be dire: 100 mg given three times per day can be effective as an anticonvulsant, but a single dose of 100 g is more than 10 times the lethal amount.
The measurement units for seven fundamental properties (“base units”) are listed in Table 1.2. The standards for these units are fixed by international agreement, and they are called the International System of Units or SI Units (from the French, Le Système International d’Unités). SI units have been used by the United States National Institute of Standards and Technology (NIST) since 1964. Units for other properties may be derived from these seven base units.
| Base Units of the SI System | ||
|---|---|---|
| Property Measured | Name of Unit | Symbol of Unit |
| length | meter | m |
| mass | kilogram | kg |
| time | second | s |
| temperature | kelvin | K |
| electric current | ampere | A |
| amount of substance | mole | mol |
| luminous intensity | candela | cd |
Table 1.2
SI units are named using a prefix and the name of the base unit. For example, a length of 1000 m is also called a kilometer because the prefix kilo means “one thousand,” which in scientific notation is 103 (1 km = 1000 m = 1 x 103 m). The prefixes used and the powers to which 10 are raised are listed in Table 1.3.
| Common Unit Prefixes | |||
|---|---|---|---|
| Prefix | Symbol | Factor | Example |
| femto | f | 10−15 | 1 femtosecond (fs) = 1 × 10−15 s (0.000000000000001 s) |
| pico | p | 10−12 | 1 picometer (pm) = 1 × 10−12 m (0.000000000001 m) |
| nano | n | 10−9 | 4 nanograms (ng) = 4 × 10−9 g (0.000000004 g) |
| micro | µ | 10−6 | 1 microliter (μL) = 1 × 10−6 L (0.000001 L) |
| milli | m | 10−3 | 2 millimoles (mmol) = 2 × 10−3 mol (0.002 mol) |
| centi | c | 10−2 | 7 centimeters (cm) = 7 × 10−2 m (0.07 m) |
| deci | d | 10−1 | 1 deciliter (dL) = 1 × 10−1 L (0.1 L ) |
| kilo | k | 103 | 1 kilometer (km) = 1 × 103 m (1000 m) |
| mega | M | 106 | 3 megahertz (MHz) = 3 × 106 Hz (3,000,000 Hz) |
| giga | G | 109 | 8 gigayears (Gyr) = 8 × 109 yr (8,000,000,000 yr) |
| tera | T | 1012 | 5 terawatts (TW) = 5 × 1012 W (5,000,000,000,000 W) |
Table 1.3
SI Base Units
The initial units of the metric system, which eventually evolved into the SI system, were established in France during the French Revolution. The original standards for the meter and the kilogram were adopted there in 1799 and eventually by other countries. This section introduces four of the SI base units commonly used in chemistry. Other SI units will be introduced in subsequent chapters.
Length
The standard unit of length in both the SI and original metric systems is the meter (m). A meter was originally specified as 1/10,000,000 of the distance from the North Pole to the equator. It is now defined as the distance light in a vacuum travels in 1/299,792,458 of a second. Longer distances are often reported in kilometers (1 km = 1000 m = 1 x 103 m), whereas shorter distances can be reported in centimeters (1 cm = 0.01 m = 1 x 10−2 m) or millimeters (1 mm = 0.001 m = 1 x 10−3 m).
Mass
The standard unit of mass in the SI system is the kilogram (kg). The kilogram was previously defined by the International Union of Pure and Applied Chemistry (IUPAC) as the mass of a specific reference object. This object was originally one liter of pure water, and more recently it was a metal cylinder made from a platinum-iridium alloy with a height and diameter of 39 mm. In May 2019, this definition was changed to one that is based instead on precisely measured values of several fundamental physical constants.1 The gram (g) is exactly equal to 1/1000 of the mass of the kilogram (10−3 kg).
Temperature
Temperature is an intensive property. The SI unit of temperature is the kelvin (K). The IUPAC convention is to use kelvin (all lowercase) for the word, K (uppercase) for the unit symbol, and neither the word “degree” nor the degree symbol (°). The degree Celsius (°C) is also allowed in the SI system, with both the word “degree” and the degree symbol used for Celsius measurements. Celsius degrees are the same magnitude as those of kelvin, but the two scales place their zeros in different places. Water freezes at 273.15 K (0 °C) and boils at 373.15 K (100 °C) by definition, and normal human body temperature is approximately 310 K (37 °C).
Time
The SI base unit of time is the second (s). Small and large time intervals can be expressed with the appropriate prefixes; for example, 3 ms = 0.000003 s = 3 × 10−6 and 5 Ms = 5,000,000 s = 5 × 106 s. Alternatively, hours, days, and years can be used.
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 (Figure 1.21). 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.
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 equivalent to a milliliter (mL) and is 1/1000 of a liter.
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 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. Table 1.4 shows the densities of some common substances.
| Densities of Common Substances | ||
|---|---|---|
| Solids | Liquids | Gases (at 25 °C and 1 atm) |
| ice (at 0 °C) 0.92 g/cm3 | water 1.0 g/cm3 | dry air 1.20 g/L |
| oak (wood) 0.60–0.90 g/cm3 | ethanol 0.79 g/cm3 | oxygen 1.31 g/L |
| iron 7.9 g/cm3 | acetone 0.79 g/cm3 | nitrogen 1.14 g/L |
| copper 9.0 g/cm3 | glycerin 1.26 g/cm3 | carbon dioxide 1.80 g/L |
| lead 11.3 g/cm3 | olive oil 0.92 g/cm3 | helium 0.16 g/L |
| silver 10.5 g/cm3 | gasoline 0.70–0.77 g/cm3 | neon 0.83 g/L |
| gold 19.3 g/cm3 | mercury 13.6 g/cm3 | radon 9.1 g/L |
Table 1.4
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.
Example 1.1
Calculation of Density:
Gold 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.
(We will discuss the reason for rounding to the first decimal place in the next section.)
Check Your Learning:
(a) To three decimal places, what is the volume of a cube (cm3) with an edge length of 0.843 cm?
(b) If the cube in part (a) is copper and has a mass of 5.34 g, what is the density of copper to two decimal places?
(a) 0.599 cm3; (b) 8.91 g/cm3
Key Concepts and Summary
Measurements provide quantitative information that is critical in studying and practicing chemistry. Each measurement has an amount, a unit for comparison, and an uncertainty. Measurements can be represented in either decimal or scientific notation. Scientists primarily use SI (International System) units such as meters, seconds, and kilograms, as well as derived units, such as liters (for volume) and g/cm3 (for density). In many cases, it is convenient to use prefixes that yield fractional and multiple units, such as ms (1 x 10−3 s) and MHz (1 x 106 Hz), respectively.
Key Equations
- density = mass/volume
Footnotes
- 1For details see https://www.nist.gov/pml/weights-and-measures/si-units-mass
Glossary
- Celsius (°C)
- unit of temperature; water freezes at 0 °C and boils at 100 °C on this scale
- density
- ratio of mass to volume for a substance or object
- kelvin (K)
- SI unit of temperature; 273.15 K = 0 ºC
- liter (L)
- (also, cubic decimeter) unit of volume; 1 L = 1,000 cm3
- SI units (International System of Units)
- standards fixed by international agreement in the International System of Units (Le Système International d’Unités)
- unit
- standard of comparison for measurements
