1.3: Measurements

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Learning Objectives

Explain the process of measurement and describe the four basic parts of a quantity.
Describe the properties and units of length, mass, volume, density, temperature, and time.
Recognize the common unit prefixes and use them to describe the magnitude of a measurement.
Describe and calculate the density of a substance.
Perform basic unit calculations and conversions in the International System of Units and other unit systems.

Measurements provide the macroscopic information that is the basis of most of the hypotheses, theories, and laws that describe 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 description of what is being measured (a unit); the scale of the measurement (a prefix attached to the unit); and an indication of the uncertainty of the measurement (the significant figures of the measurement). While the number and unit are explicitly represented when a quantity is written, the uncertainty is an aspect of how the measurement was made and is implicitly represented; uncertainty and significant figures will be discussed later.

The number in the measurement can be represented in different ways, including decimal form and scientific notation. For example, the maximum takeoff weight of a Boeing 777-200ER airliner is 298,000 kilograms, which can also be written in scientific notation as 2.98 \(\times\) 10⁵ kg. The mass of the average mosquito is about 0.0000025 kilograms, which can be written in scientific notation as 2.5 \(\times\) 10⁻⁶ kg.

Units, such as liters, pounds, and centimeters, are standards of comparison for measurements. When we buy a 2-liter bottle of a soft drink, we expect that the volume of the drink was measured, so it is two times larger than the volume that everyone agrees to be 1 liter. The meat used to prepare a 0.25-pound hamburger is measured so it weighs one-fourth as much as 1 pound. Specialized industries may develop their own units of measurement. Enthusiastic chili-eaters discuss the spiciness of a chili in terms of Scoville heat units (SHU). 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. Units should be detailed, until or unless context makes it clear. For example, if a recipe calls for three peppers, whether they are bell peppers (0 Scoville heat units), Poblano peppers (750 average SHU), Habanero peppers (580,000 average SHU), or Carolina Reaper peppers (1,570,000 average SHU) makes a large difference! However, if a recipe calls for salt then context should have you reaching for table salt, and not for rock salt used on roads!

We usually report the results of scientific measurements using units from a fixed standard call 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. The standards for these units are fixed by international agreement. SI units are an updated version of the metric system, using the units listed in Table \(\PageIndex{1}\). Other units can be derived from these base units.

Table \(\PageIndex{1}\): 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

Sometimes we use units that are fractions or multiples of a base unit. Ice cream is sold in quarts (a familiar, non-SI base unit), pints (0.5 quart), or gallons (4 quarts). We also use fractions or multiples of units in the SI system, These fractions or multiples reflect scale and are always powers of 10. Fractional or multiple SI units are named using a prefix and the name of the base unit. For example, a length of 1000 meters is also called a kilometer because the prefix kilo means “one thousand,” which in scientific notation is 10³ (1 kilometer = 1000 m = 10³ m). The prefixes used and the powers to which 10 are raised are listed in Table \(\PageIndex{2}\).

Table \(\PageIndex{2}\): Common Unit Prefixes
Prefix	Symbol	Factor	Example
femto	f	10⁻¹⁵	1 femtosec ond (fs) = 1 \(\times\) 10⁻¹⁵ s (0.000000000000001 s)
pico	p	10⁻¹²	1 picometer (pm) = 1 \(\times\) 10⁻¹² m (0.000000000001 m)
nano	n	10⁻⁹	4 nanograms (ng) = 4 \(\times\) 10⁻⁹ g (0.000000004 g)
micro	µ	10⁻⁶	1 microliter (μL) = 1 \(\times\) 10⁻⁶ L (0.000001 L)
milli	m	10⁻³	2 millimoles (mmol) = 2 \(\times\) 10⁻³ mol (0.002 mol)
centi	c	10⁻²	7 centimeters (cm) = 7 \(\times\) 10⁻² m (0.07 m)
deci	d	10⁻¹	1 deciliter (dL) = 1 \(\times\) 10⁻¹ L (0.1 L )
kilo	k	10³	1 kilometer (km) = 1 \(\times\) 10³ m (1000 m)
mega	M	10⁶	3 megahertz (MHz) = 3 \(\times\) 10⁶ Hz (3,000,000 Hz)
giga	G	10⁹	8 gigayears (Gyr) = 8 \(\times\) 10⁹ yr (8,000,000,000 Gyr)
tera	T	10¹²	5 terawatts (TW) = 5 \(\times\) 10¹² W (5,000,000,000,000 W)

SI Base Units

In 1742, a series of studies showed the English and French foot and pound measurements, supposedly the same units, had differences of 6% and 8%, respectively. A search began for a more universal system of units. After decades of debate, in 1791, the French General Assembly adopted the metre, 1/10,000,000 of the meridian passing through Paris from the equator to the North Pole, and the kilogram calculated from the mass of a cubic decimetre of water at its maximum density. The concurrent English and American governments decided to base their units of measurement on the length of a clock pendulum which takes one second to swing; they later also made the French metric system a legal system of measurement (the United States of America did so in 1866).

The world exhibition of Paris in 1867 saw the international community of scientists and industry recognize a greater uniformity between nations and the creation of a non-governmental 'Committee for Weights and Measures and for Money'. The French government began organizing an international commission in 1869, formative meetings began in 1872 prior to the break out of the Franco-Prussian War, and the Diplomatic Conference of the Meter finally occurred in 1875. It resulted in the Treaty of the Meter and established physical prototypes for the meter and kilogram.

a metal cylinder placed in a tightly secured glass bell jar. — Figure \(\PageIndex{2}\): This replica prototype kilogram is housed at the National Institute of Standards and Technology (NIST) in Maryland. (credit: National Institutes of Standards and Technology).

he following century and decades would see new units defined by physical constants of the universe. By 2008, the kilogram was the last unit not defined in reference to a universal constant, but by a platinum-iridium weight and six copies of it at the Bureau International des Poids et Mesures near Paris, France and 90 calibration standards across the world. Research into a precise, new kilogram began in 2014. The research also led to new definitions of the kelvin (temperature), mole (quantity), and ampere (electric current) units. The new definitions were ratified in 2018 and adopted by the international community in 2019. The new definitions helped establish a more complete model of measurement and precision. However, they are not practically different for everyday use, so don't go throwing out your meter sticks and scales!

Length

The standard unit of length in both the SI and original metric systems is the meter (m). It is now defined by taking the fixed numerical value of the speed of light in vacuum c to be 299,792,458 when expressed in the unit m/s, where the second is defined in terms of the fixed unperturbed ground-state hyperfine transition frequency of the caesium-133 atom. This results in a meter length that is approximately the distance light in a vacuum travels in 1/299,792,458 of a second. A meter is about 3 inches longer than a yard (Figure \(\PageIndex{1}\)); one meter is about 39.37 inches or 1.094 yards. Longer distances are often reported in kilometers (1 km = 1000 m = 10³ m), whereas shorter distances can be reported in centimeters (1 cm = 0.01 m = 10⁻² m) or millimeters (1 mm = 0.001 m = 10⁻³ m).

Mass

The standard unit of mass in the SI system is the kilogram (kg). It is now defined by taking the fixed value of Planck's constant (6.62607015 x 10⁻³⁴) when expressed in the unit Joule•second, which is equal to kg•m²•s^-1, where the meter and the second are defined in terms of the fixed speed of light in a vacuum and the fixed unperturbed ground-state hyperfine transition frequency of the caesium-133 atom. One kilogram is about 2.2 pounds. 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). The conversion between these two units and the Fahrenheit scale will be discussed later in this chapter.

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 microseconds = 0.000003 s = 3 \(\times\) 10⁻⁶ and 5 megaseconds = 5,000,000 s = 5 \(\times\) 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 \(\PageIndex{3}\)). 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.

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. Table \(\PageIndex{3}\) shows the densities of some common substances.

Table \(\PageIndex{3}\): 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

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.

\[\mathrm{density=\dfrac{mass}{volume}} \nonumber \]

Example \(\PageIndex{1}\)

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.

\[\mathrm{volume\: of\: lead\: cube=2.00\: cm\times2.00\: cm\times2.00\: cm=8.00\: cm^3} \nonumber \]

\[\mathrm{density=\dfrac{mass}{volume}=\dfrac{90.7\: g}{8.00\: cm^3}=\dfrac{11.3\: g}{1.00\: cm^3}=11.3\: g/cm^3} \nonumber \]

(We will discuss the reason for rounding to the first decimal place in the next section.)

Exercise \(\PageIndex{1}\)

1. To three decimal places, what is the volume of a cube (cm³) with an edge length of 0.843 cm?

2. 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?

Answer

1. 0.599 cm³

2. 8.91 g/cm³

Example \(\PageIndex{2}\): Using Displacement of Water to Determine Density

This PhET simulation illustrates another way to determine density, using displacement of water. Determine the density of the red and yellow blocks.

Solution

When you open the density simulation and select Same Mass, you can choose from several 5.00-kg colored blocks that you can drop into a tank containing 100.00 L water. The yellow block floats (it is less dense than water), and the water level rises to 105.00 L. While floating, the yellow block displaces 5.00 L water, an amount equal to the weight of the block. The red block sinks (it is more dense than water, which has density = 1.00 kg/L), and the water level rises to 101.25 L.

The red block therefore displaces 1.25 L water, an amount equal to the volume of the block. The density of the red block is:

\[\mathrm{density=\dfrac{mass}{volume}=\dfrac{5.00\: kg}{1.25\: L}=4.00\: kg/L} \nonumber \]

Note that since the yellow block is not completely submerged, you cannot determine its density from this information. But if you hold the yellow block on the bottom of the tank, the water level rises to 110.00 L, which means that it now displaces 10.00 L water, and its density can be found:

\[\mathrm{density=\dfrac{mass}{volume}=\dfrac{5.00\: kg}{10.00\: L}=0.500\: kg/L} \nonumber \]

Remove all of the blocks from the water and add the green block to the tank of water, placing it approximately in the middle of the tank. Determine the density of the green block.

Answer: 2.00 kg/L

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 the SI (International System) or metric systems. We use base SI 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, we find it convenient to use unit prefixes that yield fractional and multiple units, such as microseconds (10⁻⁶ seconds) and megahertz (10⁶ hertz), respectively.

Key Equations

\(\mathrm{density=\dfrac{mass}{volume}}\)

Exercise \(\PageIndex{1}\)

What are the four most commonly measured properties and their SI units?
For each of the following prefixes, what is the power of ten?
1. deci-
2. milli-
3. kilo-
4. micro-
5. mega-
6. nano-
7. centi-
For each of the following, what would be the appropriate SI unit to use for the measurement?
1. The time it takes to run 10 meter.
2. The height of a tree.
3. The weight of a puppy.
4. The temperature of boiling water.

Answer

What are the four most commonly measured properties and their SI units? For each of the following prefixes, what is the power of ten?

deci- 10¹
milli- 10^-3
kilo- 10³
micro- 10^-6
mega- 10⁶
nano- 10^-9
centi- 10^-2

For each of the following, what would be the appropriate SI unit to use for the measurement?

The time it takes to run 10 meter. Second
The height of a tree. Meter
The weight of a puppy. Kilogram
The temperature of boiling water. Degree celsius or kelvin

Glossary

Celsius (°C): unit of temperature; water freezes at 0 °C and boils at 100 °C on this scale

cubic centimeter (cm³ or cc): volume of a cube with an edge length of exactly 1 cm

cubic meter (m³): SI unit of volume

density: ratio of mass to volume for a substance or object

kelvin (K): SI unit of temperature; 273.15 K = 0 ºC

kilogram (kg): standard SI unit of mass; 1 kg = approximately 2.2 pounds

length: measure of one dimension of an object

liter (L): (also, cubic decimeter) unit of volume; 1 L = 1,000 cm³

meter (m): standard metric and SI unit of length; 1 m = approximately 1.094 yards

milliliter (mL): 1/1,000 of a liter; equal to 1 cm³

second (s): SI unit of time

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
volume: amount of space occupied by an object

References

Robild, Rosengren, and Bergdahl. Chilis: How to Grow, Harvest, and Cook with Your Favorite Hot Peppers, with 200 Varieties and 50 Spicy Recipes. Skyhorse Publishing., 2019.
Page, Chester H. and Vigoureux, Paul, eds. The International Bureau of Weights and Measures 1875-1975. U.S. Government Printing Office, 1975.
Ritter, Stephen K. “Redefining The Kilogram.” Chemical & Engineering News, vol. 86, issue 21, 2008.
Kemsley, Jyllian. “Rethinking The Mole And Kilogram.” Chemical & Engineering News, vol. 92, issue 34, 2014.
Howes, Laura. “New definitions for the kilogram and mole” Chemical & Engineering News, November 16, 2018, cen.acs.org/analytical-chemistry/New-definitions-kilogram-mole/96/web/2018/11 (accessed July 12, 2022).
The International System of Units, 9th edition, Bureau International des Poids et Mesures, 2019, ISBN 978-92-822-2272-0.