2.2: Units for Measurements

Learning Objectives

• Express quantities properly, using a number and a unit.
• State the different measurement systems used in chemistry.

Measurements

A coffee maker’s instructions tell you to fill the coffeepot with 4 cups of water and to use 3 scoops of coffee. When you follow these instructions, you are measuring. When the nurse checks your temperature, height, weight, and blood pressure at the doctor’s office, (Figure \(\PageIndex{1}\)), the nurse is also measuring.

Figure \(\PageIndex{1}\): Measuring Blood Pressure. A nurse or a doctor measuring a patient’s blood pressure is taking a measurement. (GFDL; Pia von Lützau).

Chemists measure the properties of matter and express these measurements as quantities. A quantity is an amount of something and consists of a number and a unit. The number tells us how many (or how much), and the unit tells us what the scale of measurement is. For example, when a distance is reported as “5 kilometers,” we know that the quantity has been expressed in units of kilometers and that the number of kilometers is 5. If you ask a friend how far they walk from home to school, and the friend answers “12” without specifying a unit, you do not know whether your friend walks 12 kilometers, 12 miles, 12 furlongs, or 12 yards. Both a number and a unit must be included to express a quantity properly.

Example \(\PageIndex{1}\)

Identify the number and the unit in each quantity.

1. one dozen eggs
2. 2.54 centimeters
3. a box of pencils
4. 88 meters per second

Solution

1. The number is one, and the unit is dozen eggs.
2. The number is 2.54, and the unit is centimeter.
3. The number 1 is implied because the quantity is only a box. The unit is box of pencils.
4. The number is 88, and the unit is meters per second. Note that in this case the unit is actually a combination of two units: meters and seconds.

Units

Measurements provide the macroscopic information that serves as the basis for 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 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 and 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 as 2.98 \(\times\) 10⁵ kg. The mass of the average mosquito is about 0.0000025 kilograms, which can be written 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 the volume of the drink to be measured, so it is twice as large as the volume that everyone agrees is 1 liter. The meat used to prepare a 0.25-pound hamburger is measured so it weighs one-fourth as much as 1 pound. 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.

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 adopted 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.

Unfortunately, The Metric System is Not Ubiquitous

The map below illustrates the adoption of the SI units in countries worldwide. The United States has legally adopted the metric system for measurements, but it is not widely used in everyday practice. Great Britain and much of Canada use a combination of metric and imperial units.

Figure \(\PageIndex{2}\): Areas of world using metric system (in green). Only a few countries are slow or resistant to adoption including the United States.

These units are frequently encountered in science. We usually report the results of scientific measurements in SI units, an updated version of the metric system, using the units listed in Table \(\PageIndex{1}\). Other units can be derived from these base units. 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.

Table \(\PageIndex{1}\): Base Units of the SI System

Property Measured	Name of Unit	Symbol of the Unit
length	meter	m
mass	kilogram	kg
time	second	s
temperature	kelvin	K
amount of substance	mole	mol

Metric Prefixes

Conversions between metric system units are straightforward because the system is based on powers of ten. For example, meters, centimeters, and millimeters are all metric units of length. There are 10 millimeters in 1 centimeter and 100 centimeters in 1 meter. Metric prefixes are used to distinguish between units of different size, these fractions or multiples 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
angstrom	Å	10^−10	5 angstrom (Å) =5 \(\times\) 10^−10 m (0.0000000001 m)
nano	n	10^−9	1 nanogram (ng) = 1 \(\times\) 10^−9 g (0.000000001 g)
micro	µ (mc)	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	10^3	1 kilometer (km) = 1 \(\times\) 10^3 m (1000 m)
mega	M	10^6	3 megahertz (MHz) = 3 \(\times\) 10^6 Hz (3,000,000 Hz)
giga	G	10^9	1 Gigabyte (Gb) = 1 \(\times\) 10^9 b (1,000,000,000 b)

Note

In the field of medicine, the abbreviation for the metric prefix micro is mc.

Exercise \(\PageIndex{1}\)

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

1. kilometer
2. milligram
3. nanosecond
4. centiliter

Answer a:

a. km. b. mg. c. ns. d. cL

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 travels in a vacuum 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).

Figure \(\PageIndex{3}\): 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.

Mass

The standard unit of mass in the SI system is the kilogram (kg). A kilogram was originally defined as the mass of a liter of water (a cube of water with an edge length of exactly 0.1 meter). In 1889, it was redefined by a certain cylinder of platinum-iridium alloy, which was kept in France (Figure \(\PageIndex{2}\)). Any object with the same mass as this cylinder was said to have a mass of 1 kilogram (which can lead to uncertainties unacceptable to the precision of modern instrumentation). 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). Over the past 100 years, the IPK has lost 50 millionths of a gram - a seemingly negligible amount, but something that has caused it to be lighter - or all standard replicas to be heavier - and changing the definition of a kilogram in the process. As all balances in the world are standardized to this value, it is important that this value, itself, be standard. On May 20, 2019, a new definition will be used for the kilogram, based on the unchanging Planck's constant._¹

Figure \(\PageIndex{4}\): This replica prototype kilogram is housed at the National Institute of Standards and Technology (NIST) in Maryland. (credit: National Institute of Standards and Technology).

Video \(\PageIndex{2}\): For more information on the new definition of the kilogram, check out this video!

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.

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.

Figure \(\PageIndex{5}\): (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.

Temperature

Temperature is a measure of how hot or cold an object is relative to another object (its thermal energy content), whereas heat is the flow of thermal energy between objects with different temperatures. Temperature is an intensive property. Temperature is an important parameter in chemistry. When a substance changes from solid to liquid, it is because there is an increase in the temperature of the material. Chemical reactions usually proceed faster if the temperature is increased. Many unstable materials (such as enzymes) will be viable longer at too low or too high temperatures.

The conversion between these two units and the Fahrenheit scale will be discussed later in this chapter.

The Fahrenheit Scale

The first thermometers were glass and contained alcohol, which expanded and contracted as the temperature changed. The German scientist, Daniel Gabriel Fahrenheit used mercury in the tube, an idea put forth by Ismael Boulliau. The Fahrenheit scale was first developed in 1724 and tinkered with for some time after that. The freezing point of water was defined as \(32^\text{o} \text{F}\) and the boiling point as \(212^\text{o} \text{F}\). The Fahrenheit scale is typically not used for scientific purposes.

The Celsius Scale

The degree Celsius (°C) is also allowed in the SI system, with both the word “degree” and the degree symbol used for Celsius measurements. The Celsius scale sets the freezing point and boiling point of water at \(0^\text{o} \text{C}\) and \(100^\text{o} \text{C}\) respectively. The distance between those two points is divided into 100 equal intervals, each of which is one degree. Another term sometimes used for the Celsius scale is "centigrade" because there are 100 degrees between the freezing and boiling points of water on this scale. However, the preferred term is "Celsius".

The Kelvin Scale

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 (°). It is based on molecular motion, with the temperature of \(0 \: \text{K}\), also known as absolute zero, being the point where all molecular motion ceases. The freezing point of water on the Kelvin scale is \(273.15 \: \text{K}\), while the boiling point is \(373.15 \: \text{K}\). 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).

Notice that there is no "degree" used in the temperature designation. Unlike the Fahrenheit and Celsius scales, where temperatures are referred to as "degrees \(\text{F}\)" or "degrees \(\text{C}\)", we simply designate temperatures in the Kelvin scale as kelvins.

Figure \(\PageIndex{6}\): A Comparison of the Fahrenheit, Celsius, and Kelvin Temperature Scales. Because the difference between the freezing point of water and the boiling point of water is 100° on both the Celsius and Kelvin scales, the size of a degree Celsius (°C) and a kelvin (K) are precisely the same. In contrast, both a degree Celsius and a kelvin are 9/5 the size of a degree Fahrenheit (°F). (CC BY-SA-NC 3.0; anonymous)

Converting Between Temperature Scales

The Kelvin is the same size as the Celsius degree, so measurements are easily converted from one to the other. The freezing point of water is 0°C = 273.15 K; the boiling point of water is 100°C = 373.15 K. The Kelvin and Celsius scales are related as follows:

\[T \,\text{(in °C)} + 273.15 = T \, \text{(in K)} \tag{1.7.1} \label{1.7.1}\]

\[T \; \text{ (in K)} − 273.15 = T \; \text{(in °C)} \tag{1.7.2} \label{1.7.2} \]

Degrees on the Fahrenheit scale, however, are based on an English tradition of using 12 divisions, just as 1 ft = 12 in. The relationship between degrees Fahrenheit and degrees Celsius is as follows: where the coefficient for degrees Fahrenheit is exact. (Some calculators have a function that allows you to convert directly between °F and °C.) There is only one temperature for which the numerical value is the same on both the Fahrenheit and Celsius scales: −40°C = −40°F. The relationship between the scales are as follows:

\[°C = \dfrac{(°F-32)}{1.8} \tag{1.7.3} \label{1.7.3}\]

\[°F = 1.8 \times (°C)+32 \tag{1.7.4} \label{1.7.4} \]

Example \(\PageIndex{1}\): Temperature Conversions

A student is ill with a temperature of 103.5°F. What is her temperature in °C and K?

Solution

Converting from Fahrenheit to Celsius requires the use of Equation \ref{1.7.3}:

(103.5°F - 32) / 1.8 = 39.7 °C

Converting from Celsius to Kelvin requires the use of Equation \ref{1.7.1}:

39.7 °C + 273.15 = 312.9 K

Exercise \(\PageIndex{1}\)

Convert each temperature to °C and °F.

1. the temperature of the surface of the sun (5800. K)
2. the boiling point of gold (3080. K)
3. the boiling point of liquid nitrogen (77.36 K)

Answer (a)

5527 ^oC, 9981. °F

Answer (b)

2807 °C, 5084 °F

Answer (c)

-195.79 °C, -320.42 °F

Units of Energy

Energy is measured in one of two common units: the calorie and the joule. The joule \(\left( \text{J} \right)\) is the SI unit of energy. The calorie is familiar because it is commonly used when referring to the amount of energy contained within food. A calorie \(\left( \text{cal} \right)\) is the quantity of heat required to raise the temperature of 1 gram of water by \(1^\text{o} \text{C}\). For example, raising the temperature of \(100 \: \text{g}\) of water from \(20^\text{o} \text{C}\) to \(22^\text{o} \text{C}\) would require 100 g x 2^oC = 200 cal.

Calories contained within food are actually kilocalories \(\left( \text{kcal} \right)\). In other words, if a certain snack contains 85 food calories, it actually contains \(85 \: \text{kcal}\) or \(85,000 \: \text{cal}\). In order to make the distinction, the dietary calorie is written with a capital C.

\[1 \: \text{kilocalorie} = 1 \: \text{Calorie} = 1000 \: \text{calories}\]

To say that the snack "contains" 85 Calories means that \(85 \: \text{kcal}\) of energy are released when that snack is processed by your body.

Heat changes in chemical reactions are typically measured in joules rather than calories. The conversion between a joule and a calorie is shown below.

1 cal=4.184 J

We can calculate the amount of heat released in kilojoules when a 400 Calorie hamburger is digested.

\[400 \: \text{Cal} = 400 \: \text{kcal} \times \dfrac{4.184 \: \text{kJ}}{1 \: \text{kcal}} = 1.67 \times 10^3 \: \text{kJ}\]