Measurement

How Measurements Are Made

Units of measurement must be based on objective, physical, unchanging standards.

In every scientific discipline, scientists must have a way to make quantitative observations that can be communicated to other scientists. These quantitative observations are normally called measurements. Familiar measurements include distance, volume, mass, time, and temperature.

In order for the measurements to be understood by anyone and everyone, there must be a system that defines the size of standard units of measurement. The size of a given standard unit is arbitrary, but it works as long as everyone agrees on what it is. The standard unit of length, for example, is called a meter (m), and it is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. There is no particular reason requiring a meter to be this length; it could just as easily be defined as the distance light travels in 1/300,000,000 of a second. The key is that the scientific community must agree on what length will be called 1 meter and that there be an objective, unchanging reference for that distance. In this way, scientific measurements can be shared and communicated, understood by all.

For example, the length of a meter was previously defined as the distance between two lines marked on a bar of platinum-iridium alloy kept at the International Bureau of Weights and Measures (BIPM, Bureau International des Poids et Mesures) in France. The rod was kept in a carefully controlled atmosphere, supported on cylinders placed exactly 571 mm from each other. These precautions were taken to preserve the bar and keep it from being deformed in any way, because any change to the length of the rod would have then redefined the length of a meter. Changing the reference standard to be based on the speed of light in a vacuum is a more secure way to ensure that the definition of a meter does not change.

Systems of Measurement

The International System of Units (SI), abbreviated from French Système Internationale d'Unités, is used by all branches of science. This system provides a standardized set of units that can be used by scientists to ensure precise and objective definitions.

A set of units in which some units are defined by their relationship to other units in the system and not by a physical standard is called a system of measurement. There are many systems of measurement in use today. One familiar system is the British Imperial System, which contains units such as the inch (distance), the pound (weight), and the gallon (volume). The British system evolved from units people used as far back as the Middle Ages, which were based on references handy to them. The foot, for example, is so named because it was supposed to be equal to the length of a human foot.

By the 19th century, it was realized that the system needed to be standardized, and Britain passed the Weights and Measures Act of 1824, which created precise and objective definitions. The definitions were updated in 1963, so today an imperial gallon, for example, equals the space occupied by 10 pounds of distilled water at a density of 0.998859 g/mL.

Although its units are now precisely defined, the British Imperial System is not a good choice for scientific measurements. The units are related to one another in inconsistent ways. For example, 1 mile = 1760 yards, 1 yard = 3 feet, and 1 foot = 12 inches. These relationships are difficult to remember and tricky to use in calculations. A better system would have consistent relationships and names that follow simple rules. Fortunately, this system exists. The International System of Units (SI), which in French is Système Internationale d'Unités, is the system of units used by the global scientific community, based on seven fundamental units. A more general form of the International System of Units is the metric system. It improves on the Imperial British System because the units within a particular type of measurement, such as length or mass, are all based on powers of 10 and are therefore more easily related to each other.

SI Base Units

Unit Abbreviation Use
Meter m Length
Kilogram kg Mass
Second s Time
Ampere A Electric current
Kelvin K Temperature
Mole mol Quantity of matter
Candela cd Luminous intensity

Electric current is a measure of a flow of electric charge. Luminous intensity is the quantity of visible light emitted per unit time per unit solid angle. A mole is the amount of a substance that contains as many particles as 12 grams of pure carbon-12.

Each of the seven fundamental units of measurements in the International System of Units is an SI base unit. All other SI units can be derived from these seven base units and are therefore defined by their relationship to the base units. All the SI base units are defined by objective physical phenomena. Most of these units are expressed as a mathematical expression of the base units. For example, volume is simply measured in cubic meters (m3), but some derived units are specially named. The SI unit of force, for example, is the newton (N). The newton is defined to be the unit of force necessary to accelerate 1 kilogram at a rate of 1 meter per second per second, or 1N=1kgm/s2 1\;\rm{N}=1\;\rm{kg}\cdot\rm{m/s}^2. The SI unit of energy is the joule (J), which is equal to the energy transferred by 1 N of force acting on an object for the distance of 1 m. The expression relating these units is therefore 1J=1Nm=1kgm2/s2 1\;\rm J=1\;\rm N\cdot\rm m=1\;\rm{kg}\cdot\rm m^2/\rm s^2.

Examples of Derived SI Units with Names

Unit Abbreviation Equivalent to Use
Newton N kgm/s2{\rm{kg}} \cdot {\rm{m/s}}^2 Force
Pascal Pa N/m2=kg/(ms2){\rm{N/m}}{^2} = {\rm{kg/}}\left( {{\rm{ms}}^2} \right) Pressure
Joule J Nm=kgm2/s2{\rm{N}} \cdot {\rm{m}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/}}{{\rm{s}}^2} Energy
Watt W J/s=kgm2/s3{\rm{J/s}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/s}}^3 Power
Coulomb C As{\rm{A}} \cdot {\rm{s}} Electric charge
Volt V W/A=kgm2/(s3A){\rm{W/A}} = {\rm{kg}} \cdot {\rm{m}}{^2}{\rm{/}}\left( {{{\rm{s}}^3}{\rm{A}}} \right) Electric potential difference

SI base units can be combined to describe commonly measured properties in science. Examples of equivalencies between units show how units may be derived in different ways.

Finally, prefixes are used to define the magnitude of a given unit. The meter is a length that is convenient to use when describing the width of a room or the height of a person. Instead of creating an entirely new name for a unit of length that is appropriate for describing the distance between two distant cities, the prefix kilo-, meaning 1,000, is added to the base unit meter. The symbol for kilo- is k, so 1 kilometer (km) is equal to 1,000 m. Currently, there are 20 SI prefixes ranging from yocto (y, 1024) to yotta (Y, 1024).

SI Prefixes

Larger than the Base Unit
Prefix Abbreviation Factor
Yotta Y 1024
Zetta Z 1021
Exa E 1018
Peta P 1015
Tera T 1012
Giga G 109
Mega M 106
Kilo k 103
Hecta h 102
Deca da 101
Smaller than the Base Unit
Prefix Abbreviation Factor
Deci d 10–1
Centi c 10–2
Milli m 10–3
Micro µ 10–6
Nano n 10–9
Pico p 10–12
Femto f 10–15
Atto a 10–18
Zepto z 10–21
Yocto v 10–24

Prefixes are used in the metric system to increase or decrease units by powers of ten. The prefixes are never used in combination.

Mass, Weight, and Volume

Matter can be described by its mass, weight, and volume. Mass is the amount of matter in an object. Weight is the force of gravity acting on an object. Volume is the three-dimensional space occupied by a gas, a liquid, or a solid.
Mass (m) is a measure of the amount of matter in an object. The kilogram (kg) is the base SI unit of mass. Historically the kilogram has been defined as equal to the mass of the international prototype of the kilogram (IPK). The IPK is a polished cylinder made out of a platinum-iridium alloy, stored under three bell jars at the International Bureau of Weights and Measures in Sèvres, France. It must be stored carefully to protect it from contamination or particles that might settle on it and increase its mass and from physical damage that might chip or mar the prototype and possibly decrease its mass. The kilogram is the only base unit that has a prefix. The masses of materials used in chemistry labs are typically small enough to be measured in grams (g), not kilograms.

The International Prototype of the Kilogram

The international prototype of the kilogram (IPK) is stored under three bell jars in an atmosphere-controlled environment. The IPK is 39 mm in height and 39 mm in diameter.
Mass is not the same thing as weight. Weight (w) is a measure of the force of gravity acting on an object. Because weight is a force, the SI unit of weight is the newton (N). The weight of an object is equal to the mass (m) of the object times the acceleration due to gravity (g):
w=mgw=mg
The gravitational acceleration on the moon is 1/6 that of Earth, so 1 kg would weigh 1/6 as much on the moon as it does on Earth:
wmoon=mgmoon=m(16gEarth)=16wEarthw_{\rm{moon}}=mg_{\rm{moon}}=m\left(\frac{1}{6}\,g_{\rm{Earth}}\right)=\frac{1}{6}\,{w_{\rm{Earth}}}
Volume (V) is the amount of space occupied by a given mass. The SI unit for volume is a cubic meter (m3). In chemistry a more common unit of volume is the liter (L), which is equal to 0.001 m3. Although the liter is not officially an SI unit (it is part of the metric system), it is defined by an exact mathematical relationship to a base SI unit and is acceptable to use within SI. The standard SI prefixes can all be applied to the liter, so 1mL=0.001L1\;\rm{mL}=0.001\;\rm{L}, 1kL=1,000L1\;\rm{kL}=1,000\;\rm{L}, and so forth. A liter is a greater volume than is usually needed for a typical laboratory experiment. Chemists work more often with volumes in the range of 1–1,000 mL. A useful conversion is 1mL=1cm31\;\rm{ mL}=1\;\rm{ cm}^3. The amount of matter contained in a given volume is density (ρ\rho). Density is an intensive physical property of matter, which means it does not depend on the amount of matter present. In contrast, mass is an extensive physical property of matter, which means it depends on the amount of matter present. Density is therefore a useful way to characterize a substance. It is calculated by dividing the mass of a substance by its volume and is usually reported in g/mL or g/cm3. For example, the International Prototype of the Kilogram (IPK) is a cylinder with a height (h) of 3.9 cm. It also has a diameter of 3.9 cm, so its radius (r) is 1.95 cm. The density of the prototype can be calculated by dividing its mass by its volume (V):
ρprototype=mV=1kgπr2h=1,000gπ(1.95cm)2(3.9cm)21g/cm3=21g/mL\rho_{\rm{prototype}}=\frac{m}{V}=\frac{1\;{\rm{kg}}}{\pi{r^2h}}=\frac{1,\!000\;{\rm{g}}}{\pi(1.95\;{\rm{cm}})^2(3.9\;{\rm{cm}})}\approx21\;{\rm{g}}/{\rm{cm}^3}=21\;{\rm{g}}/{\rm{mL}}

Temperature

Common temperature scales are the Fahrenheit scale, the Celsius scale, and the Kelvin scale. Unlike the Fahrenheit and Celsius scales, the Kelvin scale has no negative temperatures, and no degree symbol is used when temperatures are reported in kelvins.

Two familiar temperature scales in everyday use are the Fahrenheit and Celsius scales. The Fahrenheit temperature scale is based on a freezing point of water of 32°F and a boiling point of water of 212°F at sea level. The interval between the freezing point of water and the boiling point of water is divided into 180 parts, and the size of each part is equal to one Fahrenheit degree. In developing the scale, the Polish-Dutch physicist Daniel Gabriel Fahrenheit (1686–1736) originally designated the freezing point of water to be 30°F and normal human body temperature to be 90°F, but based on more accurate measurements, these values eventually were adjusted.

The Celsius temperature scale is based on a freezing point of water of 0°C and a boiling point of water of 100°C at sea level. The interval is divided into 100 parts so that each part is equal to one Celsius degree. Because there are fewer divisions between the freezing and boiling points of water on the Celsius scale than on the Fahrenheit scale, a Celsius degree is larger than a Fahrenheit degree. Although the Fahrenheit scale is commonly used in the United States, the Celsius scale is appropriate for scientific work. Conversions between the two scales are made according to the following equations. (Note: 5/9 is a reduced fraction of 100/180, which accounts for the different size of a Fahrenheit degree and a Celsius degree.)
°C=59(°F32)°F=(95×°C)+32\begin{aligned}{}\degree\rm C&=\frac59(\degree\rm F-32)\\{}\degree\rm F&=\left(\frac95\times\degree\!\rm C\right)+32\end{aligned}
As an example of using these conversions, to determine whether the predicted high temperature of 8°C in Montreal, Canada, is warmer or colder than the predicted high of 45°F in Bangor, Maine, convert 8°C to Fahrenheit.
°F=(95×8)+32=46.4°F\degree\rm F=\left(\frac95\times8\right)+32=46.4\,\degree\rm F
Alternatively, convert 45°F to Celsius.
°C=59(4532)=7.2°C\degree\rm C=\frac59(45-32)=7.2\,\degree\rm C
Both methods show that Montreal will be slightly warmer than Bangor. The SI unit of temperature is the kelvin. The Kelvin temperature scale is an absolute temperature scale based on the Celsius scale but shifted so that the lowest possible temperature is 0 K (–273.15°C). The temperature 0 K, called absolute zero, is the minimum possible temperature theoretically achievable, at which there is no particle motion. There are no negative Kelvin temperatures. Notice that no degree symbol is used when temperatures are reported in kelvins. Because a kelvin is the same size as a Celsius degree, converting between them is a simple matter of addition or subtraction.
K=°C+273.15\rm K=\degree\rm C+273.15

The Fahrenheit, Celsius, and Kelvin Temperature Scales

Fahrenheit, Celsius, and Kelvin temperature scales can all be defined according to the freezing point and boiling point of water. The size of a degree on the Celsius and Kelvin scales is the same. The size of a degree on the Fahrenheit scale is smaller because more degrees are included in the range of temperatures.