0.1: Units of Measurement

Learning Objectives

• To identify the basic units of measurement of the seven fundamental properties

The natural sciences begin with observation, and this usually involves numerical measurements of quantities such as length, volume, density, and temperature. Most of these quantities have units of some kind associated with them, and these units must be retained when you use them in calculations. Measuring units can be defined in terms of a very small number of fundamental ones that, through "dimensional analysis", provide insight into their derivation and meaning, and must be understood when converting between different unit systems.

Units of measure

Have you ever estimated a distance by “stepping it off”— that is, by counting the number of steps required to take you a certain distance? Or perhaps you have used the width of your hand, or the distance from your elbow to a fingertip to compare two dimensions. If so, you have engaged in what is probably the first kind of measurement ever undertaken by primitive mankind. The results of a measurement are always expressed on some kind of a scale that is defined in terms of a particular kind of unit. The first scales of distance were likely related to the human body, either directly (the length of a limb) or indirectly (the distance a man could walk in a day).

As civilization developed, a wide variety of measuring scales came into existence, many for the same quantity (such as length), but adapted to particular activities or trades. Eventually, it became apparent that in order for trade and commerce to be possible, these scales had to be defined in terms of standards that would allow measures to be verified, and, when expressed in different units (bushels and pecks, for example), to be correlated or converted.

Figure 0.1.1: Units of length used in the modern world.

Over the centuries, hundreds of measurement units and scales have developed in the many civilizations that achieved some literate means of recording them. Some, such as those used by the Aztecs, fell out of use and were largely forgotten as these civilizations died out. Other units, such as the various systems of measurement that developed in England, achieved prominence through extension of the Empire and widespread trade; many of these were confined to specific trades or industries. The examples shown here are only some of those that have been used to measure length or distance. The history of measuring units provides a fascinating reflection on the history of industrial development.

The most influential event in the history of measurement was undoubtedly the French Revolution and the Age of Rationality that followed. This led directly to the metric system that attempted to do away with the confusing multiplicity of measurement scales by reducing them to a few fundamental ones that could be combined in order to express any kind of quantity. The metric system spread rapidly over much of the world, and eventually even to England and the rest of the U.K. when that country established closer economic ties with Europe in the latter part of the 20th Century. The United States is presently the only major country in which “metrication” has made little progress within its own society, probably because of its relative geographical isolation and its vibrant internal economy.

Science, being a truly international endeavor, adopted metric measurement very early on; engineering and related technologies have been slower to make this change, but are gradually doing so. Even within the metric system, however, a variety of units were employed to measure the same fundamental quantity; for example, energy could be expressed within the metric system in units of ergs, electron-volts, joules, and two kinds of calories. This led, in the mid-1960s, to the adoption of a more basic set of units, the Systeme Internationale (SI) units that are now recognized as the standard for science and, increasingly, for technology of all kinds.

The SI base units

In principle, any physical quantity can be expressed in terms of only seven base units. Each base unit is defined by a standard which is described in the NIST Web site.

Table 0.1.1: The Seven Base units of the SI system

Physical Quantity	Base Unit	symbol
length	meter	m
mass	kilogram	kg
time	second	s
temperature (absolute)	kelvin	K
amount of substance	mole	mol
electric current	ampere	A
luminous intensity	candela	cd

A few special points about some of these units are worth noting:

• The base unit of mass is unique in that a decimal prefix (see below) is built-in to it; i.e., the base SI unit is not the gram.
• The base unit of time is the only one that is not metric. Numerous attempts to make it so have never garnered any success; we are still stuck with the 24:60:60 system that we inherited from ancient times. The ancient Egyptians of around 1500 BC invented the 12-hour day, and the 60:60 part is a remnant of the base-60 system that the Sumerians used for their astronomical calculations around 100 BCE.
• Of special interest to Chemistry is the mole, the base unit for expressing the quantity of matter. Although the number is not explicitly mentioned in the official definition, chemists define the mole as Avogadro’s number (approximately 6.0210²³) of anything.

The SI decimal prefixes

Owing to the wide range of values that quantities can have, it has long been the practice to employ prefixes such as milli and mega to indicate decimal fractions and multiples of metric units. As part of the SI standard, this system has been extended and formalized.

Table 0.1.2: The SI Decimal Prefixes

prefix	abbreviation	multiplier	--	prefix	abbreviation	multiplier
peta	P	10^18		deci	s	10^–1
tera	T	10^12		centi	c	10^–2
giga	G	10^9		milli	m	10^–3
mega	M	10^6		micro	μ	10^–6
kilo	k	10^3		nano	n	10^–9
hecto	h	10^2		pico	p	10^–12
deca	da	10		femto	f	10^–15

Useful Non-Si Units

There is a category of units that are “honorary” members of the SI in the sense that it is acceptable to use them along with the base units defined above. These include such mundane units as the hour, minute, and degree (of angle), etc., but the three shown here are of particular interest to chemistry, and you will need to know them.

Table 0.1.3: Three Useful non-SI units in common use

liter (litre)	L	1 L = 1 dm^3 = 10^–3 m^3
metric ton	t	1 t = 10^3 kg
united atomic mass unit	u	1 u = 1.66054×10^–27 kg

Derived units and dimensions

Most of the physical quantities we actually deal with in science and also in our daily lives, have units of their own: volume, pressure, energy and electrical resistance are only a few of hundreds of possible examples. It is important to understand, however, that all of these can be expressed in terms of the SI base units; they are consequently known as derived units. In fact, most physical quantities can be expressed in terms of one or more of the following five fundamental units:

• mass M
• length L
• time T
• electric charge Q
• temperature Θ (theta)

Consider, for example, the unit of volume, which we denote as V. To measure the volume of a rectangular box, we need to multiply the lengths as measured along the three coordinates:

\[V = x \cdot y \cdot z \tag{0.1.1}\]

We say, therefore, that volume has the dimensions of length-cubed:

dim.V = L³

Thus the units of volume will be m³ (in the SI) or cm³, ft³ (English), etc. Moreover, any formula that calculates a volume must contain within it the L³ dimension; thus the volume of a sphere is 4/3 πr³. The dimensions of a unit are the powers which M, L, T, Q and Θ must be given in order to express the unit. Thus, dim.V = M⁰L³T⁰Q⁰ Θ⁰, as given above.

Example 0.4.1

Find the dimensions of energy.

Solution

When mechanical work is performed on a body, its energy increases by the amount of work done, so the two quantities are equivalent and we can concentrate on work. The latter is the product of the force applied to the object and the distance it is displaced. From Newton’s law, force is the product of mass and acceleration, and the latter is the rate of change of velocity, typically expressed in meters per second per second. Combining these quantities and their dimensions yields the result shown here.

E has units of kg m² s^–2

dim.E = M L² T^–2

Table 0.1.4: Dimensions of units commonly used in Chemistry

Q	M	L	T	quantity	SI unit, other typical units
1				electric charge	coulomb
	1			mass	kilogram, gram, metric ton, pound
		1		length	meter, foot, mile
			1	time	second, day, year
		3		volume	liter, cm^3, quart, fluidounce
	1	–3		density	kg m^–3, g cm^–3
	1	1	–2	force	newton, dyne
	1	–1	–2	pressure	pascal, atmosphere, torr
	1	2	–2	energy	joule, erg, calorie, electron-volt
	1	2	–3	power	watt
1	1	2	–2	electric potential	volt
1			–1	electric current	ampere
1	1	1	–2	electric field intensity	volt m^–1
–2	1	2	–1	electric resistance	ohm
2	1	3	–1	electric resistivity
2	–1	–2	1	electric conductance	siemens, mho

Why are unit Dimensions useful?

There are several reasons why it is worthwhile to consider the dimensions of a unit.

1. Perhaps the most important use of dimensions is to help us understand the relations between various units of measure and thereby get a better understanding of their physical meaning. For example, a look at the dimensions of the frequently confused electrical terms resistance and resistivity should enable you to explain, in plain words, the difference between them.
2. By the same token, the dimensions essentially tell you how to calculate any of these quantities, using whatever specific units you wish. (Note here the distinction between dimensions and units.)
3. Just as you cannot add apples to oranges, an expression such as a = b + cx² is meaningless unless the dimensions of each side are identical. (Of course, the two sides should work out to the same units as well.)
4. Many quantities must be dimensionless— for example, the variable x in expressions such as log x, ex, and sin x. Checking through the dimensions of such a quantity can help avoid errors.

The formal, detailed study of dimensions is known as dimensional analysis and is a topic in any basic physics course.

Unit Conversions

Dimensional analysis is widely employed when it is necessary to convert one kind of unit into another, and chemistry students often use it in "chemical arithmetic" calculations, in which context it is also known as the "Factor-Label" method.

Units and their ranges in Chemistry

In this section, we will look at some of the quantities that are widely encountered in Chemistry, and at the units in which they are commonly expressed. In doing so, we will also consider the actual range of values these quantities can assume, both in nature in general, and also within the subset of nature that chemistry normally addresses. In looking over the various units of measure, it is interesting to note that their unit values are set close to those encountered in everyday human experience

Mass vs. Weight

These two quantities are widely confused. Although they are often used synonymously in informal speech and writing, they have different dimensions: weight is the force exerted on a mass by the local gravational field:

\[f = m a = m g \tag{0.1.2}\]

where g is the acceleration of gravity. While the nominal value of the latter quantity is 9.80 m s^–2 at the Earth’s surface, its exact value varies locally. Because it is a force, the SI unit of weight is properly the newton, but it is common practice (except in physics classes!) to use the terms "weight" and "mass" interchangeably, so the units kilograms and grams are acceptable in almost all ordinary laboratory contexts.

Figure 0.1.1: Mass scales used from atomic to cosmic levels. Note that in this diagram and in those that follow, the numeric scale represents the logarithm of the number shown. For example, the mass of the electron is 10^–30 kg.

The range of masses spans 90 orders of magnitude, more than any other unit. The range that chemistry ordinarily deals with has greatly expanded since the days when a microgram was an almost inconceivably small amount of material to handle in the laboratory; this lower limit has now fallen to the atomic level with the development of tools for directly manipulating these particles. The upper level reflects the largest masses that are handled in industrial operations, but in the recently developed fields of geochemistry and enivonmental chemistry, the range can be extended indefinitely. Flows of elements between the various regions of the environment (atmosphere to oceans, for example) are often quoted in teragrams.

Length

Chemists tend to work mostly in the moderately-small part of the distance range. Those who live in the lilliputian world of crystal- and molecular structures and atomic radii find the picometer a convenient currency, but one still sees the older non-SI unit called the Ångstrom used in this context; 1Å = 10^–10 m = 100pm. Nanotechnology, the rage of the present era, also resides in this realm. The largest polymeric molecules and colloids define the top end of the particulate range; beyond that, in the normal world of doing things in the lab, the centimeter and occasionally the millimeter commonly rule.

Figure 0.1.2: Length scales used from atomic to cosmic levels. Note that the numeric scale represents the logarithm of the number shown.

Time

For humans, time moves by the heartbeat; beyond that, it is the motions of our planet that count out the hours, days, and years that eventually define our lifetimes. Beyond the few thousands of years of history behind us, those years-to-the-powers-of-tens that are the fare for such fields as evolutionary biology, geology, and cosmology, cease to convey any real meaning for us. Perhaps this is why so many people are not very inclined to accept their validity.

Figure 0.1.3: Time scales used from atomic to cosmic levels. Note that the numeric scale represents the logarithm of the number shown.

Most of what actually takes place in the chemist’s test tube operates on a far shorter time scale, although there is no limit to how slow a reaction can be; the upper limits of those we can directly study in the lab are in part determined by how long a graduate student can wait around before moving on to gainful employment. Looking at the microscopic world of atoms and molecules themselves, the time scale again shifts us into an unreal world where numbers tend to lose their meaning. You can gain some appreciation of the duration of a nanosecond by noting that this is about how long it takes a beam of light to travel between your two outstretched hands. In a sense, the material foundations of chemistry itself are defined by time: neither a new element nor a molecule can be recognized as such unless it lasts around sufficiently long enough to have its “picture” taken through measurement of its distinguishing properties.

Temperature

Temperature, the measure of thermal intensity, spans the narrowest range of any of the base units of the chemist’s measure. The reason for this is tied into temperature’s meaning as a measure of the intensity of thermal kinetic energy. Chemical change occurs when atoms are jostled into new arrangements, and the weakness of these motions brings most chemistry to a halt as absolute zero is approached. At the upper end of the scale, thermal motions become sufficiently vigorous to shake molecules into atoms, and eventually, as in stars, strip off the electrons, leaving an essentially reaction-less gaseous fluid, or plasma, of bare nuclei (ions) and electrons.

Figure 0.1.4: Temperature scales used from atomic to cosmic levels. Note that the numeric scale represents the logarithm of the number shown.

Temperature Scales: the Degree

The degree is really an increment of temperature, a fixed fraction of the distance between two defined reference points on a temperature scale.

Note

Although rough means of estimating and comparing temperatures have been around since AD 170, the first mercury thermometer and temperature scale were introduced in Holland in 1714 by Gabriel Daniel Fahrenheit. Fahrenheit established three fixed points on his thermometer. Zero degrees was the temperature of an ice, water, and salt mixture, which was about the coldest temperature that could be reproduced in a laboratory of the time. When he omitted salt from the slurry, he reached his second fixed point when the water-ice combination stabilized at "the thirty-second degree." His third fixed point was "found as the ninety-sixth degree, and the spirit expands to this degree when the thermometer is held in the mouth or under the armpit of a living man in good health.

• After Fahrenheit died in 1736, his thermometer was recalibrated using 212 degrees, the temperature at which water boils, as the upper fixed point. Normal human body temperature registered 98.6 rather than 96.

• In 1743, the Swedish astronomer Anders Celsius devised the aptly-named centigrade scale that places exactly 100 degrees between the two reference points defined by the freezing- and boiling points of water.

Temperature comparisons and conversions

When we say that the temperature is so many degrees, we must specify the particular scale on which we are expressing that temperature. A temperature scale has two defining characteristics, both of which can be chosen arbitrarily:

• The temperature that corresponds to 0° on the scale;
• The magnitude of the unit increment of temperature– that is, the size of the degree.

To express a temperature given on one scale in terms of another, it is necessary to take both of these factors into account.

The key to temperature conversions is easy if you bear in mind that between the so-called ice- and steam points of water there are 180 Fahrenheit degrees, but only 100 Celsius degrees, making the F° 100/180 = 5/9 the magnitude of the C° Note the distinction between “°C” (a temperature) and “C°” (a temperature increment).

Because the ice point is at 32°F, the two scales are offset by this amount. If you remember this, there is no need to memorize a conversion formula; you can work it out whenever you need it.

Absolute Temperature scales

Near the end of the 19th Century when the physical significance of temperature began to be understood, the need was felt for a temperature scale whose zero really means zero— that is, the complete absence of thermal motion. This gave rise to the absolute temperature scale whose zero point is –273.15 °C, but which retains the same degree magnitude as the Celsius scale. This eventually got renamed after Lord Kelvin (William Thompson); thus the Celsius degree became the kelvin. Thus we can now express an increment such as five C° as “five kelvins”

Note

In 1859 the Scottish engineer and physicist William J. M. Rankine proposed an absolute temperature scale based on the Fahrenheit degree. Absolute zero (0° Ra) corresponds to –459.67°F. The Rankine scale has been used extensively by those same American and English engineers who delight in expressing heat capacities in units of BTUs per pound per F°.

The importance of absolute temperature scales is that absolute temperatures can be entered directly in all the fundamental formulas of physics and chemistry in which temperature is a variable.

Pressure

Pressure is the measure of the force exerted on a unit area of surface. Its SI units are therefore newtons per square meter, but we make such frequent use of pressure that a derived SI unit, the pascal, is commonly used:

\[1\; Pa = \dfrac{1\; N}{m^2}\]

Atmosphere Pressure

The concept of pressure first developed in connection with studies relating to the atmosphere and vacuum that were first carried out in the 17th century

Figure 0.1.5: Origin of atmospheric pressure results from the combined force of all the air directly above an object.

The molecules of a gas are in a state of constant thermal motion, moving in straight lines until experiencing a collision that exchanges momentum between pairs of molecules and sends them bouncing off in other directions. This leads to a completely random distribution of the molecular velocities both in speed and direction— or it would in the absence of the Earth’s gravitational field which exerts a tiny downward force on each molecule, giving motions in that direction a very slight advantage. In an ordinary container this effect is too small to be noticeable, but in a very tall column of air the effect adds up: the molecules in each vertical layer experience more downward-directed hits from those above it. The resulting force is quickly randomized, resulting in an increased pressure in that layer which is then propagated downward into the layers below.

At sea level, the total mass of the sea of air pressing down on each 1-cm² of surface is about 1034 g, or 10340 kg m^–2. The force (weight) that the Earth’s gravitional acceleration g exerts on this mass is

f = ma = mg = (10340 kg)(9.81 m s^–2) = 1.013 × 105 kg m s^–2 = 1.013 × 105 newtons

resulting in a pressure of 1.013 × 10⁵ n m^–2 = 1.013 × 10⁵ pa. The actual pressure at sea level varies with atmospheric conditions, so it is customary to define standard atmospheric pressure as 1 atm = 1.013 x 10⁵ pa or 101 kpa. Although the standard atmosphere is not an SI unit, it is still widely employed. In meteorology, the bar, exactly 1.000 × 10⁵ = 0.967 atm, is often used.

Figure 0.1.6: Pressure scales. Note that the numeric scale represents the logarithm of the number shown.

The barometer

In the early 17th century, the Italian physicist and mathematician Evangalisto Torricelli invented a device to measure atmospheric pressure. The Torricellian barometer consists of a vertical glass tube closed at the top and open at the bottom. It is filled with a liquid, traditionally mercury, and is then inverted, with its open end immersed in the container of the same liquid. The liquid level in the tube will fall under its own weight until the downward force is balanced by the vertical force transmitted hydrostatically to the column by the downward force of the atmosphere acting on the liquid surface in the open container. Torricelli was also the first to recognize that the space above the mercury constituted a vacuum, and is credited with being the first to create a vacuum.

One standard atmosphere will support a column of mercury that is 760 mm high, so the “millimeter of mercury”, now more commonly known as the torr, has long been a common pressure unit in the sciences: 1 atm = 760 torr.

Summary

Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially imortant that you know the precise meanings of all the highlighted terms in the context of this topic.

• Describe the names and abbreviations of the SI base units and the SI decimal prefixes.
• Define the liter and the metric ton in these units.
• Explain the meaning and use of unit dimensions; state the dimensions of volume.
• State the quantities that are needed to define a temperature scale, and show how these apply to the Celsius, Kelvin, and Fahrenheit temperature scales.
• Explain how a Torricellian barometer works.

Contributions

Stephen Lower, Professor Emeritus (Simon Fraser U.) Chem1 Virtual Textbook