Getting started

From CK12 - Flexbooks

Classification and properties of matter

Matter is “anything that has mass and occupies space,” we were taught in school. True enough, but not very satisfying. A more complete answer is unfortunately far beyond the scope of this course, but we will offer a hint of it in the later section on atomic structure. For the moment, let’s side-step definition of matter and focus on the chemist’s view: matter is what chemical substances are composed of. But what do we mean by chemical substances? How do we organize our view of matter and its roperties? These will be the subjects of this lesson.

Observable properties of matter

The science of chemistry developed from observations made about the nature and behavior of different kinds of matter, which we refer to collectively as the properties of matter.

The properties we refer to in this lesson are all macroscopic properties: those that can be observed in bulk matter. At the microscopic level, matter is of course characterized by its structure: the spatial arrangement of the individual atoms in a molecular unit or an extended solid.

The study of matter begins with the study of its properties

By observing a sample of matter and measuring its vicharious properties, we gradually acquire enough information to characterize it; to distinguish it from other kinds of matter. This is the first step in the development of chemical science, in which interest is focused on specific kinds of matter and the transformations between them.

Extensive and intensive properties

If you think about the various observable properties of matter, it will become apparent that these fall into two classes. Some properties, such as mass and volume, depend on the quantity of matter in the sample we are studying. Clearly, these properties, as important as they may be, cannot by themselves be used to characterize a kind of matter; to say that “water has a mass of ” is nonsense, although it may be quite true in a particular instance. Properties of this kind are called extensive properties of matter.

This definition of the density illustrates an important general rule: the ratio of two extensive properties is always an intensive property.

Suppose we make further measurements, and find that the same quantity of water whose mass is also occupies a volume of . We have measured two extensive properties (mass and volume) of the same sample of matter. This allows us to define a new quantity, the quotient m/V which defines another property of water which we call the density. Unlike the mass and the volume, which by themselves refer only to individual samples of water, the density (mass per unit volume) is a property of all samples of pure water at the same temperature. Density is an example of an intensive property of matter.

Intensive properties are extremely important, because every possible kind of matter possesses a unique set of intensive properties that distinguishes it from every other kind of matter. Some intensive properies can be determined by simple observations: color (absorption spectrum), melting point, density, solubility, acidic or alkaline nature, and density are common examples. Even more fundamental, but less directly observable, is chemical composition.

The more intensive properties we know, the more precisely we can characterize a sample of matter.

Intensive properties are extremely important, because every possible kind of matter possesses a unique set of intensive properties that distinguishes it from every other kind of matter. In other words, intensive properties serve to characterize matter. Many of the intensive properties depend on such variables as the temperature and pressure, but the ways in which these properties change with such variables can themselves be regarded as intensive properties.

Classify each of the following as an extensive or intensive property.

The volume of beer in a mug ext; depends on size of the mug.

The percentage of alcohol in the beer int; same for any same-sized sample.

The number of calories of energy you derive from eating a banana ext; depends on size and sugar content of the banana.

The number of calories of energy made available to your body when you consume of sugar int; same for any portion of sugar.

The mass of iron present in your blood ext; depends on volume of blood in the body.

The mass of iron present in of your blool int; the same for any sample.

The electrical resistance of a piece of 22-gauge copper wire. ext; depends on length of the wire.

The electrical resistance of a length of 22-gauge copper wire int; same for any length of the same wire.

The pressure of air in a bicycle tire pressure itself is intensive, but is also dependent on the quantity of air in the tire.

The last example shows that not everything is black or white!

But we often encounter matter whose different parts exhibit different sets of intensive properties. This brings up another distinction that we address immediately below.

Classification of matter

One useful way of organizing our understanding of matter is to think of a hierarchy that extends down from the most general and complex to the simplest and most fundamental. The orange-colored boxes represent the central realm of chemistry, which deals ultimately with specific chemical substances, but as a practical matter, chemical science extends both above and below this region.

Alternatively, it is sometimes more useful to cast our classification into two dimensions:

• Homogeneous vs. heterogeneous
• Pure substance vs. mixture

Both dimensions are defined in terms of intensive properties, so if you are not sure what these are, be sure to re-read the material in the preceding section. We will begin by looking at the distinction represented in the top line of the diagram.

Homogeneous and heterogeneous: it's a matter of phases

Homogeneous matter (from the Greek homo = same) can be thought of as being uniform and continuous, whereas heterogeneous matter (hetero = different) implies non-uniformity and discontinuity. To take this further, we first need to define “uniformity” in a more precise way, and this takes us to the concept of phases.

A phase is a region of matter that possesses uniform intensive properties throughout its volume. A volume of water, a chunk of ice, a grain of sand, a piece of copper— each of these constitutes a single phase, and by the above definition, is said to be homogeneous.

A sample of matter can contain more than a single phase; a cool drink with ice floating in it consists of at least two phases, the liquid and the ice. If it is a carbonated beverage, you can probably see gas bubbles in it that make up a third phase.

Phase boundaries

Each phase in a multiphase system is separated from its neighbors by a phase boundary, a thin region in which the intensive properties change discontinuously. Have you ever wondered why you can easily see the ice floating in a glass of water although both the water and the ice are transparent? The answer is that when light crosses a phase boundary, its direction of travel is slightly bent, and a portion of the light gets reflected back; it is these reflected and distorted light rays emerging from that reveal the chunks of ice floating in the liquid.

If, instead of visible chunks of material, the second phase is broken into tiny particles, the light rays usually bounce off the surfaces of many of these particles in random directions before they emerge from the medium and are detected by the eye. This phenomenon, known as scattering, gives multiphase systems of this kind a cloudy appearance, rendering them translucent instead of transparent. Two very common examples are ordinary fog, in which water droplets are suspended in the air, and milk, which consists of butterfat globules suspended in an aqueous solution.

Getting back to our classification, we can say that

Homogeneous matter consists of a single phase throughout its volume; heterogeneous matter contains two or more phases.

Dichotomies (“either-or” classifications) often tend to break down when closely examined, and the distinction between homogeneous and heterogeneous matter is a good example; this really a matter of degree, since at the microscopic level all matter is made up of atoms or molecules separated by empty space! For most practical purposes, we consider matter as homogeneous when any discontinuities it contains are too small to affect its visual appearance.

How large must a molecule or an agglomeration of molecules be before it begins to exhibit properties of a being a separate phase? Such particles span the gap between the micro and macro worlds, and have been known as colloids since they began to be studied around 1900. But with the development of nanotechnology in the 1990s, this distinction has become even more fuzzy.

Pure substances and mixtures

The air around us, most of the liquids and solids we encounter, and all too much of the water we drink consists not of pure substances, but of mixtures. You probably have a general idea of what a mixture is, and how it differs from a pure substance; what is the scientific criterion for making this distinction?

To a chemist, a pure substance usually refers to a sample of matter that has a distinct set of properties that are common to all other samples of that substance. A good example would be ordinary salt, sodium chloride. No matter what its source (from a mine, evaporated from seawater, or made in the laboratory), all samples of this substance, once they have been purified, possess the same unique set of properties.

A pure substance is one whose intensive properties are the same in any purified sample of that same substance.

A mixture, in contrast, is composed of two or more substances, and it can exhibit a wide range of properties depending on the relative amounts of the components present in the mixture. For example, you can dissolve up to of salt in one litre of water at room temperature, making possible an infinite variety of “salt water” solutions. For each of these concentrations, properties such as the density, boiling and freezing points, and the vapor pressure of the resulting solution will be different.

Is anything really pure?

“ Pure: It Floats”

This description of Ivory Soap is a classic example of junk science from the 19th century. Not only is the term “pure” meaningless when applied to an undefined mixture such as hand soap, but the implication that its ability to float is evidence of this purity is deceptive. The low density is achieved by beating air bubbles into it, actually reducing the “purity” of the product and in a sense cheating the consumer.

We all prefer to drink “pure” water, but we don't usually concern ourselves with the dissolved atmospheric gases and ions that are present in most drinking waters. These same substances could seriously interfere with certain uses to which we put water in the laboratory, were we customarily use distilled or de-ionized water. But even this still contains some dissolved gases and occasionally some silica, but their small amounts and relative inertness make these impurities insignificant for most purposes. When water of the highest obtainable purity is required for certain types of exacting measurements, it is commonly filtered, de-ionized, and triple-vacuum distilled. But even this “chemically pure” water is a mixture of isotopic species: there are two stable isotopes of both hydrogen and , often denoted by D) and oxygen and which give rise to combinations such as , , etc., all of which are readily identifiable in the infrared spectra of water vapor. (Interestingly, the ratio of in water varies enough from place to place that it is now possible to determine the source of a particular water sample with some precision.) And to top this off, the two hydrogen atoms in water contain protons whose magnetic moments can be parallel or antiparallel, giving rise to ortho- and para-water, respectively.

The bottom line: To a chemist, the term “pure” has meaning only in the context of a particular application or process.

Operational and conceptual classifications

Since chemistry is an experimental science, we need a set of experimental criteria for placing a given sample of matter in one of these categories. There is no single experiment that will always succeed in unambiguously deciding this kind of question. However, there is one principle that will always work in theory, if not in practice. This is based on the fact that the various components of a mixture can, in principle, always be separated into pure substances.

Consider a heterogeneous mixture of salt water and sand. The sand can be separated from the salt water by the mechanical process of filtration. Similarly, the butterfat contained in milk may be separated from the water by a mechanical process known as centrifugation, which depends on differences in density between the two components. These examples illustrate the general principle that heterogeneous matter may be separated into homogeneous matter by mechanical means. Turning this around, we have an operational definition of heterogeneous matter: if, by some mechanical operation we can separate a sample of matter into two or more other kinds of matter, then our original sample was heterogeneous.

To find a similar operational defnition for homogeneous mixtures, consider how we might separate the two components of a solution of salt water. The most obvious way would be to evaporate off the water, leaving the salt as a solid residue. Thus a homogeneous mixture can be separated into pure substances by undergoing appropriate changes of state— that is, by evaporation, freezing, etc. If a sample of matter remains unchanged by carrying out operations of this kind, then it could be a pure substance.

Some common methods of separating homogeneous mixtures into their components are outlined below.

Distillation. A liquid is partly boiled away; the first portions of the condensed vapor will be enriched in the lower-boiling component.

Fractional crystallization. A hot saturated solution of a solid in a liquid is allowed to cool slowly; the first solid that crystallizes out tends to be of higher purity.

Liquid-liquid extraction. Two mutually-insoluble liquids, one containing two or more solutes (dissolved substances), are shaken together. Each solute will concentrate in the liquid in which it is more soluble.

Chromatography. As a liquid or gaseous mixture flows along a column containing an adsorbant material, the more strongly-adsorbed components tend to move more slowly and emerge later than the less-strongly adsorbed components.

Physical and chemical properties

Since chemistry is partly the study of the transformations that matter can undergo, we can also assign to any substance a set of chemical properties that express the various changes of composition the substance is known to undergo. Chemical properties also include the conditions of temperature, etc., required to bring about the change, and the amount of energy released or absorbed as the change takes place.

The properties that we described above are traditionally known as physical properties, and are to be distinguished from chemical properties that usually refer to changes in composition that a substance can undergo. For example, we can state some of the more distinctive physical and chemical properties of sodium:

physical properties	chemical properties
appearance: a soft, shiny metal density: melting point: boiling point:	forms an oxide and a hydride NaH burns in air to form sodium peroxide reacts violently with water to release hydrogen gas dissolves in liquid ammonia to form a deep blue solution

Problem Example

Classify each of the statements as a physical or chemical property, and explain the basis for your answer.

Chlorine is a greenish-yellow gas at room temperature.

This is another way of stating that the boiling point (a physical property) is below .

Liquid oxygen is attracted by a magnet.

Even under the influence of the magnet, the oxygen is still the same substance, , so the effect is purely a physical property.

Gold is highly resistant to corrosion.

Corrosion involves the reaction of a metal with oxygen and water, so corrosion (and by extension, resistance to corrosion) is definitely a chemical property.

Hydrogen cyanide is an extremely poisonous gas.

Most poisonous substances act by combining chemically with substances that interfere with some aspect of cellular biochemistry, so we can consider this to be a chemical property of .

Sugar is a high-energy food.

The chemical energy contained in a food or fuel can be released only through a chemical reaction leading to lower-energy products. The “high-energy” part might be considered a physical property, since this depends on the quantity of energy obtainable from a given mass of the substance.

Since chemistry is partly the study of the transformations that matter can undergo, we can also assign to any substance a set of chemical properties that express the various changes of composition the substance is known to undergo. Chemical properties also include the conditions of temperature, etc., required to bring about the change, and the amount of energy released or absorbed as the change takes place.

Another dubious dichotomy

The more closely one looks at the distinction between physical and chemical properties, the more blurred this distinction becomes. For example, the high boiling point of water compared to that of methane, , is a consequence of the electrostatic attractions between bonds in adjacent molecules, in contrast to those between bonds; at this level, we are really getting into chemistry! So although you will likely be expected to “distinguish between” physical and chemical properties on an exam, don't take it too seriously.

What you should be able to do

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.

• Give examples of extensive and intensive properties of a sample of matter. Which kind of property is more useful for describing a particular kind of matter?
• Explain what distinguishes heterogeneous matter from homogeneous matter.
• Describe the following separation processes: distillation, crystallization, liquid-liquid extraction, chromatography.
• To the somewhat limited extent to which it is meaningful, classify a given property as a physical or chemical property of matter.

Understanding Density and Buoyancy

The density of an object is one of its most important and easily-measured physical properties. Densities are widely used to identify pure substances and to characterize and estimate the composition of many kinds of mixtures. The purpose of this lesson is to show how densities are defined, measured, and utilized, and to make sure you understand the closely-related concepts of buoyancy and specific gravity

Defining density

You didn't have to be in the world very long to learn that the mass and volume of a given substance are directly proportional, although you certaintly did not first learn it in these words which are now the words of choice now that you have become a scholar.

These plots show how the masses of three liquids vary with their volumes. Notice that

• the plots all have the same origin of : if the mass is zero, so is the volume;
• the plots are all straight lines, which signify direct proportionality.

The only difference between these plots is their slopes. Denoting mass and volume by m and V respectively, we can write the equation of each line as , where the slope (rho) is the proportionality constant that relates mass to volume. This quantity is known as the density, which is usually defined as the mass per unit volume: .

The volume units millilitre (mL) and cubic centimetre are almost identical and are used interchangably in this course.

Density can be expressed in any combination of mass and volume units; the most commonly seen units are grams per mL , or kilograms per litre.

The general meaning of density is the amount of anything per unit volume. What we conventionally call the “density” is more precisely known as the “mass density”.

Problem Example 1

Ordinary commercial nitric acid is a liquid having a density of , and contains by weight. a) Calculate the mass of in of nitric acid. b) What volume of acid will contain of ?

Solution: The mass of of the acid is . The weight of acid that contains of is and will have a volume of .

Specific volume

It is sometimes more convenient to express the volume occupied by a unit mass of a substance. This is just the inverse of the density and is known as the specific volume.

Problem Example 2

A glass bulb weighs when evacuated, and when filled with xenon gas at . The bulb can hold of water. Find the density and specific volume of xenon under these conditions.

Solution: The mass of xenon is found by difference: . The density . The specific volume is .

Specific gravity

A quantity that is very closely related to density, and which is frequently used in its place, is specific gravity.

Specific gravity is the ratio of the mass of a material to that of an equal volume of water. Because the density of water is about , the specific gravity is numerically very close to that of the density, but being a ratio, it is dimensionless.

The presence of “volume” in this definition introduces a slight complication, since volumes are temperature-dependent owing to thermal expansion. At , water has its maximum density of almost exactly , so if the equivalent volume of water is assumed to be at this temperature, then the density and specific gravity can be considered numerically identical. In making actual comparisons, however, the temperatures of both the material being measured and of the equivalent volume of water are frequently different, so in order to specify a specific gravity value unambiguously, it is necessary to state the temperatures of both the substance in question and of the water.

Thus if we find that a given volume of a substance at weighs as much as the same volume of water measured at , we would express its specific gravity as

Although most chemists find density to be more convenient to work with and consider specific gravity to be rather old-fashioned, the latter quantity is widely used in many industrial and technical fields ranging from winemaking to urinalysis.

Densities of substances and materials

Solids, liquids and gases

Densities of gases in g/l

chlorine -

carbon dioxide -

air (dry) -

methane -

hydroden -

In general, gases have the lowest densities, but these densities are highly dependent on the pressure and temperature which must always be specified. To the extent that a gas exhibits ideal behavior (low pressure, high temperature), the density of a gas is directly proportional to the masses of its component atoms, and thus to its molecular weight. Measurement of the density of a gas is a simple experimental way of estimating its molecular weight (more here).

Densities of liquids in g/ml at

mercury -

sulfuric acid -

water -

olive oil -

ethanol -

Liquids encompass an intermediate range of densities. Mercury, being a liquid metal, is something of an outlier. Liquid densities are largely independent of pressure, but they are somewhat temperature-sensitive.

Densities of solids in g/ml at

osmium -

gold -

iron -

aluminum -

sodium chloride -

sugar -

h.d polyurethane -

wood (pine) -

lithium -

The density range of solids is quite wide. Metals, whose atoms pack together quite compactly, have the highest densities, although that of lithium, the lighest metallic element, is quite low. Composite materials such as wood and high-density polyurethane foam contain void spaces which reduce the average density.

How the temperature affects density

All substances tend to expand as they are heated, causing the same mass to occupy a greater volume, and thus lowering the density. For most solids, this expansion is relatively small, but it is far from negligible; for liquids, it is greater. The volumes of gases, as you may already know (see here for details), are highly temperature-sensitive, and so, of course, are their densities.

What is the cause of thermal expansion? As molecules aquire thermal energy, they move about more vigorously. In condensed phases (liquids and solids), this motion has the character of an irregular kind of bumping or jostling that causes the average distances between the molecules to increase, thus leading to increased volume and smaller density.

Densities of the elements

One might expect the densities of the chemical elements to increase uniformly with atomic weight, but this is not what happens; density depends on the volume as well as the mass, and the volume occupied by a given mass of an element, and these volumes can vary in a non-uniform way for two reasons:

The sizes (atomic radii) follow the zig-zag progression that characterizes the other periodic properties of the elements, with atomic volumes diminishing with increasing nuclear charge across each period (more here).

The atoms comprising the different solid elements do not pack together in the same way. The non-metallic solids are often composed of molecules that are more spread out in space, and which have shapes that cannot be arranged as compactly. so they tend to form more open crystal lattices than do the metals, and therefore have lower densities.

The plot below is taken from the popular WebElements site.

Density of water

Nature has conveniently made the density of water at ordinary temperatures almost exactly . Water is subject to thermal expansion just as are all other liquids, and throughout most of its temperature range, the density of water diminishes with temperature. But water is famously exceptional over the temperature range , where raising the temperature causes the density to increase, reaching its greatest value at about .

This density maximum is one of many “anomalous” behaviors of water. As you may know, the molecules in liquid and solid water are loosely joined together through a phenomenon known as hydrogen bonding. Any single water molecule can link up to four other molecules, but this occurs only when the molecules are locked into place within an ice crystal. This is what leads to a relatively open lattice arrangement, and thus to the relatively low density of ice.

Below are three-dimensional views of a typical local structure of liquid water (right) and of ice (left). Notice the greater openness of the ice structure which is necessary to ensure the strongest degree of hydrogen bonding in a uniform, extended crystal lattice. The more crowded and jumbled arrangement in liquid water can be sustained only by the greater amount thermal energy available above the freezing point.

Figure  ice^[1]

Figure  water^[1]

When ice melts, thermal energy begins to overcome the hydrogen-bonding forces so that each molecule, instead of being permanently connected to four neighbors, is now only linked to an average of three other molecules through hydrogen bonds that continually break and re-form. With fewer hydrogen bonds, the geometrical requirements that formerly mandated a more open structural arrangement now diminish, so the entire network tends to collapse, rendering the water more dense. As the temperature rises, the fraction of molecules that occupy ice-like clusters diminishes, contributing to the rise in density that is seen between and .

The density maximum of water corresponds to the temperature at which the breakup of ice-like clusters (leading to higher density) and thermal expansion (leading to lower density) achieve a balance.

Whenever a continuously varying quantity such as density passes through a maximum or a minimum value as the temperature or some other variable is changing, you know that two opposing effects are at work.

Problem Example 3

Suppose that you place of pure water at in the refrigerator and that it freezes, producing ice at 0C. What will be the volume of the ice?

Solution: From the graph above, the density of water at is , and that of ice at .

Some environmental consequences of water's density maximum

The density maximum at has some interesting consequences in the aquatic ecology of lakes. In all but the most shallow lakes, the water tends to be stratified, so that for most of the year, the denser water remains near the bottom and mixes very little with the less-dense waters above. Because water has its density maximum at , the waters of deep lakes (and of the oceans) usually stay around at all times of the year. In the summer this will be the coldest water, but in the winter, the surface waters lose heat to the atmosphere and if they cool below , they will be colder than the more dense waters below.

When the weather turns cold in the fall, the surface waters lose heat and cool to . This more dense layer of water sinks to the bottom, displacing the water below, which rises to the surface and restores nutrients that were removed when dead algae sank to the bottom. This “fall turnover” renews the lake for the next season.

Buoyancy

What do an ice cube and a block of wood have in common? Throw either material into water, and it will float. Well, mostly; each object will have its bottom part immersed, but the upper part will ride high and dry. People often say that wood and ice float because they are “lighter than water”, but this of course is nonsense unless we compare the masses of equal volumes of the substances. In other words, we need to compare the masses-per-unit-volume, meaning the densities, of each material with that of water. So we would more properly say that objects capable of floating in water must have densities smaller than that of water.

The apparent weight of an object immersed in a fluid will be smaller than its “true” weight (Archimedes' principle). The latter is the downward force exerted by gravity on the object. Within a fluid, however, this downward force is partially opposed by a net upward force that results from the displacement of this fluid by the object. The difference between these two weights is known as the buoyancy.

The displaced fluid is of course not really confined to the “phantom volume” shown at the bottom of the diagram; it spreads throughout the container and exerts forces on all surfaces of the object and increase with depth, combining to produce the net buoyancy force as shown. See here for another diagram that shows this more clearly.

Dynamics of buoyancy - an interesting physics-mechanics treatment

Problem Example 4

An object weighs in air and has a volume of . What will be its apparent weight when immersed in water?

Solution: When immersed in water, the object is buoyed up by the mass of the water it displaces, which of course is the mass of of water. Taking the density of water as unity, the upward (buoyancy) force is just .

The apparent weight will be .

Air is of course a fluid, and buoyancy can be a problem when weighing a large object such as an empty flask. The following problem illustrates a more extreme case:

Problem Example 5

A balloon having a volume of is placed on a sensitive balance which registers a weight of . What is the “true weight” of the balloon if the density of the air is ?

Solution: The mass of air displaced by the balloon exerts a buoyancy force of . Thus the true weight of the balloon is this much greater than the apparant weight: .

Problem Example 6

A piece of metal weighs in air, in water, and when immersed in gasoline. a) What is the density of the metal? b) What is the density of the gasoline?

Solution: When immersed in water, the metal object displaces of water whose volume is . The density of the metal is thus .

The metal object displaces of gasoline, whose density must therefore be .

Floating – “the tip of the iceberg”

When an object floats in a liquid, the portion of it that is immersed has a volume that depends on the mass of this same volume of displaced liquid.

Problem Example 7

A cube of ice that is on each side floats in water. How many cm does the top of the cube extend above the water level? (Density of ice =

Solution: The volume of the ice is and its mass is . The ice is supported by an upward force equivalent to this mass of displaced water whose volume is . Since the cross section of the ice cube is , it must sink by in order to displace of water. Thus the height of cube above the water is .

... hence the expression, “the tip of the iceberg”, implying that 90% of its volume is hidden under the surface of the water.

How is density measured?

The most obvious way of finding the density of a material is to measure its mass and its volume. This is the only option we have for gases, but observing the mass of a fixed volume of a liquid is time-consuming and awkward, and measuring the volumes of solids whose shapes are irregular or which are finely divided is usually impractical.

Liquids: the hydrometer

The traditional hydrometer is a glass tube having a weighted bulb near the bottom. The hydrometer is lowered into a container of the liquid to be measured, and comes to a rest with the upper part protruding above the liquid surface at a height (read from a calibrated scale) that depends on the density of the liquid. This will only work, of course, if the overall density of the hydrometer itself is smaller than the density of the liquid to be measured. For this reason, hydrometers intended for general use come in sets. Because liquid densities are temperature dependent, hydrometers intended for precise measurements also contain an internal thermometer so that this information can be collected in the event that temperature corrections will be made.

Owing to the ease with which they can be observed, densities are widely employed to estimate the composition or quality of liquid mixtures or solutions, and in some cases determine their commercial value. This has given rise to many kinds of hydrometers that are specialized for specific uses:

• Saccharometer – used by winemakers and brewers to measure the sugar content of a liquid
• Alcoholometer – measures the alcoholic content of a liquid
• Salinometer – measures the “salinity” (salt content) of brine or seawater
• Lactometer - measures the specific gravity of milk products

Figure  Battery hydrometer - theory^[1]

Figure  build-it-yourself drinking-straw hydrometer^[1]

Figure  Aquarium salinity hydrometer^[1]

Figure  A boat with depth markings on its body can be thought of as a gigantic hydrometer!^[1]

Figure  ...for pressurized-liquids^[1]

Figure  Sugar and syrup hydrometer^[1]

Don't confuse them!

A hydrometer measures the density or specific gravity of a liquid

a hygrometer measures the relative humidity of the air

Hydrometer scales

Hydrometers for general purpose use are normally calibrated in units of specific gravity, but often defined at temperatures other than . A very common type of calibration is in “degrees” on various arbitrary scales, of which the best known are the Baumé scales. Special-purpose hydrometer scales can get quite esoteric; thus alcohol hydrometers may directly mesure percentage alcohol by weight on a scale, or “proof” (twice the volume-percent of alcohol) on a scale.

Solids

Measuring the density of a solid that is large enough to weigh accurately is largely a matter of determining its volume. For an irregular solid such as a rock, this is most easily done by observing the amount of water it displaces.

A small vessel having a precisely determined volume can be used to determine the density of powdered or granular samples. The vessel (known as a pycnometer) is weighed while empty, and again when filled; the density is found from the weight difference and the calibrated volume of the pycnometer. This method is also applicable to liquids and gases.

In forensic work it is often necessary to determine the density of very small particles such as fibres, flakes of paint or metal, or grains of sand. Neither the weight nor volumes of such samples can be determined directly, so the simplest solution is to place the sample in a series of liquids of different densities, and see if it floats, sinks, or remains suspended within the liquid. A more sophisticated method is to layer two liquids in a vertical glass tube and allow them to slowly mix, creating a density gradient. When a particle is dropped into the tube, it sinks to a depth that matches its density.

This reference provides a brief summary of some of the modern methods of determining density.

Some applications of density

Archimedes' principle

The most famous application of buoyancy is due to Archimedes of Syracuse around 250 BC. He was asked to determine whether the new crown that King Hiero II had commissioned contained all the gold that he had provided to the goldsmith for that purpose; apparently he suspected that the smith might have set aside some of the gold for himself and substituted less-valuable silver instead. According to legend, Archimedes devised the principle of the “hydrostatic balance” after he noticed his own apparent loss in weight while sitting in his bath. The story goes that he was so enthused with his discovery that he jumped out of his bath and ran through the town, shouting “eureka” to the bemused people.

Problem Example 8

If the weight of the crown when measured in air was and its weight in water was , what was the density of the crown?

Solution: The volume of the crown can be found from the mass of water it displaced, and thus from its buoyancy: .

Given the densities of the pure metals: silver = 10.5, gold = , it would appear that the king did get ripped off!

The Golden Crown; an interesting commentary on this story

What is the size of an atom?

One of the delights of chemical science is to find way of using the macroscopic properties of bulk matter to uncover information about the microscopic world at the atomic level. The following problem example is a good illustration of this.

Problem Example 9

Estimate the diameter of the neon atom from the following information:

Density of liquid neon: ; molar mass of neon: .

Solution: This problem can be divided into two steps.

1 - Estimate the volume occupied by each atom. One mole (6.02E23 atoms) of neon occupy a volume of . If this space is divided up equally into tiny boxes, each just large enough to contain one atom, then the volume allocated to each atom is given by: .

2 - Find the length of each box, and thus the atomic diameter. Each atom of neon has a volume of about . If we re-express this volume as and fudge the “28” a bit, we can come up with a reasonably good approximation of the diameter of the neon atom without even using a calculator. Taking the volume as allows us to find the cube root, , which corresponds to the length of the box and thus to the diameter of the atom it encloses.

The accepted [van der Waals] atomic radius of neon is , corresponding to a diameter of about . This estimate is suprisingly good, since the atoms of a liquid are not really confined to orderly little boxes in the liquid.

What you should be able to do

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.

• Given two of the following values: mass - volume - density, find the value of the third.
• Define specific volume and specific gravity, and explain the significance of expessing the latter in a form such as .
• Describe the two factors responsible for the density maximum of water.
• Explain why weighing a solid object suspended in a fluid yields a smaller value than its “true” weight. Be able to find this difference when given the volume of the solid and the density of the fluid.
• Describe the purpose of a hydrometer and explain how it works.

Concept Map

Energy, Heat and Temperature: an introduction

All chemical changes are accompanied by the absorption or release of heat. The intimate connection between matter and energy has been a source of wonder and speculation from the most primitive times; it is no accident that fire was considered one of the four basic elements (along with earth, air, and water) as early as the fifth century BCE. This unit will cover only the very basic aspects of the subject, just enough to get you started; a much more complete set of tutorial lessons can be found here.

Energy

Energy is one of the most fundamental and universal concepts of physical science, but one that is remarkably difficult to define in way that is meaningful to most people. This perhaps reflects the fact that energy is not a “thing” that exists by itself, but is rather an attribute of matter (and also of electromagnetic radiation) that can manifest itself in various ways. It can be observed and measured only indirectly through its effects on matter that acquires, loses, or possesses it.

You will recall from earlier science courses that energy can take many forms: mechanical, chemical, electrical, radiation (light), and thermal. You also know that energy is conserved; it can be passed from one “system” to another, but it can never simply disappear.

Kinetic energy and potential energy

In the 17th Century, the great mathematician Gottfried Leibniz (1646-1716) suggested the distinction between vis viva (“live energy”) and vis mortua (“dead energy”), which later became known as kinetic energy and potential energy.

Whatever energy may be, there are basically two kinds: kinetic and potential. Kinetic energy is associated with the motion of an object; a body with a mass m and moving at a velocity v possesses the kinetic energy .

Potential energy is energy a body has by virtue of its location in a force field— a gravitational, electrical, or magnetic field. For example, if an object of mass m is raised off the floor to a height h, its potential energy increases by mgh, where g is a proportionality constant known as the acceleration of gravity. Similarly, the potential energy of a particle having an electric charge q depends on its location in an electrostatic field.

A nicely-done elementary tutorial on energy

Thermal energy and chemical energy

All molecules at temperatures above absolue zero are in a continual state of motion, and they therefore possess kinetic energy. But unlike the motion of a massive body such as a baseball or a car that is moving along a uniform trajectory, the motions of individual atoms or molecules are random and chaotic, forever changing in magnitude and direction as they collide with each other or (in the case of a gas,) with the walls of the container.

The sum total of all of this microscopic-scale randomized kinetic energy within a body is given a special name, thermal energy. [Animation link]

Atoms and molecules also possess potential energy in the form of the relative positions of electrons in the elctrostatic fields of their positively-charged nuclei. The potential energies of electrons in the force field created by two or more nuclei can be thought of as “chemical energy”, which gives rise to the effects we know as chemical bonding.

Most practical applications of energy involve both kinetic and potential components. For example, a vibrating guitar string exhibits both kinds of energy. It would therefore be more correct to say that chemical energy is mostly potential energy, and thermal energy is mostly kinetic energy.

Molecules are thus both vehicles for storing and transporting energy, and the means of converting it from one form to another when the formation, breaking, or rearrangement of the chemical bonds within them is accompanied by the uptake or release of energy, most commonly in the form of heat.

See the Chem1 Chemical Energetics site for a full treatment of the subject.

Energy scales are always arbitrary

You might at first think that a book sitting on the table has zero kinetic energy since it is not moving. In truth, however, that the earth itself is moving; it is spinning on its axis, it is orbiting the sun, and the sun itself is moving away from the other stars in the general expansion of the universe. Since these motions are normally of no interest to us, we are free to adopt an arbitrary scale in which the velocity of the book is measured with respect to the table; on this so-called laboratory coordinate system, the kinetic energy of the book can be considered zero.

We do the same thing with potential energy. If we define the height of the table top as the zero of potential energy, then an object having a mass m suspended at a height h above the table top will have a potential energy of mgh. Now let the object fall; as it accelerates in the earth's gravitational field, its potential energy changes into kinetic energy. An instant before it strikes the table top, this transformation is complete and the kinetic energy is identical with the original mgh. As the object comes to rest, its kinetic energy appears as heat (in both the object itself and in the table top) as the kinetic energy becomes randomized as thermal energy.

The chemical connection

The same principle applies to chemical substances; we can arbitrarily assign an energy of zero to a mixture of hydrogen and oxygen at . When they react, a quantity of heat is given off, and the energy of the resulting molecules is reduced by that amount. The fact that this energy is negative (with respect to the original and ) simply reflects the particular energy scale we have chosen.

Energy units

Energy is measured in terms of its ability to perform work or to transfer heat. Mechanical work is done when a force f displaces an object by a distance d: . The basic unit of energy is the joule. One joule is the amount of work done when a force of 1 newton acts over a distance of ; thus . The newton is the amount of force required to accelerate a mass by , so the basic dimensions of the joule are .The other two units in wide use. the calorieand the BTU (British thermal unit) are defined in terms of the heating effect on water. For the moment, we will confine our attention to the joule and calorie.

Heat and work

Heat and work are both measured in energy units, but they do not constitute energy itself. As we will explain below, they refer to processes by which energy is transfered to or from something— a block of metal, a motor, or a cup of water.

Heat

When a warmer body is brought into contact with a cooler body, thermal energy flows from the warmer one to the cooler until their two temperatures are identical. The warmer body loses a quantity of thermal energy , and the cooler body acquires the same amount of energy. We describe this process by saying that “ joules of heat has passed from the warmer body to the cooler one.” It is important, however, to understand that

Heat is the transfer of energy due to a difference in temperature

We often refer to a “flow” of heat, recalling the 18th-century notion that heat was an actual substance called “caloric” that could flow like a liquid.

In other words, heat is a process; it is not something that can be contained or stored in a body. It is important that you understand this, because the use of the term in our ordinary conversation (“the heat is terrible today”) tends to make us forget this distinction.

Work

Work is the transfer of energy by any process other than heat.

Work, like energy, can take various forms: mechanical, electrical, gravitational, etc. All have in common the fact that they are the product of two factors, an intensity term and a capacity term.For example, the simplest form of mechanical work arises when an object moves a certain distance against an opposing force. Electrical work is done when a body having a certain charge moves through a potential difference.

type of work	intensity factor	capacity factor	formula
mechanical	force	change in distance
gravitational	gravitational potential (a function of height)	mass	mgh
electrical	potential difference	quantity of charge

Performance of work involves a transformation of energy; thus when a book drops to the floor, gravitational work is done (a mass moves through a gravitational potential difference), and the potential energy the book had before it was dropped is converted into kinetic energy which is ultimately dispersed as thermal energy.

Mechanical work is the product of the force exerted on a body and the distance it is moved:

(Illustration from the Ben Wiens Energy site)

Heat and work are best thought of as processes by which energy is exchanged, rather than as energy itself. That is, heat “exists” only when it is flowing, work “exists” only when it is being done.

When two bodies are placed in thermal contact and energy flows from the warmer body to the cooler one,we call the process “heat”. A transfer of energy to or from a system by any means other than heat is called “work”.

So you can think of heat and work as just different ways of accomplishing the same thing: the transfer of energy from one place or object to another.

To make sure you understand this, suppose you are given two identical containers of water at . Into one container you place an electrical immersion heater until the water has absorbed of heat. The second container you stir vigorously until of work has been performed on it. At the end, both samples of water will have been warmed to the same temperature and will contain the same increased quantity of thermal energy. There is no way you can tell which contains “more work” or “more heat”.

An important limitation on energy conversion

This limitation is the essence of the Second Law of Thermodynamics which we will get to much later in this course

Thermal energy is very special in one crucial way. All other forms of energy are interconvertible: mechanical energy can be completely converted to electrical energy, and the latter can be completely converted to thermal, as in the water-heating example described above. So although work can be completely converted into thermal energy, complete conversion of thermal energy into work is impossible. A device that partially accomplishes this conversion is known as a heat engine; a steam engine, a jet engine, and the internal combusion engine in a car are well-known examples.

Temperature and its meaning

We all have a general idea of what temperaure means, and we commonly associate it with “heat”, which, as we noted above, is a widely mis-understood word.

Both relate to what we described above as thermal energy—the randomized kinetic energy associated with the various motions of matter at the atomic and molecular levels.

Heat, you will recall, is not something that is “contained within” a body, but is rather a process in which [thermal] energy enters or leaves a body as the result of a temperature difference. So if we place of water on a stove until it has absorbed of heat, for example, then we can say that the water has aquired of energy.

Temperature

And as we all know, the temperature of the water will rise. Temperature is a measure of the average kinetic energy of the molecules within the water. You can think of temperature as an expression of the “intensity” with which the thermal energy in a body manifests itself in terms of chaotic, microscopic molecular motion.

Heat is the quantity of thermal energy that enters or leaves a body.

Temperature measures the average translational kinetic energy of the molecules in a body.

You will notice that we have sneaked the the word “translational” into this definition of temperature. Translation refers to a change in location: molecules moving around in random directions. This is the major form of thermal energy under ordinary conditions, but molecules can also undergo other kinds of motion, namely rotations and internal vibrations. These latter two forms of thermal energy are not really “chaotic” and do not contribute to the temperature.

Energy is measured in joules, and temperature in degrees. This difference reflects the important distinction between energy and temperature:

• We can say that the of water we heated on the stove now contains more energy than it did before. And because energy is an extensive quantity, we know that a portion of this warmer water contains more energy than it did originally.
• Temperature, by contrast, is not a measure of quantity; being an intensive property, it is more of a “quality” that describes the “intensity” with which thermal energy manifests itself.

Temperature scales

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 , the temperature at which water boils, as the upper fixed point.Normal human body temperature registered rather than 96.

Temperature is measured by observing its effect on some temperature-dependent variable such as the volume of a liquid or the electrical resistance of a solid. In order to express a temperature numerically, we need to define a scale which is marked off in uniform increments which we call degrees. The nature of this scale— its zero point and the magnitude of a degree, are completely arbitrary.

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.

For reasons best known to Celsius, he assigned to the freezing point of water and to its boiling point, resulting in an inverted scale that nobody liked. After his death a year later, the scale was put the other way around. The revised centigrade scale was quickly adopted everywhere except in the English-speaking world, and became the metric unit of temperature. In 1948 it was officially renamed as the Celsius scale; it is now used in every developed country except in the U.S.A., where students still have to learn to convert between the two scales.

The key to this conversion 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 degrees, making the the magnitude of the Note the distinction between “” (a temperature) and “” (a temperature increment).

Because the ice point is at , 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 , 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. It is now common to express an increment such as five as “five kelvins”

In 1859 the Scottish engineer and physicist William J. M. Rankine proposed an absolute temperature scale based on the Fahrenheit degree. Absolute zero corresponds to . 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 .

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. Perhaps the most common example, known to all beginning students, is the ideal gas equation of state

.

Heat capacity

As a body loses or gains heat, its temperature changes in direct proportion to the amount of thermal energy transferred:

The proportionality constant C is known as the heat capacity. If is expressed in kelvins (degrees) and in joules, the units of C are . In other words, the heat capacity tells us how many joules of energy it takes to change the temperature of a body by . The greater the value of C, the the smaller will be the effect of a given energy change on the temperature.

It should be clear that C is an extensive property— that is, it depends on the quantity of matter. Everyone knows that a much larger amount of heat is required to bring about a change in the temperature of of water compared to of water. For this reason, it is customary to express C in terms of unit quantity, such as per gram, in which case it becomes the specific heat capacity, commonly referred to as the “specific heat” and has the units .

Note: you are expected to know the units of specific heat. The advantage of doing so is that you need not learn a “formula” for solving specific heat problems.

Problem Example 1

How many joules of heat must flow into of water at to raise its temperature to ?

Solution: The mass of the water is . The specific heat of water is . From the definition of specific heat, the quantity of energy is .

How can I rationalize this procedure? It should be obvious that the greater the mass of water and the greater the temperature change, the more heat will be required, so these two quantities go in the numerator. Similarly, the energy required will vary invrsely wih the specific heat, which therefore goes in the denominator.

Specific heat capacities of some common substances

substance
Aluminum
Copper
Lead
Mercury
Zinc
Alcohol (ethanol)
Water
Ice
Gasoline (n-octane)
Glass
Carbon (graphite/diamond)
Sodium chloride
Rock (granite)
Air

Note especially the following:

• The molar heat capacities of the metallic elements are almost identical. This is the basis of the Law of Dulong and Petit, which served as an important tool for estimating the atomic weights of some elements.
• The intermolecular hydrogen bonding in water and alcohols results in anomalously high heat capacities for these liquids; the same is true for ice, compared to other solids.
• The values for graphite and diamond are consistent with the principle that solids that are more “ordered” tend to have larger heat capacities.

Problem Example 2

A piece of nickel weighing is heated to , and is then dropped into of water at . The temperature of he metal falls and that of the water rises until thermal equilibrium is attained and both are at . What is the specific heat of the metal?

Solution: The mass of the water is . The specific heat of water is and its temperature increased by , indicating that it absorbed of energy. The metal sample lost this same quantity of energy, undergoing a temperature drop of as the result. The specific heat capacity of the metal is .

Notice that no “formula” is required here as long as you know the units of specific heat; you simply place the relevant quantities in the numerator or denominator to make the units come out correctly.

What you should be able to do

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.

• Explain the difference between kinetic energy and potential energy.
• Define chemical energy and thermal energy.
• Define heat and work, and describe an important limitation in their interconversion.
• Describe the physical meaning of temperature.
• Explain the meaning of a temperature scale and describe how a particular scale is defined.
• Convert a temperature expressed in Fahrenheit or Celsius to the other scale.
• Describe the Kelvin temperature scale and its special significance.
• Define heat capacity and specific heat, and explain how they can be measured.

Concept Map

Units and dimensions

The measure of matter

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.

All 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).

Figure  Leonardo da Vinci - Vitruvian Man^[1]

Wikipedia article on the history of measurement

Scales and units

Figure  ^[1]

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.

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 the 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.

Brief history of the SI - NIST Reference on the SI

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.

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; that is, it is not the gram, as you might expect.
• 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 ) 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.

prefix	abbreviation	multiplier	--	prefix	abbreviation	multiplier
peta	P			deci	s
tera	T			centi	c
giga	G			milli	m
mega	M			micro
kilo	k			nano	n
hecto	h			pico	p
deca	da	10		femto	f

Units outside the SI

liter (litre)	L
metric ton	t
united atomic mass unit	u

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.

Wikipedia article on SI derived units

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	temperature (theta)

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

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

Thus the units of volume will be (in the SI) or , (English), etc. Moreover, any formula that calculates a volume must contain within it the dimension; thus the volume of a sphere is .

The dimensions of a unit are the powers which and must be given in order to express the unit.

Thus, , as given above.

Problem Example

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.

Dimensions of units commonly used in Chemistry

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

Why are unit dimensions useful?

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

• 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.
• 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.)
• Just as you cannot add apples to oranges, an expression such as is meaningless unless the dimensions of each side are identical. (Of course, the two sides should work out to the same units as well.)
• Many quantities must be dimensionless— for example, the variable x in expressions such as , and . 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 empoyed 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.

A nice tutorial on unit conversions - another tutorial site

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 and 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:

where is the acceleration of gravity. While the nominal value of the latter quantity is 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  Please 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 .^[1]

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 = . 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.

Time

Time present and time past Are both perhaps present in time future And time future contained in time past. If all time is eternally present All time is unredeemable.

T.S. Eliott

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.

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.

Wikipedia article on time and its measurement

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.

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.

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

In order 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 the magnitude of the Note the distinction between “” (a temperature) and “” (a temperature increment).

69, Fault 1: 'class 'modtex.execution.ExecutionError':(70, "This is pdfeTeXk, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
 file:line:error style messages enabled.
entering extended mode
(./math.tex
LaTeX2e 2003/12/01
Babel v3.8d and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/local/teTeX/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/local/teTeX/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/local/teTeX/share/texmf-dist/tex/latex/stmaryrd/stmaryrd.sty)
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/local/teTeX/share/texmf-dist/tex/latex/cancel/cancel.sty)
(/usr/local/teTeX/share/texmf-dist/tex/latex/graphics/color.sty
(/usr/local/teTeX/share/texmf-dist/tex/latex/graphics/color.cfg)
(/usr/local/teTeX/share/texmf-dist/tex/latex/graphics/dvips.def)
(/usr/local/teTeX/share/texmf-dist/tex/latex/graphics/dvipsnam.def))
No file math.aux.
(/usr/local/teTeX/share/texmf-dist/tex/latex/stmaryrd/ustmry.fd)
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/local/teTeX/share/texmf-dist/tex/latex/amsfonts/umsb.fd)
./math.tex:10: Undefined control sequence.
l.10 \\begin{math}\\rightleftarrow
                                \\end{math}
No pages of output.
Transcript written on math.log.
")'

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 , 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 as “five kelvins”

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

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. Perhaps the most common example, known to all beginning students, is the ideal gas equation of state, .

Why do we have so many temperature scales? (a StraightDope article)

A short history of the thermometor and temperature scales

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:

Pressure of the atmosphere

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 [link].

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 of surface is about , or . The force (weight) that the Earth’s gravitional acceleration exerts on this mass is

resulting in a pressure of pa. The actual pressure at sea level varies with atmospheric conditions, so it is customary to define standard atmospheric pressure as pa or 101 kpa. Although the standard atmosphere is not an SI unit, it is still widely employed. In meteorology, the bar, exactly , is often used.

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 high, so the “millimeter of mercury”, now more commonly known as the torr, has long been a common pressure unit in the sciences: .

See here for more on gas pressure and the atmosphere

Torricelli and the invention of the barometer

What you should be able to do

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.

Concept Map

The meaning of measure

The exact distance between the upper lip and the tip of the dorsal fin will forever be hidden in a fog of uncertainty. The angle at which we hold the calipers and the force with which we close them on the object will never be exactly reproducible. A more fundamental limitation occurs whenever we try to compare a continuously-varying quantity such as distance with the fixed invervals on a measuring scale; between and there is the same infinity of distances that exists between and !

The “true value” of a measured quantity, if it exists at all, will always elude us; the best we can do is learn how to make meaningful use (and to avoid mis-use!) of the numbers we read off of our measuring devices.

Image by Stephen Winsor; used with permission of the artist.

Uncertainty is certain!

In science, there are numbers and there are “numbers”. What we ordinarily think of as a “number” and will refer to here as a pure number is just that: an expression of a precise value. The first of these you ever learned were the counting numbers, or integers; later on, you were introduced to the decimal numbers, and the rational numbers, which include numbers such as and that cannot be expressed as exact decimal values.

The other kind of numeric quantity that we encounter in the natural sciences is a measured value of something– the length or weight of an object, the volume of a fluid, or perhaps the reading on an instrument. Although we express these values numerically, it would be a mistake to regard them as the kind of pure numbers described above.

Confusing? Suppose our instrument has an indicator such as you see here. The pointer moves up and down so as to display the measured value on this scale. What number would you write in your notebook when recording this measurement? Clearly, the value is somewhere between 130 and 140 on the scale, but the graduations enable us to be more exact and place the value beteween 134 and 135. The indicator points more closely to the latter value, and we can go one more step by estimating the value as perhaps 134.8, so this is the value you would report for this measurement.

Now here’s the important thing to understand: although “134.8” is itself a number, the quantity we are measuring is almost certainly not 134.8— at least, not exactly. The reason is obvious if you note that the instrument scale is such that we are barely able to distinguish between 134.7, 134.8, and 134.9. In reporting the value 134.8 we are effectively saying that the value is probably somewhere with the range 134.75 to 134.85. In other words, there is an uncertainty of unit in our measurement.

All measurements of quantities that can assume a continuous range of values (lengths, masses, volumes, etc.) consist of two parts: the reported value itself (never an exactly known number), and the uncertainty associated with the measurement.

Scatter and error in measured values

All measurements are subject to error which contributes to the uncertainty of the result. By “error”, we do not mean just outright mistakes, such as incorrect use of an instrument or failure to read a scale properly; although such gross errors do sometimes happen, they usually yield results that are sufficiently unexpected to call attention to themselves.

Random error

When you measure a volume or weight, you observe a reading on a scale of some kind, such as the one illustrated just above. Scales, by their very nature, are limited to fixed increments of value, indicated by the division marks. The actual quantities we are measuring, in contrast, can very continuously, so there is an inherent limitation in how finely we can discriminate between two values that fall between the marked divisions of the measuring scale. The same problem remains if we substitute an instrument with a digital display; there will always be some point at which some value that lies between the two smallest divisions must arbitrarily toggle between two numbers on the readout display. This introduces an element of randomness into the value we observe, even if the “true” value remains unchanged.

The more sensitive the measuring instrument, the less likely it is that two successive measurements of the same sample will yield identical results. In the example we discussed above, distinguishing between the values 134.8 and 134.9 may be too difficult to do in a consistent way, so two independent observers may record different values even when viewing the same reading. Each measurement is also influenced by a myriad of minor events, such as building vibrations, electrical fluctuations, motions of the air, and friction in any moving parts of the instrument. These tiny influences consititute a kind of “noise” that also has a random character. Whether we are conscious of it or not, all measured values contain an element of random error.

Systematic error

Suppose that you weigh yourself on a bathroom scale, not noticing that the dial reads “” even before you have placed your weight on it. Similarly, you might use an old ruler with a worn-down end to measure the length of a piece of wood. In both of these examples, all subsequent measurements, either of the same object or of different ones, will be off by a constant amount. Unlike random error, which is impossible to eliminate, these systematic errors are usually quite easy to avoid or compensate for, but only by a conscious effort in the conduct of the observation, usually by proper zeroing and calibration of the measuring instrument. However, once systematic error has found its way into the data, it is can be very hard to detect.

More than one answer in replicate measurements

If you wish to measure your height to the nearest centimetre or inch, or the volume of a liquid cooking ingredient to the nearst “cup”, you can probably do so without having to worry about random error. The error will still be present, but its magnitude will be such a small fraction of the value that it will not be detected. Thus random error is not something we worry about too much in our daily lives.

If we are making scientific observations, however, we need to be more careful, particularly if we are trying to exploit the full sensitivity of our measuring instruments in order to achieve a result that is as reliable as possible. If we are measuring a directly observable quantity such as the weight or volume of an object, then a single measurement, carefully done and reported to a precision that is consistent with that of the measuring instrument, will usually be sufficient.

More commonly, however, we are called upon to find the value of some quantity whose determination depends on several other measured values, each of which is subject to its own sources of error. Consider a common laboratory experiment in which you must determine the percentage of acid in a sample of vinegar by observing the volume of sodium hydroxide solution required to neutralize a given volume of the vinegar. You carry out the experiment and obtain a value. Just to be on the safe side, you repeat the procedure on another identical sample from the same bottle of vinegar. If you have actually done this in the laboratory, you will know it is highly unlikely that the second trial will yield the same result as the first. In fact, if you run a number of replicate (that is, identical in every way) determinations, you will probably obtain a scatter of results.

To understand why, consider all the individual measurements that go into each determination; the volume of the vinegar sample, your judgement of the point at which the vinegar is neutralized, and the volume of solution used to reach this point. And how accurately do you know the concentration of the sodium hydroxide solution, which was made up by dissolving a measured weight of the solid in water and then adding more water until the solution reaches some measured volume. Each of these many observations is subject to random error; because such errors are random, they can occasionally cancel out, but for most trials we will not be so lucky– hence the scatter in the results.

A similar difficulty arises when we need to determine some quantity that describes a collection of objects. For example, a pharmaceutical researcher will need to determine the time required for half of a standard dose of a certain drug to be eliminated by the body, or a manufacturer of light bulbs might want to know how many hours a certain type of light bulb will operate before it burns out. In these cases a value for any individual sample can be determined easily enough, but since no two samples (patients or light bulbs) are identical, we are compelled to repeat the same measurement on multiple samples, and once again, are faced with a scattering of results.

As a final example, suppose that you wish to determine the diameter of a certain type of coin. You make one measurement and record the results. If you then make a similar measurement along a different cross-section of the coin, you will likely get a different result. The same thing will happen if you make successive measurements on other coins of the same kind.

Here we are faced with two kinds of problems. First, there is the inherent limitation of the measuring device: we can never reliably measure more finely than the marked divisions on the ruler. Secondly, we cannot assume that the coin is perfectly circular; careful inspection will likely reveal some distortion resulting from a slight imperfection in the manufacturing process. In these cases, it turns out that there is no single, true value of either quantity we are trying to measure.

The mean and its meaning

When we obtain more than one result for a given measurement (either made repeatedly on a single sample, or more commonly, on different samples), the simplest procedure is to report the mean, or average value. The mean is defined mathematically as the sum of the values, divided by the number of measurements:

If you are not familiar with this notation, don’t let it scare you! Take a moment to see how it expresses the previous sentence; if there are n measurements, each yielding a value , then we sum over all i and divide by n to get the mean value . For example, if there are only two measurements, and , then the mean is .

Problem Example

Calculate the mean value of the set of eight measurements illustrated here.

Solution:

There are eight data points (10.4 was found in three trials, 10.5 in two), so . The mean is

.

Accuracy and precision

We tend to use these two terms interchangeably in our ordinary conversation, but in the context of scientific measurement, they have very different meanings:

Accuracy refers to how closely the measured value of a quantity corresponds to its “true” value.

Precision expresses the degree of reproducibility, or agreement between repeated measurements.

Accuracy, of course, is the goal we strive for in scientific measurements. Unfortunately, however, there is no obvious way of knowing how closely we have achieved it; the “true” value, whether it be of a well-defined quantity such as the mass of a particular object, or an average that pertains to a collection of objects, can never be known– and thus we can never recognize it if we are fortunate enough to find it.

Thus we cannot distinguish between the four scenarios illustrated above by simply examining the results of the two measurements. We can, however, judge the precision of the results, and then apply simple statistics to estimate how closely the mean value is likely to reflect the true value in the absence of systematic error.

You would not want to predict the outcome of the next election on the basis of interviews with only two or three voters; you would want a sample of ten to twenty at a minimum, and if the election is an important national one, a fair sample would require hundreds to thousands of people distributed over the entire geographic area and representing a variety of socio-economic groups. Similarly, you would want to test a large number of light bulbs in order to estimate the mean lifetime of bulbs of that type. Statistical theory tells us that the more samples we have, the greater will be the chance that the mean of the results will correspond to the “true” value, which in this case would be the mean obtained if samples could be taken from the entire population (of people or of light bulbs.)

This point can be better appreciated by examining the two sets of data shown here. The set on the left consists of only three points (shown in orange), and gives a mean that is quite far removed from the “true” value, which is arbitrarily chosen for this example.

In the data set on the right, composed of nine measurements, the deviation of the mean from the true value is much smaller.

Deviation of the mean from the “true value” becomes smaller when more measurements are made.

Absolute and relative uncertainty

If you weigh out of a solid sample on a laboratory balance that is accurate to within , then the actual weight of the sample is likely to fall somewhere in the range of 74.0 to ; the absolute uncertainty in the weight you observe is , or . If you use the same balance to weigh out of another sample, the actual weight is between and , and the absolute uncertainty is still .

Although the absolute uncertainties in these two examples are identical, we would probably consider the second measurement to be more precise because the uncertainty is a smaller fraction of the measured value. The relative uncertainties of the two results would be

(about 3 parts in 1000 (PPT), or )

(about 0.8 PPT , or )

Relative uncertainties are widely used to express the reliability of measurements, even those for a single observation, in which case the uncertainty is that of the measuring device. Relative uncertainties can be expressed as parts per hundred (percent), per thousand (PPT), per million, (PPM), and so on.

What you should be able to do

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.

• Give an example of a measured numerical value, and explain what distinguishes it from a “pure” number.
• Give examples of random and systematic errors in measurements.
• Find the mean value of a series of similar measurements.
• State the principal factors that affect the difference between the mean value of a series of measurements, and the “true value” of the quantity being measured.
• Calculate the absolute and relative precisions of a given measurement, and explain why the latter is generally more useful.
• Distinguish between the accuracy and the precision of a measured value, and on the roles of random and systematic error.

Concept Map

Significant figures and rounding off

The numerical values we deal with in science (and in many other aspects of life) represent measurements whose values are never known exactly. Our pocket-calculators or computers don't know this; they treat the numbers we punch into them as “pure” mathematical entities, with the result that the operations of arithmetic frequently yield answers that are physically ridiculous even though mathematically correct.

The purpose of this unit is to help you understand why this happens, and show you what to do about it.

Digits: significant and otherwise

“The population of our city is 157,872.”

“The number of registered voters as of Jan 1 was 27,833.”

Consider the two statements shown here. Which of these would you be justified in dismissing immediately? Certainly not the second one, because it probably comes from a database which contains one record for each voter, so the number is found simply by counting the number of records.

The first statement cannot possibly be correct. Even if a city’s population could be defined in a precise way (Permanent residents? Warm bodies?), how can we account for the minute-by minute changes that occur as people are born and die, or move in and move away?

Making sure that numbers make sense

What is the difference between the two population numbers stated above? The first one expresses a quantity that cannot be known exactly– that is, it carries with it a degree of uncertainty. It is quite possible that the last census yielded precisely 157,872 records, and that this might be the “population of the city” for legal purposes, but it is surely not the “true” population. To better reflect this fact, one might list the population (in an atlas, for example) as 157,900 or even 158,000. These two quantities have been rounded off to four and three significant figures, respectively, and the have the following meanings:

• 157900 (the significant digits are underlined here) implies that the population is believed to be within the range of about 157850 to about 157950. In other words, the population is 15790050. The “plus-or-minus 50” appended to this number means that we consider the absolute uncertainty of the population measurement to be . We can also say that the relative uncertainty is , which we can also express as 1 part in 1579, or , or about 0.06 percent.
• The value 158000 implies that the population is likely between about 157500 and 158500, or 158000500. The absolute uncertainty of 1000 translates into a relative uncertainty of or 1 part in 158, or about 0.6 percent.

Which of these two values we would report as “the population” will depend on the degree of confidence we have in the original census figure; if the census was completed last week, we might round to four significant digits, but if it was a year or so ago, rounding to three places might be a more prudent choice. In a case such as this, there is no really objective way of choosing between the two alternatives.

This illustrates an important point: the concept of significant digits has less to do with mathematics than with our confidence in a measurement. This confidence can often be expressed numerically (for example, the height of a liquid in a measuring tube can be read to ), but when it cannot, as in our population example, we must depend on our personal expereience and judgement.

So, what is a significant digit? According to the usual definition, it is all the numerals in a measured quantity (counting from the left) whose values are considered as known exactly, plus one more whose value could be one more or one less:

• In “157900” (four significant digits), the leftmost three digits are known exactly, but the fourth digit, “9” could well be “8” if the “true value” is within the implied range of 157850 to 157950.
• In “158000” (three significant digits), the leftmost two digits are known exactly, while the third digit could be either “7” or “8” if the true value is within the implied range of 157500 to 158500.

Although rounding off always leads to the loss of numeric information, what we are getting rid of can be considered to be “numeric noise” that does not contribute to the quality of the measurement.

The purpose in rounding off is to avoid expressing a value to a greater degree of precision than is consistent with the uncertainty in the measurement.

Implied uncertainty

If you know that a balance is accurate to within , say, then the uncertainty in any measurement of mass carried out on this balance will be . Suppose, however, that you are simply told that an object has a length of , with no indication of its precision. In this case, all you have to go on is the number of digits contained in the data. Thus the quantity “” is specified to in 0 42, or one part in 42 . The implied relative uncertainty in this figure is 1/42, or about . The precision of any numeric answer calculated from this value is therefore limited to about the same amount.

Round-off error

It is important to understand that the number of significant digits in a value provides only a rough indication of its precision, and that information is lost when rounding off occurs.

Suppose, for example, that we measure the weight of an object as on a balance believed to be accurate to within . The resulting value of tells us that the true weight of the object could be anywhere between and . The absolute uncertainty here is , and the relative uncertainty is 1 part in 32.8, or about 3 percent.

How many significant digits should there be in the reported measurement? Since only the leftmost “3” in “3.28” is certain, you would probably elect to round the value to . So far, so good. But what is someone else supposed to make of this figure when they see it in your report? The value “” suggests an implied uncertainty of , meaning that the true value is likely between and . This range is below that associated with the orginal measurement, and so rounding off has introduced a bias of this amount into the result. Since this is less than half of the uncertainty in the weighing, it is not a very serious matter in itself. However, if several values that were rounded in this way are combined in a calculation, the rounding-off errors could become significant.

Rules for rounding off numeric values

The standard rules for rounding off are well known. Before we set them out, let us agree on what to call the various components of a numeric value.

• The most significant digit is the leftmost digit (not counting any leading zeros which function only as placeholders and are never significant digits.)
• If you are rounding off to n significant digits, then the least significant digit is the nth digit from the most significant digit.The least significant digit can be a zero.
• The first non-significant digit is the th digit.

The rules for rounding off are shown here.

• If the first non-significant digit is less than 5, then the least significant digit remains unchanged.
• If the first non-significant digit is greater than 5, the least significant digit is incremented by 1.
• If the first non-significant digit is 5, the least significant digit can either be incremented or left unchanged (see below!)
• All non-significant digits are removed.

Fantasies about fives

Students are sometimes told to increment the least significant digit by 1 if it is odd, and to leave it unchanged if it is even. One wonders if this reflects some idea that even numbers are somehow “better” than odd ones! (The ancient superstition is just the opposite, that only the odd numbers are “lucky”.)

In fact, you could do it equally the other way around, incrementing only the even numbers. If you are only rounding a single number, it doesn’t really matter what you do. However, when you are rounding a series of numbers that will be used in a calculation, if you treated each first-nonsignificant 5 in the same way, you would be over- or underestating the value of the rounded number, thus accumulating round-off error. Since there are equal numbers of even and odd digits, incrementing only the one kind will keep this kind of error from building up.

You could do just as well, of course, by flipping a coin!

Examples of rounding-off

number to round / no. of sig. digits	result	comment
		First non-significant digit (1) is less than 5, so number is simply truncated.
	or	First non-significant digit is 5, so least sig. digit can either remain unchanged or be incremented.
		Crossing “decimal boundary”, so all numbers change.
		The two zeros are just placeholders
		The two leading zeros are not significant digits.

Rounding up the nines

Suppose that an object is found to have a weight of . This would place its true weight somewhere in the range of to . In judging how to round this number, you count the number of digits in “3.98” that are known exactly, and you find none! Since the “4” is the leftmost digit whose value is uncertain, this would imply that the result should be rounded to one significant figure and reported simply as . An alternative would be to bend the rule and round off to two significant digits, yielding . How can you decide what to do?

In a case such as this, you should look at the implied uncertainties in the two values, and compare them with the uncertainty associated with the original measurement.

rounded value	implied max	implied min	absolute uncertainty	relative uncertainty
			or	1 in 400, or
			or	1 in 4,
			or	1 in 40,

Clearly, rounding off to two digits is the only reasonable course in this example.

The same kind of thing could happen if the original measurement was . Again, the true value is believed to be in the range of to . The fact that no digit is certain here is an artifact of decimal notation. The absolute uncertainty in the observed value is , so the value itself is known to about 1 part in 100, or . Rounding this value to three digits yields with an implied uncertainty of , or 1 part in 100, consistent with the uncertainty in the observed value.

Observed values should be rounded off to the number of digits that most accurately conveys the uncertainty in the measurement.

• Usually, this means rounding off to the number of significant digits in in the quantity; that is, the number of digits (counting from the left) that are known exactly, plus one more.
• When this cannot be applied (as in the example above when addition of subtraction of the absolute uncertainty bridges a power of ten), then we round in such a way that the relative implied uncertainty in the result is as close as possible to that of the orbserved value.

Rounding off the results of calculations

In science, we frequently need to carry out calculations on measured values. For example, you might use your pocket calculator to work out the area of a rectangle:

rounded value	precision
	1 part in 158, or
	1 part in 16, or

Comment: Your calculator is of course correct as far as the pure numbers go, but you would be wrong to write down as the answer. Two possible options for rounding off the calculator answer are shown below:

It is clear that neither option is entirely satisfactory; rounding to 3 significant digits leaves the answer too precisely specified, whereas following the rule and rounding to 2 digits has the effect of throwing away some precision. In this case, it could be argued that rounding to three digits is justified because the implied relative uncertainty in the answer, , is more consistent with those of the two factors.

The above example is intended to point out that the rounding-off rules, although convenient to apply, do not always yield the most desirable result.

Other examples of rounding calculated values based on measurements are given below.

calculator result	rounded	remarks
	1.6	Rounding to two significant figures yields an implied uncertainty of or , three times greater than that in the least-preciseely known factor. This is a good illustration of how rounding can lead to the loss of information.
	1.9E6	The “3.1” factor is specified to 1 part in 31, or . In the answer 1.9, the value is expressed to 1 part in 19, or . These precisions are comparable, so the rounding-off rule has given us a reasonable result.
A certain book has a thickness of ; find the height of a stack of 24 identical books:		The “24” and the “1” are exact, so the only uncertain value is the thickness of each book, given to 3 significant digits. The trailing zero in the answer is only a placeholder.
	10.4	In addition or subtraction, look for the term having the smallest number of decimal places, and round off the answer to the same number of places.
		[see below]

The last of the examples shown above represents the very common operation of converting one unit into another. There is a certain amount of ambiguity here; if we take “9 in” to mean a distance in the range 8.5 to 9.5 in, then the uncertainty is in, which is 1 part in 18, or about . The relative uncertainty in the answer must be the same, since all the values are multiplied by the same factor, . In this case we are justified in writing the answer to two significant digits, yielding an uncertainty of about ; if we had used the answer “” (one significant digit), its implied uncertainty would be , or .

Some Web links

The excellent Significant figures tutorial by David Dice allows you to test your understanding as you go along.

For a somewhat different take on the subject, see Brad Thompson's “Good enough for Chemistry: a tragedy in three scenes”.

What you should be able to do

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.

• Give an example of a measurement whose number of significant digits is clearly too great, and explain why.
• State the purpose of rounding off, and describe the information that must be known to do it properly.
• Round off a number to a specified number of significant digits.
• Explain how to round off a number whose second-most-significant digit is 9.
• Carry out a simple calculation that involves two or more observed quantities, and express the result in the appropriate number of significant figures.

Concept Map