Classification and properties of matter - Saylor...

Classification and properties of matter

⇐index matter density | energy & heat | units & dimensions | measurement error | significant figures

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On this page: Observable properties of matter

Classification of matter Pure substances and mixtures

Physical and chemical properties What you should be able to do

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 properties? 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 various properties, we gradually acquire enough information tocharacterize it; to distinguish it from other kinds of matter. This is the first step in the development of chemical science, in which interest is focussed 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 2 kg” 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 anintensive property.

Suppose we make further measurements, and find that the same quantity of water whose mass is 2.0 kg also occupies a volume of 2.0 litres. 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 thedensity. 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 10.0 g of sugar int; same for any 10-g 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 5 mL of your blool int; the same for any 5-mL sample.

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

The electrical resistance of a 1-km length of 22-gauge copper wire int; same for any 1-km 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, whereasheterogeneous 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 ordinaryfog, 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 357 g 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?

"9944100% 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 (H1 and H2, often denoted by D) and oxygen (O16 and O18) which give rise to combinations such as H2O18, HDO16, etc., all of which are readily identifiable in the infrared spectra of water vapor. (Interestingly, the ratio of O18/O16 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 (25°C) chemical properties

appearance: a soft, shiny metal

density: 0.97 g cm3

melting point: 97.5°C

boiling point: 960°C

forms an oxide Na2O and a hydride NaH

burns in air to form sodium peroxide Na2O2

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 20°C.

Liquid oxygen is attracted by a magnet. Even under the influence of the magnet, the oxygen is still the same substance, O2, 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 HCN.

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, CH4, is a consequence of the electrostatic attractions between O-H bonds in adjacent molecules, in contrast to those between C-H 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

⇐index | matter density energy & heat | units & dimensions | measurement error | significant figures


On this page: Defining density

Densities of substances Buoyancy

Density measurement Some applications

What you should be able to do Concept Map

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

1 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 (0,0): 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 asm = ρV, where the slope ρ (rho) is the proportionality constant that relates mass to volume. This quantity ρ is known as the density, which is usually defined as the mass per unit volume: ρ = m/V.

The volume units millilitre (mL) and cubic centimetre (cm3) are almost identical and are used interchangably in this course.

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".

Density can be expressed in any combination of mass and volume units; the most commonly seen units are grams per mL (g mL–1, g cm–3), or kilograms per litre.

Problem Example 1

Ordinary commercial nitric acid is a liquid having a density of 1.42 g mL–1, and contains 69.8% HNO3 by weight. a) Calculate the mass of HNO3 in 800 ml of nitric acid. b) What volume of acid will contain 100 g of HNO3?

Solution: The mass of 800 mL of the acid is (1.42 g mL–1) × (800 mL) = 1140 g. The weight of acid that contains 100 g of HNO3is (100 g) / (0.698) = 143 g and will have a volume of (143 g) / (1.42 g mL–1) = 101 mL.

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 66.3915 g when evacuated, and 66.6539 g when filled with xenon gas at 25°C. The bulb can hold 50.0 mL of water. Find the density and specific volume of xenon under these conditions.

Solution: The mass of xenon is found by difference: (66.6539 – 66.3915)g = 0.2624 g. The density ρ = m/V = (0.2624 g)/(0.050 L) = 5.248 g L–1. The specific volume is 1/(5.248 g L–1 = 0.190 L g–1.

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 1.00 g mL–1, 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 4°C, water has its maximum density of almost exactly 1.000 g mL–1, 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 20°C weighs 1.11 times as much as the same volume of water measured at 4°C, 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.

2 Densities of substances and materials Solids, liquids and gases

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).

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.

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 1.000 g/mL ( 1 kg/L). 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 0-4° C, where raising the temperature causes the density to increase, reaching its greatest value at about 4°C.

This 4°C density maximum is one of many "anomalous" behaviors of water. As you may know, the H2O 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 H2O 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.

ice

water

When ice melts, thermal energy begins to overcome the hydrogen-bonding forces so that each H2O 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 H2O molecules that occupy ice-like clusters diminishes, contributing to the rise in density that is seen between 0° and 4°.

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.

The 4° 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.

Problem Example 3

Suppose that you place 1000 mL of pure water at 25°C 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 25°C is 0.9997 kg L–1, and that of ice at 0°C 0.917 g L–1.

Some environmental consequences of water's density maximum

The density maximum at 4°C 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 4°C, the waters of deep lakes (and of the oceans) usually

stay around 4°C 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 4°, 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 4°C. 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.

3 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 36 g in air and has a volume of 8.0 cm3. 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 8 cm3 of water. Taking the density of water as unity, the upward (buoyancy) force is just 8 g.

The apparent weight will be (36 g) – (8 g) = 28 g.

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 5.000 L is placed on a sensitive balance which registers a weight of 2.833 g. What is the "true weight" of the balloon if the density of the air is 1.294 g L–1?

Solution: The mass of air displaced by the balloon exerts a buoyancy force of (5.000 L) / (1.294 g L –1) = 3.860 g. Thus the true weight of the balloon is this much greater than the apparent weight: (2.833 + 3.860) g = 6.69 g.

Problem Example 6

A piece of metal weighs 9.25 g in air, 8.20 g in water, and 8.36 g 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 (9.25 – 8.20) g = 1.05 g of water whose volume is (1.05 g) / (1.00 g cm–3) = 1.05 cm3. The density of the metal is thus (9.25 g) / (1.05 cm3) = 8.81 g cm–3.

The metal object displaces (9.25 - 8.36) g = 0.89 g of gasoline, whose density must therefore be (0.89 g) / (1.05 cm3) = 0.85 g cm–3.

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 10 cm on each side floats in water. How many cm does the top of the cube extend above the water level? (Density of ice = 0.917 g cm–3.)

Solution: The volume of the ice is (10 cm)3 = 1000 cm3 and its mass is (1000 cm3) x (0.917 g cm–3) = 917 g. The ice is supported by an upward force equivalent to this mass of displaced water whose volume is (917 g) / (1.00 g cm–3) = 917 cm3 . Since the cross section of the ice cube is 100-cm2, it must sink by 9.17 cm in order to displace 917 cm3 of water. Thus the height of cube above the water is (10 cm – 9.17 cm) = 0.83 cm.

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

4 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

Battery hydrometer - theory

build-it-yourself drinking-straw hydrometer

Aquarium salinity hydrometer

A boat with depth markings on its body can be thought of as a gigantic hydrometer!

...for pressurized-liquids

Sugar and syrup hydrometer

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 25°C. 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 0–100% scale, or "proof" (twice the volume-percent of alcohol) on a 0-200 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.

5 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 4.876 kg and its weight in water was 4.575 kg, 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: (4876 – 4575) g / (1.00 g cm–3) = 301 cm3. The density is then (4876 g) / (301 cm3) = 16.2 g cm–3

The densities of the pure metals: silver = 10.5, gold = 19.3 g cm–3,

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: 1.204 g cm–3; molar mass of neon: 20.18 g.

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 (20.18 g) / (1.204 g cm–3) = 16.76 cm3. 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: (16.76 cm3 mol–1) / (6.02E23 atom mol–1) = 2.78E–23 cm3 atom–1.

2 - Find the length of each box, and thus the atomic diameter. Each atom of neon has a volume of about 2.8E–23 cm3. If we re-express this volume as 28E–24 cm3 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 27E–24 cm3 allows us to find the cube root, 3.0E–8 cm = 3.0E–10 m = 300 pm, 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 154 pm, corresponding to a diameter of about 310 pm. This estimate is suprisingly good, since the atoms of a liquid are not really confined to orderly little boxes in the liquid.


• 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 1.1525/4.

• Describe the two factors responsible for the 4°C 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 for beginning chemistry

⇐index | matter | density energy & heat units & dimensions | measurement error | significant figures


On this page: Kinetic and potential energy

Thermal and chemical energy Energy scales and units

Heat and work Temperature Heat capacity

What you should be able to do Concept Map

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.

1 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 GottfriedLeibniz (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 mv2/2.

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 ½mv2is 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 25°C. When they react, a quantity of heat ΔH is given off, and the energy of the resulting H2O molecules is reduced by that amount. The fact that this energy is negative (with respect to the original H2 and O2) 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 fdisplaces an object by a distance d: w = f × 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 1 m; thus 1 J = 1 N-m. The newton is the amount of force required to accelerate a 1-kg mass by 1 m/sec2, so the basic dimensions of the joule are kg m2 s–2.The other two units in wide use. the calorie and 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.

2 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 ΔE, and the cooler body acquires the same amont of energy. We describe this process by saying that "ΔE 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 distincion.

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 f Δx

gravitational gravitational potential (a function of height) mass mgh

electrical potential difference quantity of charge QΔV

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: 1 N-m = 1 J

(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 25°C. Into one container you place an electrical immersion heater until the water has absorbed 100 joules of heat. The second container you stir vigorously until 100 J 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 areinterconvertible: 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 thatpartially 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.

3 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 10 g of water on a stove until it has absorbed 100 J of heat, for example, then we can say that the water has aquired 100 J 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 10 g of water we heated on the stove now contains 100 J more energy than it did before. And because energy is an extensive quantity, we know that a 5-g portion of this warmer water contains 50 J 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 212 degrees, the temperature at which water boils, as the upper fixed point.Normal human body temperature registered 98.6 rather than 96.

Belize and the U.S.A. are the only countries that still use the Fahrenheit scale.

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 100 degrees to the freezing point of water and 0° 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.

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

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

Absolute temperature scales

Near the end of the 19th Century when the physical significance of temperature began to be understood, the need was felt for a temperature scale whose zero really means zero— that is, the complete absence of thermal motion. This gave rise to the absolute temperature scale whose zero point is –273.15 °C, but which retains the same degree magnitude as the Celsius scale. This eventually got renamed after Lord Kelvin (William Thompson); thus the Celsius degree became the kelvin. It is now common to express an increment such as five C° 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 (0° Ra) corresponds to –459.67°F. The Rankine scale has been used extensively by those same American and English engineers who delight in expressing heat capacities in units of BTUs per pound per F°.

The importance of absolute temperature scales is that absolute temperatures can be entered directly in all the fundamental formulas of physics and chemistry in which temperature is a variable. Perhaps the most common example, known to all beginning students, is the ideal gas equation of state

PV = nRT.

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

q = C ΔT

The proportionality constant C is known as the heat capacity

C = ΔT  /  q

If ΔT is expressed in kelvins (degrees) and q in joules, the units of C are J K–1. In other words, the heat capacity tells us how many joules of energy it takes to change the temperature of a body by 1 C°. 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 10-C° change in the temperature of 1 L of water compared to 10 mL 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 J K–

1g–1.

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 150 mL of water at 0° C to raise its temperature to 25° C?

Solution: The mass of the water is (150 mL)/(1.00 g mL–1) = 150 g. The specific heat of water is 4.18 J K–1 g–

1. From the definition of specific heat, the quantity of energy q = ΔE is (150 g)(25.0 K)(4.18 J K–1 g–1) = 16700 J.

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 C, J /g-K

Aluminum 0.900 Note especially the following:

Copper 0.386 Lead 0.128 Mercury 0.140 Zinc .387 Alcohol (ethanol) 2.4 Water 4.18 Ice (–10° C) 2.05 Gasoline (n-‐octane) .53 Glass .84 Carbon (graphite/diamond) .710 / .509 Sodium chloride 0.854 Rock (granite) .790 Air 1.01

• 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 2.40 g is heated to 200.0° C, and is then dropped into 10.0 mL of water at 15.0° C. The temperature of he metal falls and that of the water rises until thermal equilibrium is attained and both are at 18.0° C. What is the specific heat of the metal?

Solution: The mass of the water is (10 mL) × (1.00 g mL–1) = 10 g. The specific heat of water is 4.18 J K–1 g–

1 and its temperature increased by 3.0 C°, indicating that it absorbed (10 g)(3 K)(4.18 J K–1 g–1) = 125 J of energy. The metal sample lost this same quantity of energy, undergoing a temperature drop of 182 C° as the result. The specific heat capacity of the metal is (125 J) / (2.40 g)(182 K) = 0.287 J K–1 g–1.

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.


• 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

The measure of matter Understanding the units of scientific measurement

⇐index | matter | density | energy & heat units & dimensions measurement error | significant figures


Farther down on this page:

• The SI Base units • The SI decimal prefixes • Units outside the SI

• Derived units and dimensions • Units and their ranges in Chemistry • What you should be able to do

• Concept map

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.

1 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).

Leonardo da Vinci - Vitruvian Man

Wikipedia article on the history of measurement

Scales and units

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 systemthat 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

2 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.

length meter m

mass kilogram kg

time second s

temperature (absolute) kelvin K

amount of substance mole mol

electric curent ampere A

luminous intensity candela cd

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

• The base unit of mass is unique in that a decimal prefix (see below) is built-in to it; 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 Sumeriansused for their astronomical calculations around 100 BCE.)

• Of special interest to Chemistry is the mole, the base unit for expressing the quantity of matter. Although the number is not explicitly mentioned in the official definition, chemists define the mole as Avogadro’s number (approximately 6.02 1023) of anything.

3 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 1018 deci s 10–1

tera T 1012 centi c 10–2

giga G 109 milli m 10–3

mega M 106 micro μ 10–6

kilo k 103 nano n 10–9

hecto h 102 pico p 10–12

deca da 10 femto f 10–15

4 Units outside the SI

liter (litre) L 1 L = 1 dm3 = 10–3 m3

metric ton t 1 t = 103 kg

united atomic mass unit u 1 u = 1.66054×10–27 kg

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

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

In fact, most physical quantities can be expressed in terms of one or more of the following five fundamental units:

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

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

V = x · y · z

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

dim.V = L3

Thus the units of volume will be m3 (in the SI) or cm3, ft3 (English), etc. Moreover, any formula that calculates a volume must contain within it the L3 dimension; thus the volume of a sphere is 4/3 πr3.

The dimensions of a unit are the powers which M, L, T, Q and Θ must be given in order to express the unit.

Thus, dim.V = M0L3T0Q0 Θ0, as given above.

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

Q M L T quantity SI unit, other typical units

1 electric charge coulomb

1 mass kilogram, gram, metric ton, pound

1 length meter, foot, mile

1 time second, day, year

3 volume liter, cm3, quart, fluidounce

1 –3 density kg m–3, g cm–3

1 1 –2 force newton, dyne

1 –1 –2 pressure pascal, atmosphere, torr

1 2 –2 energy joule, erg, calorie, electron-‐volt

1 2 –3 power watt

1 1 2 –2 electric potential volt

1 –1 electric current ampere

1 1 1 –2 electric field intensity volt m–1

–2 1 2 –1 electric resistance ohm

2 1 3 –1 electric resistivity 2 –1 –2 1 electric conductance siemens, mho

Why are unit dimensions useful?

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

• 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 a = b + cx2 is meaningless unless the dimensions of each side are identical. (Of course, the two sides should work out to the same units as well.)

• Many quantities must be dimensionless— for example, the variable x in expressions such as log x, ex, and sin x. Checking through the dimensions of such a quantity can help avoid errors.

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

Unit conversions

Dimensional analysis is widely 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

6 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:

f = m a = m g

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

Please note that in this diagram and in those that follow, the numeric scale represents the logarithmof the number shown. For example, the mass of the electron is 10–30 kg.

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

Length

Chemists tend to work mostly in the moderately-small part of the distance range. Those who live in the lilliputian world of crystal- and molecular structures and atomic radii find the picometer a convenient currency, but one still sees the older non-SI unit called the Ångstrom used in this context; 1Å = 10–10 m = 100pm. Nanotechnology, the rage of the present era, also resides in this realm. The largest polymeric molecules and colloids define the top end of the particulate range; beyond that, in the normal world of doing things in the lab, the centimeter and occasionally themillimeter 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 212 degrees, the temperature at which water boils, as the upper fixed point. With this change, normal human body temperature registered 98.6 rather than 96.

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

Temperature comparisons and conversions

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

• The temperature that corresponds to 0° on the scale;

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

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 F° 100/180 = 5/9 the magnitude of the C° Note the distinction between “°C” (atemperature) and “C°” (a temperature increment).

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

Absolute temperature scales

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

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

The importance of absolute temperature scales is that absolute temperatures can be entered directly in all the fundamental formulas of physics and chemistry in which temperature is a variable. Perhaps the most common example, known to all beginning students, is the ideal gas equation of state, PV = nRT.

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:

1 Pa = 1 N m–2

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 1-cm2 of surface is about 1034 g, or 10340 kg m–2. The force (weight) that the Earth’s gravitional acceleration g exerts on this mass is

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

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

The barometer

In the early 17th century, the Italian physicist and mathematician Evangalisto Torricelli invented a device to measure atmospheric pressure. The Torricellian barometerconsists 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 76 cm high, so the “millimeter of mercury”, now more commonly known as the torr, has long been a common pressure unit in the sciences: 1 atm = 760 torr.

See here for more on gas pressure and the atmosphere

Torricelli and the invention of the barometer


• 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, andFahrenheit temperature scales.

• Explain how a Torricellian barometer works.

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