Journey of the Universe

The Ancient World - Cantor

Bestselling author Norman Cantor delivers this compact but magisterial survey of the ancient world -- from the birth of Sumerian civilization around 3500 B.C. in the Tigris-Euphrates valley (present-day Iraq) to the fall of the Roman Empire in A.D. 476. In Antiquity, Cantor covers such subjects as Classical Greece, Judaism, the founding of Christianity, and the triumph and decline of Rome.

In this fascinating and comprehensive analysis, the author explores social and cultural history, as well as the political and economic aspects of his narrative. He explains leading themes in religion and philosophy and discusses the environment, population, and public health. With his signature authority and insight, Cantor highlights the great books and ideas of antiquity that continue to influence culture today.

Author's Note

This book is an attempt to communicate to the educated reader and to students of history some basic knowledge about antiquity from 2.5 million years ago -- the dawn of humanity -- to the fall of the Roman Empire in the fifth century A.D.  I have tried to convey this long history in simple language, keeping names, places, and dates to a minimum, and focusing on the Mediterranean and Western Europe.

Part I, Basic Narrative, lays out the fundamental knowledge about antiquity that every educated person should possess.

Part II, Societies and Cultures, focuses in more detail on the most important aspects of antiquity.

I have used the conventional, European Christian dating of B.C. (Before Christ) and A.D. (Anno Domini, "in the year of the Lord) rather than B.C.E. (Before the Common Era) and C.E. -- not for religious reasons, but simply because that is what most readers are familiar with.

One

Basic Narrative

Chapter One

Hydraulic Despotisms

A very long time ago, some 2.5 million years B.C., the mother of human species as we know it, our ultimate ancestor, appeared in East Africa.  She walked erect and was able to close her thumbs and forefingers to make tools for doing what her limbs were unable to do.  She was four feet tall and probably black.  This is what the science of paleontology told us during the last four decades of the twentieth century.

The earliest humans were related to primates, the apes and monkeys.  Humans and gorillas share 92 percent of their DNA.  The genetic conformity between humans and chimpanzees is significantly greater. Humans and chimpanzees share 98 percent of their DNA. It is possible that humans and chimpanzees are descended from the same species of animal long ago extinct. Or that humans evolved out of chimpanzees who gave up swinging from trees in order to find food on the ground. Like chimps, humans tend to migrate in colonies. Gorillas are more individualistic, but they, too, are often found to be migrating and living in small groups.

That humans are a species of primate is indisputable.  The earliest humans lived and traveled in small groups.  They were hunters and gatherers.  They gathered fruits and vegetables growing wild in the then-great forests and savannas of East Africa and they hunted animals that they could kill and eat.

Like most humans today, they were carnivorous, but not entirely so.  As long as there were vegetables and fruits in abundance, they were satisfied with a vegetarian diet.  But fresh meat, eaten both raw and cooked, appealed to them.

Out of stones and bones, they made weapons to kill animals.  Their throats and larynxes could utter sounds that allowed for communication between these humans, and over time, these sounds were shaped into organized languages.

Frequently on the move in search of food, the humans very slowly drifted northward and moved up the great rivers that flowed together to form the Nile valley.  Around a hundred thousand years ago, the humans reached the Nile delta and the Mediterranean Sea and began to spread east and north from there. By this time, they had learned to be farmers, to plant seeds, to irrigate their croplands, and to build villages and town, drawing their sustenance from the cultivated earth.  But they did not cease to hunt and gather.

The humans reached Europe – at first, the territory adjacent to the Crimea and the northern shores of the Black Sea – about 10,000 B.C.

Based on their excavations, archaeologist tell us later, around 6000 B.C., two centers of rich and highly developed civilization had emerged in the Near East – in the northern extremity of Egypt and in the Tigris-Euphrates valley, in what is today southern Iraq.

A Guide to Further Reading

Prehistory

There is a gap of 2.5 million years between the emergence of recognizable humans in East Africa and the appearance of literate civilizations in the Nile and Tigris-Euphrates valleys around 3500 B.C. What is known from archaeological data about this period of so-called prehistory is summarized in:

Colin Tudge, The Time Before History, 1996

Donald Johanson and Blake, Edgar, From Lucy to Language, 1996

Virginia Morrell, Ancestral Passions, 1995, a fascinating biography of the famous Leakey family of paleontologists.

The availability of irrigation systems to water the land and produce grain and other food crops was the material foundation for these two great river-valley societies, Egypt and Iraq. They were hydraulic despotisms, in which a small ruling class, with the aid of the soldiers and priests, commanded the material resources that gave sustenance to these civilizations and allowed them to build cities, palaces, and tombs.

The soldiers made sure that the peasants and laborers did what had to be done to maintain irrigation systems, harvest crops, and erect buildings. The priests assured the masses that this forced-labor system was dictated by the gods, who were represented on earth by kings.

The Nile Valley was for the most part a natural irrigation system, in which the great river overflowed once a year, covering the land with rich silt brought from East Africa, but pharaohs, as the Nile kings were called, also built some major canals to improve upon natural irrigation.  The Tigris-Euphrates valley was a scene of massive and complicated irrigation systems built by human labor to pull the water inland from the rivers.

We know about these two large and prosperous settlements of Iraq and northern Egypt exclusively from the material records offered by archaeology.  It was not until around 3500 B.C. that writing emerged in both societies.  Each developed its own distinctive forms of writing.

For another millennium, these written records consisted entirely of state business-accounts and letters, and panegyrics to the mightiness and divinity of kings. (By the dawn of writing, a half-million people were settled in each of the Nile and Tigris-Euphrates valleys, where huge temples and palaces were erected for kings, priests, and aristocrats, while the common people lived in small houses made of sundried bricks, or in tents.)

The forces for change in the two great societies of Egypt and Iraq were, with one exception, external rather than internal.  That one exception was the attempt by Pharaoh Akhenaton (around 1330 B.C.) to create a new monotheistic religion (with similarities to Judaism) and eliminate the power of the traditional priests who served a multitude of deities.  This theological revolution was immediately reversed after Akhenaton’s death.

Otherwise, what happened in the two river valley civilizations was determined by wars spurred by invasions from without.  In Egypt, dynasties enduring for centuries presided over irrigation and cultivation, huge edifices built by forced labor, and the manufacture of exquisite paintings and jewelry.  For a century, around 1100 B.C., Egypt was invaded and ruled by a “sea people,” from western Asia, but then effective power returned to a native dynasty.

The history of the ancient Tigris-Euphrates valley was shaped by a series of invasions from the north. As Sumerians, Chaldeans, Assyrians, and Babylonians succeeded one another, the structure of severely class-ridden societies and agriculture-based economies did not change.  The series of invasions and conquest ended about 500 B.C. with Iraq absorbed into the expanding Iranian (Persian) empire to the east.

In the first century B.C., Egypt was absorbed into the expanding Roman empire, and remained its wealthiest province until the Muslim Arabs took it over in the seventh century A.D.

?? * * *??

In both instances the hydraulic despotisms were marked by the tremendous capacity for millennia-long continuities in government, religion, urban environment, social structure, and economy. Life expectancy was short, never rising above forty years of age until well into the twentieth century A.D. Peasants and urban laborers constituted 95 percent of the population.

Monarchy was considered divine.  The priests unequivocally supported the king. Aristocrats derived their wealth from royal largess. Whether they were legally slaves or subject to some other variant of forced labor, the role of peasants and urban workers was to serve the king, priests, and aristocrats until death.

The physical remains of the Iraqi monarchies have survived only in small fragments, the most impressive being huge Assyrian statues and bas-reliefs now found in the Metropolitan Museum of Art in New York and the British Museum in London. Many of the famous pyramid tombs of the Egyptian rules have survived, and some are still unexcavated. The higher one was on the social scale while alive, the more likely one was to enjoy a happy afterlife, accompanied by one’s prized possessions. The contents of the excavated pyramids can be seen in museums in Cairo, in principal Western European cities, particularly London and Paris, and in Brooklyn, New York. Egyptian craftsmen were highly skilled, especially in the making of colorful murals and gold jewelry. Many Egyptian jewelry designs are still popular today.

No despotism, not even the technocratic monstrosities of the twentieth century A.D., can entirely shut off the human predilection for law, art, and charitable ethics.  The Iraqis were prone to compiling law codes, such as that of Hammurabi, which slightly mitigated the harshness of everyday life for the masses. The Iraqis also developed myths about the creation of the world, including a story about a Great Flood, nowadays believed to have actually occurred on the shores of the Black Sea. Those writings may have influenced the authors of the Hebrew Bible, who lived in Iraq around 500 B.C.

The Egyptians produced attractive murals of domestic life and idealized depictions of animals, and a host of myths connected to Re, the Sun god.

The only clear expression of intellectual dissent from hydraulic despotism occurred in the southern half of the coastal lands of the eastern Mediterranean, called variously Canaan, Palestine, Israel, Judah, and today, Israel again. Here and in a satellite Jewish colony in Iraq, between 800 and 500 B.C., visionaries (“the Prophets”) – namely Amos, Ezekiel, Isaiah (at least two different writers writing under this name, and Jeremiah – wrote elegant poems calling for social justice in the world and a freer, more open and humanitarian society.

This Promised Land of the Hebrews was on the main mercantile-and-army route that ran from Egypt to Iraq. It was the place of equalitarian ideas that contradicted the way of life of the two great empires that lay at either end of the road that traversed Israel.

Proponents of a monotheistic theology that resembled that of Pharaoh Akhenaton, and of a strong national consciousness, the Hebrew prophets called for a just and free society sharply at odds with the hydraulic despotisms, even in the severely edited form their writings were given by the priests before these writings were admitted into the slowly developing Hebrew Bible.  Some of the works attributed to Isaiah and Ezekiel were composed in Iraq.  Jeremiah spent the last years of his life in Egypt, so that there is more than local significance to the radical, democratic, humanitarian visions of the Hebrew prophets.

The text of the Hebrew Bible read in synagogues today was defined in Spain around 1000 A.D., but this version corresponds very closely to a Biblical text written in or near Jerusalem between 300 and 250 B.C. and discovered in the late 1940s in caves overlooking the Dead Sea, south of Jerusalem.

Historians and archaeologists now believe that the Hebrew Bible crystallized around 600 B.C., in the time of King Josiah, who, together with the Judean priests, undertook a religious reformation.  A strict monotheism was ordained, local shrines destroyed, and the important religious services centralized in Jerusalem.  A stern moral code and a maze of religious laws were imposed.  Shortly, after, the main part of the Jewish community was exiled by the Babylonians to Iraq, where some Jewish families continued to live in unbroken peace until 1948. After their conquest of Iraq around 500 B.C., the tolerant Persian rulers encouraged the Jews to return to Judea, and perhaps half did so, carrying with them historical myths and ethical ideas from Iraq.

Their scribes settled down to organize and compose the Bible as it has existed since 250 B.C. and later came to be part of the Christian Old Testament.  God’s covenant with the Jews, His chosen people and witnesses to His message of love and law for all mankind, became the prime theme of the Hebrew Bible, which now included the heavily edited texts of the Prophets calling for social justice at a time when a succession of petty Hebrew kings imitated the hydraulic despotisms in Iraq and Egypt.

A grand national myth was created in which the Hebrews were said to have come originally from Iraq and to have migrated to Canaan and then to Egypt, where they became slaves of the Pharaohs. They were led by Moses into freedom and a return to the Promised Land.  Archaeologists after a century of excavation know today that this is only a liberation myth; there was no exodus of the Jews from Egypt, although Passover annually celebrates this event.  The rabbis who took over from the Jerusalem priests after the Romans sacked Jerusalem in 70 A.D. placed another conservatizing layer of editing upon the final text of the Bible, making it a persistent reminder of national consciousness but further downplaying the radical call for social justice against hydraulic despotism, which the Prophets had inculcated between 800 and 500 B.C.

The First Law of Thermodynamics

(Four Laws That Drive the Universe by Peter Atkins)

2. THE FIRST LAW

The conservation of energy

The first law of thermodynamics is generally thought to be the least demanding to grasp, for it is an extension of the law of conservation of energy, that energy can be neither created nor destroyed. That is, however much energy there was at the start of the universe, so there will be that amount at the end.  But thermodynamics is a subtle subject, and the first law is much more interesting than this remark might suggest.  Moreover, like the zeroth law, which provided an impetus for the introduction of the property ‘temperature’ and its clarification, the first law motivates the introduction and helps to clarify the meaning of the elusive concept of ‘’energy’.

We shall assume at the outset that we have no inkling that there is any such property, just as in the introduction to the zeroth law we did not pre-assume that there was anything we should call temperature, and then found that the concept was forced upon us as an implication of the law.  All we shall assume is that the well-established concepts of mechanics and dynamics, like mass, weight, force, and work, are known.  In particular, we shall base the whole of this presentation on an understanding of the notion of ‘work’.

Work is motion against an opposing force. We do work when we raise a weight against the opposing force of gravity.  The magnitude of the work we do depends on the mass of the object, the strength of the gravitational pull on it, and the height through which it is raised.  You yourself might to be the weight:  you do work when you climb a ladder; the work you do is proportional to your weight and the height through which you climb.  You also do work when cycling into the wind: the stronger the wind and the further you travel the greater the work you do. You do work when you stretch or compress a spring, and the amount of work you do depends on the strength of the spring and the distance through which it is stretched or compressed.

All work is equivalent to the raising of a weight.  For instance, although we might think of stretching a spring, we could connect the stretched spring to a pulley and weight and see how far the weight is raised when the spring returns to its natural length. The magnitude of the work of raising a mass m (for instance, 50 kg) through a height h (for instance, 2.0 m) on the surface of the Earth is calculated from the formula mgh, where g is a constant known as the acceleration of free fall, which at sea level on Earth is close to 9.8 m s^-2.  Raising a 50 kg weight through 2.0 m requires work of magnitude 980 kg m^2 s^-2.  As we saw in the footnote on p. 14, the awkward combination of units ‘kilograms metre squared per second squared’ is called the joule (symbol J). So, to raise our weight, we have to do 980 joules (980 J) of work.

(p.14  1. Energy is reported in joules (J): 1 J = 1 kg m^2 s^-2. We could think of 1 J as the energy of a 2 kg ball travelling at 1 m s^-1). Each pulse of the human heart expends an energy of about 1 J.)

Work is the primary foundation of thermodynamics and in particular of the first law. Any system has the capacity to do work.  For instance, a compressed or extended spring can do work: as we have remarked, it can be used to bring about the raising of a weight.  An electric battery has the capacity to do work, for it can be connected to an electric motor which in turn can be used to raise a weight.  A lump of coal in an atmosphere of air can be used to do work by burning it as a fuel in some kind of engine.  It is not an entirely obvious point, but when we drive an electric current through a heater, we are doing work on the heater, for the same current could be used to raise a weight by passing it through an electric motor rather than the heater.  Why a heater is called a ‘heater’ and not a ‘worker’ will become clear once we have introduced the concept of heat.  That concept hasn’t appeared yet.

With work a primary concept in thermodynamics, we need a term to denote the capacity of a system to do work: that capacity we term energy. A fully stretched spring has a greater capacity to do work than the same spring only slightly stretched: the fully stretched spring has a greater energy than the slightly stretched spring.  A litre of hot water has the capacity to do more work than the same litre of cold water: a litre of hot water has a greater energy than a litre of cold water.  In this context, there is nothing mysterious about energy: it is just a measure of the capacity of a system to do work, and we know exactly what we mean by work.  

*

Now we extend these concepts from dynamics to thermodynamics.

The Zeroth Law of Thermodynamics

"About 14 billion years ago, the universe flared forth in the big bang, unfurling space, time, light, and matter in a singular, immense explosion.  Less than one minute after this "Great Radiance," protons and neutrons emerged and combined to form nuclei of the simplest elements, mostly hydrogen and helium.  About 300,000 years later, electrons combined with these nuclei for the first time, transforming matter into hydrogen and helium atoms." (Sampson)

            0 - Immense explosion

1 min. - Protons and neutrons and combined to form nuclei of simplest elements

300,000 years - Electrons combined with nuclei to form atoms

Four Laws That Drive the Universe by Peter Atkins

1.     THE ZEROTH LAW 

The concept of temperature

"The zeroth law is an afterthought. Although it had long been known that such a law was essential to the logical structure of thermodynamics, it was not dignified with a name and number until early in the twentieth century. By then, the first and second laws had become so firmly established that there was no hope of going back and renumbering them. As will become apparent, each law provides an experimental foundation for the introduction of a thermodynamic property. The zeroth law establishes the meaning of what is perhaps the most familiar but is in fact the most enigmatic of these properties: temperature.

            …slightly dusty definitions…

Thermo dynamics, like much of the rest of science, takes terms with an everyday meaning and sharpens them – some would say, hijacks them – so that they take on an exact an unambiguous meaning.  We shall see that happening throughout this introduction to thermodynamics.  It starts as soon as we enter its doors. The part of the universe that is at the centre of attention in thermodynamics is called the system.  A system may be a block of iron, a beaker of water, an engine, a human body. It may even be a circumscribed part of each of those entities.  The rest of the universe is called the surroundings. The surroundings are where we stand to make observations on the system and infer its properties.  Quite often, the actual surroundings consist of a water bath maintained at constant temperature, but that is a more controllable approximation to the true surroundings, the rest of the world. The system and its surroundings jointly make up the universe.  Whereas for us the universe is everything, for a less profligate thermodynamicist  it might consist of a beaker of water (they system) immersed in a water bath (the surroundings).

system + surroundings = universe

A system is defined by its boundary.  If matter can be added to or removed from the system, then it is said to be open. A bucket, or more refinedly an open flask, is an example, because we can just shovel in material. A system with a boundary that is impervious to matter is called closed. A sealed bottle is a closed system.  A system with a boundary that is impervious to everything in the sense that the system remains unchanged regardless of anything that happens in the surroundings is called isolated. A stoppered vacuum flask of hot coffee is a good approximation to an isolated system.

Open system

Closed system

Isolated system

The properties of a system depend on the prevailing conditions. For instance, the pressure of a gas depends on the volume it occupies, and we can observe the effect of changing that volume if the system has flexible walls. “Flexible walls” is best thought of as meaning that the boundary of the system is rigid everywhere except for a patch – a piston – that can move in and out.  Think of a bicycle pump with your finger sealing the orifice.

Properties are divided into two classes. An extensive property depends on the quantity of matter in the system – its extent.   The mass of a system is an extensive property; so is its volume.  Thus, 2 kg of iron occupies twice the volume of 1 kg of iron. An intensive property is independent of the amount of matter present.  The temperature (whatever that is) and the density are examples. The temperature of water drawn from a thoroughly stirred hot tank is the same regardless of the size of the sample. The density of iron is 8.9 g cm^-3 regardless of whether we have a 1 kg block or a 2 kg block. We shall meet many examples of both kinds of property as we unfold thermodynamics and it is helpful to keep the distinction in mind.

Depends on the amount of matter – extensive property

Mass and volume

Independent of the amount of matter – intensive property

Temperature and density

*

…the enigma of temperature and the zeroth law…

So much for these slightly dusty definitions.  Now we shall use a piston – a movable patch in the boundary of a system – to introduce one important concept that will then be the basis for introducing the enigma of temperature and the zeroth law itself.

Suppose we have two closed systems, each with a piston on one side and pinned into place to make a rigid container.  The two pistons are connected with a rigid rod so that as one moves out the other moves in.  We release the pins on the piston.  If the piston on the left drives the piston on the right into the system, we can infer that the pressure on the left was higher than that on the right, even though we have not made a direct measure of the two pressures. If the piston on the right won the battle, then we should infer that the pressure on the right was higher than that on the left.  If nothing had happened when we released the pins, we should infer that the pressures of the two systems were the same, whatever they might be. The technical expression for the condition arising from the equality of pressures is mechanical equilibrium. Thermodynamicists get very excited, or at least get very interested, when nothing happens, and this condition of equilibrium will grow in importance as we go through the laws. We need one more aspect of mechanical equilibrium: it will seem trivial at this point, but establishes the analogy that will enable us to introduce that concept of temperature. Suppose the two systems, which we shall call A and B, are in mechanical equilibrium when they are brought together and the pins are released. That is, they have the same pressure. Now suppose we break that link and establish a link between system A and a third system, C, equipped with a piston.  Suppose we observe no change: we infer that the systems A and C are in mechanical equilibrium and we can go on to say that they have the same pressure.  Now suppose be break that link and put system C in mechanical contact with system B. Even without doing the experiment, we know what will happen:  nothing.  Because systems A and B have the same pressure, and A and C have the same pressure, we can be confident that systems C and B have the same pressure, and that pressure is a universal indicator of mechanical equilibrium.

Now we move from mechanics to thermodynamics and the world of the zeroth law.  Suppose that system A has rigid walls made of metal and system B likewise.  When we put the two systems in contact, they might undergo some kind of physical change. For instance, their pressures might change or we could see a change in colour through a peephole. In everyday language we would say that ‘heat has flowed from one system to the other’ and their properties have changes accordingly.  Don’t imagine, though, that we know what heat is yet: that mystery is an aspect of the first law, and we aren’t even at the zeroth law yet.

It may be the case that no change occurs when the two systems are in contact even though they are made of metal. In that case we say that the two systems are in thermal equilibrium. Now consider three systems, just as we did when talking about mechanical equilibrium. It is found that if A is put in contact with B and found to be in thermal equilibrium, and B is put in contact with C and found to be in thermal equilibrium, the when C is put in contact with A, it is always found that the two are in thermal equilibrium. This rather trite observation is the essential content of the zeroth law of thermodynamics:

if A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C will be in thermal equilibrium with A.

The zeroth law implies that just as the pressure is a physical property that enables us to anticipate when systems will be in mechanical equilibrium when brought together regardless of their composition and size, then there exists a property that enables us to anticipate when two systems will be in thermal equilibrium regardless of their composition and size: we call this universal property the temperature. We can now summarize the statement about the mutual thermal equilibrium of the three systems simply by saying that they all have the same temperature.

We are not yet claiming that we know what temperature is, all we are doing is recognizing that the zeroth law implies the existence of a criterion of thermal equilibrium: if the temperatures of two systems are the same, then they will be in thermal equilibrium when put in contact through conducting walls and an observer of the two systems will have the excitement of noting that nothing changes.

We can now introduce two more contributions to the vocabulary of thermodynamics. Rigid walls that permit changes of state when closed systems are brought into contact – that is, in the language of Chapter 2, permit the conduction of heat – are called diathermic (from the Greek words for ‘through’ and ‘warm’). Typically, diathermic walls are made of metal, but any conducting material would do.  Saucepans are diathermic vessels. If no change occurs, then either the temperatures are the same or – if we know that they are different – then the walls are classified as adiabatic (‘impassable’).  We can anticipate that walls are adiabatic if they are thermally insulated, such as in a vacuum flask or if the system is embedded in foamed polystyrene.

The zeroth law is the basis of the existence of a thermometer, a device for measuring temperature. A thermometer is just a special case of the system B that we talked about earlier. It is a system with a property that might change when put in contact with a system with diathermic walls. A typical thermometer makes use of the thermal properties of material.  Thus, if we have a system B (‘the thermometer ‘) and put it in thermal contact with A and find that the thermometer does not change, and then we put the thermometer in contact with C and find that it still doesn’t change, then we can report that A and C are at the same temperature.

There are several scales of temperature, and how they are established is fundamentally the domain of the second law (see Chapter 3). However, it would be too cumbersome to avoid referring to these scales until then, though formally that could be done, and everyone is aware of the Celsius (centigrade) and Fahrenheit scales.  The Swedish astronomer Anders Celsius (1701-1744) after whom the former is named devised a scale on which water froze at 100^o and boiled at 0^o, the opposite of the current version of his scale (0^oC and 100^oC, respectively). The German instrument maker Daniel Fahrenheit (1686-1736) was the first to use mercury in a thermometer: he set 0^o at the lowest temperature he could reach with a mixture of salt, ice, and water, and for 100^o he chose his body temperature, a readily transportable but unreliable standard. On this scale water freezes at 32^oF and boils at 212^oF.

The temporary advantage of Fahrenheit’s scale was that with the primitive technology of the time, negative values were rarely needed. As we shall see, however, there is an absolute zero of temperature, a zero that cannot be passed and where negative temperatures have no meaning except in a certain formal sense, not one that depends on the technology of the time (se Chapter 5). It is therefore natural to measure temperatures by setting 0 at this lowest attainable zero and to refer to such absolute temperatures as the thermodynamic temperature. Thermodynamic temperatures are denoted T, and whenever that symbol is used in this book, it means the absolute temperature with T = 0 corresponding to the lowest possible temperature.  The most common scale of thermodynamic temperatures is the Kelvin scale, which uses degrees (‘kelvins’, K) of the same size as the Celsius scale. On this scale, water freezes at 273 K (that is, at 273 Celsius-sized degrees above absolute zero; the degree sign is not used on the Kelvin scale) and boils at 373 K. Put another way, the absolute zero of temperature lies at -273^oC. Very occasionally you will come across the Rankine scale, in which absolute temperatures are expressed using degrees of the same size as Fahrenheit’s.

*

In each of the first three chapters I shall introduce a property from the point of view of an external observer. Then I shall enrich our understanding by showing how that property is illuminated by thinking about what is going on inside the system. Speaking about the ‘inside’ of a system, its structure in terms of atoms and molecules, is alien to classical thermodynamics, but it adds deep insight, and science is all about insight.

Classical thermodynamics is the part of thermodynamics that emerged during the nineteenth century before everyone was fully convinced about the reality of atoms, and concerns relationships between bulk properties. You can do classical thermodynamics even if you don’t believe in atoms.  Towards the end of the nineteenth century, when most scientists accepted that atoms were real and not just an accounting device, there emerged the version of thermodynamics called statistical thermodynamics, which sought to account for the bulk properties of matter in terms of its constituent atoms. The ‘statistical’ part of the name comes from the fact that in the discussion of bulk properties we don’t need to think about the behavior of individual atoms but we do need to think about the average behaviour of myriad atoms. For instance, the pressure exerted by a gas arises from the impact of its molecules on the walls of the container; but to understand and calculate that pressure, we don’t need to calculate the contribution of every single molecule: we can just look at the average of the storm of molecules on the walls.  In short, whereas dynamics deals with the behaviour of individual bodies, thermodynamics deals with the average behaviour of vast numbers of them.

The central concept of statistical thermodynamics as far as we are concerned in this chapter is an expression derived by Ludwig Boltzmann (1844-1906) towards the end of the nineteenth century. That was not long before he committed suicide, partly because he found intolerable the opposition to his ideas from colleagues who were not convinced about the reality of atoms.  Just as the zeroth law introduces the concept of temperature from the viewpoint of bulk properties, so the expression that Boltzmann derived introduces it from the viewpoint of atoms, and illuminates its meaning.

To understand the nature of Boltzmann’s expression, we need to know that an atom can exist with only certain energies.

Less than one minute after this "Great Radiance," protons and neutrons emerged and combined to form nuclei of the simplest elements, mostly hydrogen and helium.  About 300,000 years later, electrons combined with these nuclei for the first time, transforming matter into hydrogen and helium atoms.

This is the domain of quantum mechanics, but we do not need any of that subject’s details, only that single conclusion. At a given temperature – in the bulk sense – a collection of atoms consists of some in their lowest energy state (their ‘ground state’), some in the next higher energy state, and so on, with populations that diminish in progressively higher energy states.  When the populations of the states have settled down into their ‘equilibrium’ populations, and although atoms continue to jump between energy levels there is no net change in the populations, it turns out that these populations can be calculated from a knowledge of the energies of the states and a single parameter, B (beta).

Populations of the various energy states can be calculated from a knowledge of the energies of the states and beta.

Another way of thinking about the problem is to think of a series of shelves fixed at different heights on a wall, the shelves representing the allowed energy states and their heights the allowed energies. The nature of these energies is immaterial: they may correspond, for instance, to the translational, rotational, or vibrational motion of molecules.  Then we think of tossing balls (representing the molecules) at the shelves and noting where they land.  It turns out that the most probable distribution of populations (the numbers of balls that land on each shelf) for a large number of throws, subject to the requirement that the total energy has a particular value, can be expressed in terms of that single parameter B.

The precise form of the distribution of the molecules over their allowed states, or the balls over the shelves, is called the Boltzmann distribution. This distribution is so important that it is important to see its form. To simplify matters, we shall express it in terms of the ratio of the population of a state of energy E to the population of the lowest state, energy 0:

                                    Population of state of energy E

                                    ---------------------------------------- = e^-BE

                                    Population of state of energy 0

We see that for states of progressively higher energy, the populations decrease exponentially: there are fewer balls on the high shelves than on the lower shelves.  We also see that as the parameter B increases, then the relative population of a state of given energy decreases and the balls sink down on to the lower shelves. They retain their exponential distribution, with progressively fewer balls in the upper levels, but the populations die away more quickly with increasing energy.

When the Boltzmann distribution is used to calculate the properties of a collection of molecules, such as the pressure of a gaseous sample, it turns out that it can be identified with the reciprocal of the (absolute) temperature. Specifically, B = 1/kT, where k is a fundamental constant called Boltzmann’s constant.  To bring B into line with the Kelvin temperature scale, k has the value 1.38 X 10^-23 joules per kelvin. 1 The point to remember is that, because B is proportional to 1/T, as the temperature goes up, B goes down, and vice versa.

1.     Energy is reported in joules (J): 1J = 1 kg m^2 s^-2.  We could think of 1 J as the energy of a 2kg ball travelling at 1 m s^-1. Each pulse of the human heart expends an energy of about 1 J.

There are several points worth making here. First, the huge importance of the Boltzmann distribution is that it reveals the molecular significance of temperature:  temperature is the parameter that tells us the most probable distribution of populations of molecules over the available states of a system at equilibrium. When the temperature is high (B low), many states have significant populations; when the temperature is low (B high), only the states close to the lowest state have significant populations. (Figure 4). Regardless of the actual values of the populations, they invariably follow an exponential distribution of the kind given by the Boltzmann expression.  In terms of our balls-on-shelves analogy, low temperatures (high B) corresponds to our throwing the balls weakly at the shelves so that only the lowest are occupied. High temperatures (low B) corresponds to our throwing the balls vigorously at the shelves, so that even high shelves are populated significantly. Temperature, then, is just a parameter that summarizes the relative populations of energy levels in a system at equilibrium.

Caption for Figure 4.The Boltzmann distribution is an exponentially decaying function of the energy.  As the temperature is increased, the populations migrate from lower energy levels to higher energy levels.  At absolute zero, only the lowest state is occupied; at infinite temperature, all states are equally populated.

The second point is that B is a more natural parameter for expressing temperature than T itself.  Thus, whereas later we shall see that absolute zero of temperature (T = 0) is unattainable in a finite number of steps, which may be puzzling, it is far less surprising that an infinite value of B (the value of B when T = o) is unattainable in a finite number of steps. However, although B is the more natural way of expressing temperatures, it is ill-suited to everyday use. Thus water freezes at 0^oC (273 K), corresponding to B = 2.65 X 10^20 J^-1, and boils at 100^oC (373 K), corresponding to B = 1.94 X 10^20 J^-1. These are not values that spring readily off the tongue.  Nor are the values of B that typify a cool day (10^oC, corresponding to 2.56 X 10^20 j^-1) and a warmer one (20^oC, corresponding to 2.47 X 10^20 J^-1).

The third point is that the existence and value of the fundamental constant k is simply a consequence of our insisting on using a conventional scale of temperature rather than the truly fundamental scale based on B. The Fahrenheit, Celsius, and Kelvin scales are misguided: the reciprocal of temperature, essentially B, is more meaningful, more natural, as a measure of temperature.  There is not hope, though, that it will ever be accepted, for history and the potency of simple numbers, like 0 and 100, and even 32 and 212, are too deeply embedded in our culture, and just too convenient for everyday use.

Although Boltzmann’s constant k is commonly listed as a fundamental constant, it is actually only a recovery from a historical mistake.  If Ludwig Boltzmann had done his work before Fahrenheit and Celsius had done theirs, then it would have been seen that B was the natural measure of temperature, and we might have become used to expressing temperatures in the units of inverse joules with warmer systems at low values of B and cooler systems at high values. However, conventions had become established, with warmer systems at higher temperatures than cooler systems, and k was introduced, through kB = 1/T, to align the natural scale of temperature based on B to the conventional and deeply ingrained one based on T. thus, Boltzmann’s constant is nothing but a conversion factor between a well-established conventional scale and the one that, with hindsight, society might have adopted.  Had it adopted B as its measure of temperature, Boltzmann’s constant would not have been necessary.

We shall end this section on a more positive note. We have established that the temperature, and specifically B, is a parameter that expresses the equilibrium distribution of the molecules of a system over their available energy states.  One of the easiest systems to imagine in this connection is a perfect (or ‘ideal’) gas, in which we imagine the molecules as forming a chaotic swarm, some moving fast, others slow, travelling in straight lines until one molecule collides with another, rebounding in a different direction and with a different speed, and striking the walls in a storm if impacts and thereby giving rise to what we interpret as pressure. A gas is a chaotic assembly of molecules (indeed, the words ‘gas’ and ‘chaos’ stem from the same root), chaotic in spatial distribution and chaotic in the distribution of molecular speeds.  Each speed corresponds to a certain kinetic energy, and so the Boltzmann distribution can be used to express, through the distribution of molecules over their possible translational energy states, their distribution of speeds, and to relate that distribution of speeds to the temperature.  The resulting expression is called the Maxwell-Boltzmann distribution of speeds, for James Clerk Maxwell (1831-1879) first derived it in a slightly different way.  When the calculation is carried through, it turns out that the average speed of the molecules increases as the square root of the absolute temperature.  The average speed of molecules in the air on a warm day (25^oC, 298 K) is greater by 4 per cent than their average speed on a cold day (0^oc, 273 K). Thus, we can think of temperature as an indication of the average speeds of molecules in a gas, with high temperatures corresponding to high average speeds and low temperatures to lower average speeds  (Figure 5).

Caption for Figure 5. The Maxwell-Boltzmann distribution of molecular speeds for molecules of various mass and at different temperatures.  Note that light molecules have higher average speeds than heavy molecules.  The distribution has consequences for the composition of planetary atmospheres, as light molecules (such as hydrogen and helium) may be able to escape into space.

A word or two of summary might be appropriate at this point.  From the outside, from the viewpoint of an observer stationed, as always, in the surroundings, temperature is a property that reveals whether, when closed systems are in contact through diathermic boundaries, they will be in thermal equilibrium – their temperatures are the same – or whether there will be a consequent change of state – their temperatures are different – that will continue until the temperatures have equalized. From the inside, from the viewpoint of a microscopically eagle-eyed observer within the system, one able to discern the distribution of molecules over the available energy levels, the temperature is the single parameter that expresses those populations. As the temperature is increased, that observer will see the population extending up to higher energy states, and as it is lowered, the populations relax back to the states of lower energy.  At any temperature, the relative populations of a state varies exponentially with the energy of the state. That states of higher energy are progressively populated as the temperature is raised means that more and more molecules are moving (including rotating and vibrating) more vigorously, or the atoms trapped at their locations in a solid are vibrating more vigorously about their average positions. Turmoil and temperature go hand in hand.

Notes and Journal