Three Laws of Nature

Three Laws of Nature

A Little Book on Thermodynamics

R. Stephen Berry
Published with assistance from the foundation established in memory of Philip Hamilton McMillan of the
Class of 1894, Yale College.

Copyright © 2019 by R. Stephen Berry.

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10 9 8 7 6 5 4 3 2 1
 Contents

Preface

ONE
What Is Thermodynamics?
The First Law

TWO
Why We Can’t Go Back in Time
The Second and Third Laws

THREE
How Did Classical Thermodynamics Come to Exist?

FOUR
How Do We Use (and Might We Use) Thermodynamics?

FIVE
How Has Thermodynamics Evolved?

SIX
How Can We Go Beyond the Traditional Scope of Thermodynamics?

SEVEN
What Can Thermodynamics Teach Us About Science More Generally?

Index
 Preface

This book has a very specific purpose: to use the science of thermodynamics as a
paradigm to show what science is, what science does and how we use it, how
science comes to exist, and how science can evolve as we try to reach and
address more and more challenging questions about the natural world. It is
intended primarily for people with little or no background in science, apart,
perhaps, from some exposure in school or college to “science for everyone”
courses.
The approach, quite frankly, came from three stimuli. One was a course I
gave for non-scientist undergraduates at the University of Chicago that evolved
over years. The second was an adult education course that grew out of that
undergraduate course—but was at a somewhat lower level, primarily in terms of
the mathematics. The third, first in time, which had a very crucial influence on
that evolution, was a pair of essays published in 1959 by the British scientist and
novelist C. P. Snow, titled “The Two Cultures,” based on lectures he had given at
Cambridge University. Snow said, in essence, that those who belong to the
culture of scientists know vastly more about the culture of humanities than the
reverse, and that this is a serious problem for our society. He argued,
metaphorically, that if we were to achieve a proper balance, the non-scientists
would know as much about the second law of thermodynamics as the scientists
know about Shakespeare. In a later essay, actually published in 1964 with the
first two, he retreated and replaced the second law of thermodynamics with
contemporary biology.
I might have disagreed with his change even then, but as biology has
advanced it is very clear that, in our time, he was right in the first place. Why?
To understand biology today, one must learn many, many facts, but one needs
very few facts to understand thermodynamics. It is a subject rooted in a few
concepts, far more than on systematic inferences from vast amounts of
information, as contemporary biology is now.
This book is written for people who are interested in what science is and does,
and how it comes to be, but who have little or no prior substantive knowledge of
any science. In writing for this audience, there was a clear choice to be made
regarding the order of the chapters. Would the ideas be clearer if the history
came first, or if the first thing one read was the substance of the science as it is
today? I have chosen the latter. Thus I first show, in Chapters 1 and 2, what the
science of thermodynamics is as we now use it almost constantly, and then, in
Chapter 3, address the question of how we arrived where we are.
The first two chapters develop an overview of what the traditional science of
thermodynamics is, of the concepts that embody it, and how it uses these
concepts. These chapters are built around the three laws of thermodynamics,
namely the conservation of energy, the inevitability and direction of change with
time, and the existence of a limiting absolute zero of temperature.
The third chapter presents the history of the science. There we see how the
concepts, the variables, and the tools that we call the laws of thermodynamics
evolved. This is important because some of the concepts of thermodynamics are
so ubiquitous and pervasive in our thinking and language that they seem obvious
and trivially simple to us, where, in reality, they evolved only with great
difficulty, and remain in many ways amazingly subtle and not so simple at all.
That history has been, and even still is, a tortuous but fascinating path, especially
when we understand where and what thermodynamics is today. And it offers a
rich insight into how a science evolves, often having to resolve competing,
conflicting concepts.
The fourth chapter describes the ways we apply thermodynamics, especially
how we use information from this science to improve the performances of many
things we do in everyday life. The fifth chapter describes the way the science of
thermodynamics has continued to evolve since its pioneers established its
foundations. This is, in a sense, a contemporary counterpart to Chapter 3.
Despite the elegant and in many ways the most natural presentation of the
subject, in a formal, postulational and essentially closed format, the subject has
continued to undergo evolutionary changes, expanding and strengthening as new
discoveries require that we extend our concepts. A crucial advance discussed in
this chapter is the connection between the macroscopic approach of traditional
thermodynamics and the microscopic description of the world, based on its
elementary atomic building blocks. This link comes via the application of
statistics, specifically through what is called “statistical mechanics.”
The sixth chapter examines the open challenges that we face if we wish to
build upon the thermodynamics we now have to provide us with tools to go
beyond, to other, still more difficult and usually more complex problems of how
nature functions. It addresses questions of whether and when there are limits to
what a particular science can tell us, and whether inferences we make using the
tools and concepts of a science can sometimes have limits of validity. This
chapter reveals how this science continues to offer new challenges and
opportunities, as we recognize new questions to ask.
The seventh and final chapter addresses the question of how thermodynamics,
one particular example of a kind of science, can show us what a science is and
does, and hence why we do science at all. This chapter is a kind of overview, in
which we look at how thermodynamics can serve as a paradigm for all kinds of
science, in that it reveals what scientific knowledge is, how it differs from
knowledge in other areas of human experience, how scientific knowledge, as
illustrated by thermodynamics, has become deeply integrated into how we live,
and how we can use scientific knowledge to help guide human existence.
A comment directly relevant to the intent and content of this book:
“Thermodynamics is the science most likely to be true.” This is a paraphrase of a
comment made by Albert Einstein; he was putting thermodynamics in the larger
context of all of science. Einstein certainly recognized that every science is
always subject to change, to reinterpretation, even to deep revision; he himself
was responsible for such deep changes in what had been accepted as scientific
laws. His comment reflects his belief that thermodynamics contains within itself
aspects—concepts and relations—that are unlikely to require deep revision,
unlike the way quantum theory revised mechanics and relativity revised our
concepts of space and gravitation. Was he right?

I would like to acknowledge the very helpful comments from Alexandra Oleson
and Margaret Azarian, both experienced and skilled at editing.
Three Laws of Nature
 ONE
What Is Thermodynamics?
The First Law

Thermodynamics occupies a special niche among the sciences. It is about

everything, but in another sense it is not about anything truly real—or at least,
not anything tangible. As far as we know, it applies to everything we observe in
the universe, from the smallest submicroscopic particles to entire clusters of
galaxies. Yet in its classic form, thermodynamics is about idealized systems:
systems that are in a state of equilibrium, despite the obvious fact that nothing in
the universe is ever truly in equilibrium, in the strictest sense. Everything in our
observable universe is constantly changing, of course—but much of this change
happens so slowly that we are unaware of it. Consequently, it is often very useful
—and even valid—to treat a system that is not changing in any readily
observable way as if it were in equilibrium. Here, traditional thermodynamics
shows its full power and allows us to use its concepts to describe such a system.
At this point, we need to be specific about what we mean by “equilibrium,”
and a state of equilibrium. In thermodynamics, we usually treat systems of
macroscopic size—ones that are large enough to see, at least with a magnifying
glass, or vastly larger—but not systems on an atomic scale, or even thousands of
atoms. (Strictly, we can use thermodynamics to treat macroscopic assemblages
of very small things—collections of many, many atoms or molecules, for
example—but not those small things individually.) We deal with properties of
these macroscopic systems that we can observe and measure, such as
temperature, pressure, and volume. We specifically do not include the motions
and collisions of the individual molecules or atoms that constitute the systems.
After all, thermodynamics developed before we understood the atomic and
molecular structure of matter, so we can use its concepts without any explicit
reference to the behavior of those constituent particles.
This allows us to give meaning to equilibrium, and states of equilibrium. If
the properties we use to describe the state of a macroscopic system, such as its
temperature, its composition, its volume, and the pressure it is subject to, remain
unchanged for as long as we observe that system, we say it is in equilibrium. It
may be in equilibrium only for a matter of minutes, or hours, but if, for example,
it is a system in the outdoor environment subject to changes in temperature from
day to night, then we might say the system is in equilibrium for minutes or
perhaps a few hours, but its state changes as night sets in and the temperature of
the surroundings—and hence of the system, too—drops. So in this example,
equilibrium is a state that persists only so long as the key properties remain
unchanging, perhaps for minutes or hours. But during those time periods, we say
the system is in equilibrium. Even when the temperature of the system is
changing but only very, very slowly, so slowly that we cannot measure the
changes, we can treat the system as if it is in equilibrium. Alternatively, imagine
that our observation takes months, so that we see only an average temperature,
never the daytime or nighttime temperature. This long-term observation shows
us a different kind of equilibrium from what our minute-long or hour-long
observation reveals. The key to the concept of equilibrium lies in whether we are
unable to see any changes in the properties we use to characterize the
macroscopic state of the system we are observing, which, in turn, depends on the
length of time our observation requires. Thermodynamics typically involves
measurements that we can do in minutes, which determines what we mean by
“equilibrium” in practice.
Of course this science has evolved to say some extremely important things
about phenomena that are not in equilibrium; as we shall see later, much of the
evolution of the subject in recent times has been to extend the traditional subject
in order to describe aspects of systems not in equilibrium. But the
thermodynamics of systems not in equilibrium falls outside the classic textbook
treatment and, in a sense, goes beyond the main target of our treatment here. In
the last part of this work, we shall deal with situations out of equilibrium as a
natural extension of the subject, in order to illustrate how new questions can lead
thinking and inferences into new areas, thereby extending the science.
There is an interesting paradox associated with thermodynamics as a science
of systems in equilibrium, because the essential information we extract using
traditional thermodynamics centers on what and how changes can occur when
something goes from one state—one we can call an equilibrium state—to
another such state of equilibrium. It is, in a way, a science of the rules and
properties of change. Thermodynamics provides strict rules about what changes
can occur and what changes are not allowed by nature.
The first idea that comes to mind for most people when they hear the word
“thermodynamics” is that it’s about energy. And that is certainly correct; energy
is an absolutely central concept in thermodynamics. The first rule one encounters
on meeting this subject is a law of nature, known as the first law of
thermodynamics, which states that energy can be neither created nor destroyed,
but only transformed from one form to another. We shall have much to say about
this law; examining the process of its discovery tells us much about the nature of
science, about scientific understanding, and about energy itself.
Thermodynamics uses a set of properties to describe the states of systems and
the changes between these states. Several of these are commonplace concepts
that we use in everyday experience: temperature, pressure, and volume are the
most obvious. Mass, too, may appear. So may composition; pure substances and
mixtures don’t have exactly the same properties because we can typically vary
the ratios of component substances in mixtures. The relative amounts of different
substances in a mixture becomes a property characteristic of that mixture. Other
commonplace properties, such as voltage and current, can sometimes appear
when we describe the thermodynamics of an electrical system, and there are
similar concepts for magnetic systems.
But there are also some very special concepts—we can now call them
properties—that one may not encounter as readily outside the context of
thermodynamics or its applications. The most obvious of these is entropy, which
we shall have much to say about in Chapter 2. Entropy is the central concept
associated with the second law of thermodynamics, and is the key to the
direction things must move as time passes. (It is also now a key concept in other
areas, such as information theory.) There are still other very powerful properties
we shall examine that tell us about the limits of performance of systems, whether
they be engines, industrial processes, or living systems. These are the quantities
called “thermodynamic potentials.” Then there is one more law, the third law of
thermodynamics, which relates temperature and entropy, specifically at very,
very low temperatures.
In this introductory chapter and the next, we examine the laws of
thermodynamics and how these are commonly used for a variety of practical
purposes as well as for providing fundamental descriptions of how the universe
and things in it must behave. In carrying this out, we shall frequently refer to the
many observable (and measurable) properties as variables, because
thermodynamics addresses the way that properties vary, and how they often vary
together in specific ways. It is those ways that form the essential information
content of the laws of thermodynamics.

The Most Frequently Encountered Variables and the First Law

Thermodynamics approaches the description of nature using properties of matter

and how they change, in terms we encounter in everyday experience. In some
ways, this description is very different, for example, from the one we associate
with Newton’s laws of motion or the quantum theory. The laws of motion are the
natural concepts for understanding very simple systems such as billiard balls, or
the behavior of matter at the size scale of atoms and molecules. We refer to that
approach as a microscopic description, while thermodynamics we say is at a
macroscopic level. A thermodynamic description uses properties, mostly familiar
ones, that characterize the behavior of systems composed of many, many
individual elements, be they atoms, molecules, or stars making up a galaxy. As
we said, among them are temperature, pressure, volume, and composition, the
substances that constitute the system. In contrast, properties or variables
appropriate for a microscopic description include velocity, momentum, angular
momentum, and position. Those are appropriate for describing collisions of
individual atoms and molecules, for example. We simply can’t imagine wanting
to keep track of the motion of all the individual molecules of oxygen and
nitrogen in the air in a room. The key to a thermodynamic description expresses
the changes in the macroscopic properties or variables. The variables we use in
thermodynamics (and in some other approaches as well) describe phenomena on
the scale of our daily experience, not normally at the atomic level.
Consider how we use those macroscopic variables to describe individual
macroscopic objects, such as a baseball; we can determine its volume and
temperature, for example. However, when we treat a baseball as an individual
thing, as a sphere to be driven by colliding with a bat, paying no attention to its
microscopic, atomic-level structure, we can use the same variables we would use
at the micro level, mass, position, velocity, angular momentum. The important
distinction here is not the size of the individual objects but how simplistic our
description of the object is; are we concerned with how individual components
that make up the baseball behave and interact, or with the way the system
composed of those many individual components behaves as a whole?
There is, however, one property, energy, that is appropriate and important at
both levels, so it holds a very special place in how we look at nature. Because we
study its changes, we often refer to it as a variable. When we think of energy, we
inevitably think of temperature and heat. Temperature is almost universally
abbreviated as T, and we shall very often use that symbol. At this point,
however, we shall not try to be any more precise about what temperature is than
to say it is the intensity of heat, as measured by a thermometer, that people use
different scales to measure it, and that it is how we quantify the intensity of heat.
Obviously we are all familiar with the Fahrenheit scale, in which water
freezes at 32° F and boils at 212° F—when the pressure on that water has the
precise value we call 1 atmosphere. (This scale was introduced in the early
eighteenth century by Daniel Gabriel Fahrenheit, in the Netherlands.) Most
readers will also be familiar with the Celsius, or centigrade, scale, in which
water freezes at 0° C and boils at 100° C, again at a pressure of precisely 1
atmosphere. (It was Anders Celsius, a Swedish astronomer, who introduced this
scale in 1742.) The first letter of the name of the temperature scale, capitalized,
denotes the units and scale being used to measure the degrees of temperature—F
for Fahrenheit, C for Celsius, and at least one other we’ll encounter (which will
give us a more precise and less arbitrary way of defining temperature).
Temperature is a property unrelated to the amount of material or of space, so
long as we consider things that consist of many atoms or molecules; so is
density, the mass per unit of volume. We refer to such properties as intensive, in
contrast to such properties as total mass or volume, which do depend directly on
the amount of stuff. Those are called extensive properties.
At this point, pressure is a bit easier than temperature to define precisely: it is
the force per unit of area, such as the force exerted by 1 pound of mass resting
on 1 square inch of area under the attraction of the earth’s gravity at an altitude
of sea level. Likewise, we can use the force per unit area exerted by the
atmosphere on the earth’s surface at sea level as another measure of pressure.
Volume is still easier: it is simply the amount of space occupied by whatever
concerns us, and it can be measured, for example, in either cubic inches or cubic
centimeters.
Now we address some slightly more subtle properties that serve as variables
in thermodynamics. We said that temperature is a measure of the intensity of
heat, but how do we measure the quantity of heat? The classic way is in terms of
how heat changes the temperature of water: 1 calorie is the quantity of heat
required to raise the temperature of 1 cubic centimeter (cc) of water by precisely
1° C. Anticipating, we shall soon recognize that the quantity of heat is a quantity
of one form of energy. It is a form that can be contained as a property of an
unchanging system, or as a form of energy that passes from one body to another,
the amount of energy that is exchanged between a source and a receiver or sink.
The quantity of heat is conventionally designated as Q.
Next, we come to the concept that, as we shall see, was one of those that
motivated the creation of thermodynamics—namely, work. Work, in traditional,
classical mechanics, is the process of moving something under the influence of a
force. It may be accelerating a car, or lifting a shovel full of dirt, or pushing an
electric current through some kind of resisting medium such as the filament of
an incandescent light bulb. Whatever that force is, so long as there is anything
opposing that force, such as gravity when lifting a shovel or inertia when
accelerating a car, energy must be exchanged between the source of the work
and the source of the resistance. Thus work is that energy used to make
something specific happen. It may include work against friction in addition to
work against inertia, as it is with the work of moving a piston in an engine. Work
is conventionally abbreviated as W. The explicit forms used to evaluate work are
always expressions of particular processes, whether lifting a weight against
gravity, pounding a nail into a wooden board, pushing electrons through a
filament, or whatever. Work, like heat, is a form of energy that is done by some
source on some “sink,” whether it be a flying object accelerated by the work
done by its engine or a balloon being blown up by a child.
Work is energy exchanged in a very ordered way. Consider a piston in an
engine. When work is done to accelerate the piston, all of the atoms of the piston
move in concert; the piston may or may not change temperature, but it certainly
retains its identity. If, on the other hand, we heat the piston without moving it,
the individual atoms that constitute it move more vigorously as the temperature
of the piston increases. (We shall return to how this came to be understood in
Chapter 3, when we examine the history of the concepts of thermodynamics.)
Here, we simply recognize that work is energy transferred in a very orderly way,
while heat is energy transferred in a very random, disorderly way. Of course
when a real system or machine does work, there are invariably ways that some
of the energy put into the working device goes into what we call waste—notably
friction and heat losses. Yet in order to move a piston, for example, we must
accept that we need to overcome not only its inertia, but also the friction that is
inherent between the piston and its cylinder. It is often useful for conceptual
purposes to imagine ideal systems in which those “wasteful” and unwanted
aspects are absent; such idealization is often a useful way to simplify a physical
process and make it easier to understand and analyze. We shall see such
idealization introduced when we examine the contributions of the French
engineer Sadi Carnot in Chapter 3.
One very important thing to recognize from the outset is that heat and work
can both be expressed in the same units. They are both forms of energy,
expressible in calories, the unit we introduced to measure heat, or in ergs, the
basic small unit for work in the metric system, or any of several other units, such
as joules. (One erg is sometimes said to be “one fly push-up.” One joule is
10,000,000 ergs.) Both heat and work, however, must be in units that are
essentially a mass multiplying the square of a velocity v, or a mass m, multiplied
by the square of a length unit l, and divided by the square of a time unit t. We
can express this tersely thus, using square brackets to indicate “the dimensions
of”: [Q] = [W] = [ml2/t2]. In this context, “dimension” refers to a physical
property. The dimensions of this kind that we encounter most frequently are
mass, length, and time, but there are some others, such as electric charge.
Alternatively, we can express these dimensions in terms of a velocity v, because
velocity is distance per unit of time, or distance/time, l/t. Hence [Q] = [W] =
[mv2]. When we use those dimensions, we give quantitative units for them, such
as centimeters or inches for length, grams or kilograms or pounds for mass.
Velocity involves two basic dimensions, length and time, so it is measured in
units such as miles per hour or centimeters per second.
Heat and work are the two forms of energy that are always associated with
some process, and hence are called process variables. Temperature, pressure,
mass, and volume, on the other hand, describe states of systems (in equilibrium,
in principle), and hence are called state variables. We shall be using and relating
these concepts shortly.
For two quantities to be comparable, they must be expressed in the same
dimensions, and, of course, to make that comparison, the quantities must be
expressed in the same units, the same measurement scale. We can compare
lengths if they are expressed in inches or meters, but if some are in inches and
others are in meters or light-years, we cannot compare them until we convert
them all into a common set of units. Each dimension has its own set of units.
Length has many choices, such as inches, centimeters, or kilometers. Velocity,
with dimensions of distance divided by time, or [l/t], can be, for example, miles
per hour or meters per second. There are several different units that we use for
energy: calories, ergs, joules, for example, each of which can be expressed as
some mass unit, multiplying the square of some length unit, divided by the
square of some unit measuring time. We will address these options later, as we
deal with specific examples.
Now we notice something very important. You can recall from high school
science that the kinetic energy of a moving object with mass m, moving at a
velocity v, is precisely ½ mv2. This is a hint toward the very fundamental
concept that heat, work, and kinetic energy share a common basic characteristic:
they are all forms of energy. The grand, unifying concept that encompasses these
and many other things we can experience or recognize is energy. In the third
chapter, we shall see how this concept evolved through many years of
experiment and thought and even controversy. (In fact, it is even continuing to
evolve and expand now, as you read this.) Here, we shall assume that energy is
such a widely used concept that the reader has an intuitive sense of what it is.
How precise or sophisticated that understanding may be is not important at this
point.
The most important of energy’s characteristics here is that it can appear in
many, many forms. Energy is stored in chemical bonds in coal or natural gas in a
form that can be transformed in part into heat. This is achieved simply by
subjecting the coal or gas to a chemical reaction, namely combustion—simply
burning—which replaces the original chemical bonds between carbon atoms or
carbon-hydrogen bonds with new, stronger bonds to oxygen that weren’t present
in the original fuel. In this way, we transform some chemical energy into heat.
Likewise, we can transform some heat energy into electrical energy by using
heat to make steam, and then pushing the steam, at fairly high pressure, against
the blades of a turbine that rotates a coil of wire through the field of a magnet, a
process that yields electric current. This is a rather subtle, complex process,
because energy that is heat at our starting point becomes the energy that
transforms liquid water to steam and also heats it further; then that energy
becomes mechanical energy that rotates the turbine blades, which in turn
becomes electrical energy in the form of a current moving at some chosen
voltage. This everyday phenomenon is actually a remarkable demonstration of
some of the many forms that energy may take.
Here we can introduce the first law of thermodynamics, a phenomenon of
nature that is so familiar that we often fail to recognize how astonishing it is. The
first law tells us that energy can never be created or destroyed, but only
transformed from one form to others. We say that energy is conserved.
First, however, we need to say a little more about those variables of energy,
heat, and work. If a system, any system, is in an unchanging equilibrium state,
and is isolated from interacting with anything else, then certain of its properties
are also unchanging. One of those properties is the energy of that system. The
energy of an isolated system is a property, but, because of that isolation, it cannot
be changed; it is a constant, one that at least partially characterizes the properties
of that system, so long as the system cannot interact with anything outside itself.
We can think of our entire universe as such a system; so far as we can tell, there
is nothing else with which it can interact, and the principle of conservation of
energy says that therefore the total energy of the universe cannot change. We
frequently idealize small systems, such as objects we can use for experiments, as
being isolated from their environment; thus, for practical purposes, we can, for
example, wrap enough insulation around an object that for a long time, it
effectively has no contact or interaction with its surroundings. That way, it
cannot exchange energy with those surroundings, whether losing or gaining it. In
such a case, that isolated object must have an unchanging amount of energy. For
isolated systems, therefore, we say that energy is a property of the state of the
system, or energy is a state variable. Every isolated system has a precise energy
of its own, an energy that can only change by spoiling that isolation and allowing
the system to interact—more precisely, to take in or give off energy—with
whatever other system it is interacting with.
We can envision other kinds of states of equilibrium, states in which nothing
changes with time. One of the most useful is the state of a system that is in
contact with a thermostat, a device, perhaps idealized, that maintains that system
at a constant temperature T. The temperature is a property of the state of this
system, and hence is also a state variable. The mass m, the volume V, and the
pressure p are also state variables. Some state variables, such as mass and
volume, depend directly on the size of the system; double the size of the system
and you double its mass and volume. These are called extensive variables.
Others, typically temperature, density, and pressure, do not depend at all on the
size of the system; these are called intensive variables. Color is another intensive
characteristic, but is rarely relevant to thermodynamics—although it may be
important for energy transfer, as with absorption of light.
Now we turn to work and heat. Work is something that we do, or a system
does, when it changes its state and, typically, its environment as well. Likewise
heat is a form of energy that a system exchanges with its surroundings, either
absorbing it or depositing it. Neither work nor heat is a state variable; they are
both what we can call process variables. They are the variables describing the
way the system undergoes change, from one state to another. When a system
changes its state, it does so by exchanging work, heat, or both with its
surroundings. And this distinction, between state variables and process variables,
brings us to an effective way to express the first law of thermodynamics. Work
and heat are the only two kinds of process variables that we know and use. Of
course there are many kinds of work—mechanical, electrical, and magnetic, for
example. Although heat may pass from warmer to cooler bodies in various
forms, we know only one kind of heat. Work is energy being transferred from a
source into a very specific form, such as the motion of a piston or the initiation
and maintenance of an electric current. Heat is perhaps best understood by
thinking of it at a microscopic level; at the macroscopic level, the amount of heat
in a system determines the temperature of that system. However at the
microscopic level, the heat contained in a system determines the extent to which
the component atoms and molecules move, in their random way. The more heat,
the more vigorous the motion of those tiny particles.
As a matter of convention, we will say that if heat is added to the system, then
the quantity of heat Q is positive. Likewise, if the system does work, we will say
that the amount of work W is positive. Hence positive Q increases the energy of
the system, and positive W takes energy from the system and thereby decreases
its energy. As an aside, we note that the convention for Q is the one generally
accepted. For the sign of W, however, both conventions are used by different
authors; some take positive W as work done on the system and hence increasing
its energy. Either convention is consistent and workable; we just have to decide
which we will use and stay with that.
Now we need to introduce some notation, in order to express the first law and
its consequences. We shall use the standard symbol E to represent energy,
specifically the amount of energy. Another standard representation we shall use
is the set of symbols generally taken to signify changes. To specify changes, it is
standard practice to use letters from the Greek alphabet. We denote very small
changes in any quantity, call it X, with the letter d or a Greek δ (lowercase delta),
either dX or δX. These typically can be so small that an ordinary observation
would, at best, barely detect them. We designate a very small change in pressure
p as dp or δp. Significant, observable changes in X we indicate with a Greek Δ
(uppercase delta), as ΔX. We designate a large change in the volume V of a
system as ΔV. With this bit of notation, we move toward stating the first law in a
form that makes it extremely useful.
The conservation of energy and the identification of energy as a state variable
for an isolated system tells us that if a system goes from some state, say #1, to
another state, say #2, then the change in the system’s energy is simply the
difference between the energies of states 1 and 2. We can write this as an
equation:

ΔE = E(2) – E(1).

The important point we must now recognize is that this change in energy doesn’t
depend in any way on how we bring the system from state 1 to state 2. We might
be able to make the change by heating or cooling alone, or by doing or extracting
work with no exchange of heat with an outside source, or by some combination
of the two. No matter how we take the system from state 1 to state 2, the energy
change is the same. This means that however we exchange heat or perform or
extract work, their difference must be the same fixed amount, the difference
between E(1) and E(2). We can write this as an equation, either for measurable
changes of energy, as
 ΔE = Q - W

or in terms of even the smallest changes, as

δE = δQ - δW.

These equations say that however we put in or take out heat, and however the
system does work or has work done on it, the difference between the heat taken
in and the work done is precisely equal to the change in energy of the system
from its initial to its final state. It is these equations that are our terse statement
of the first law of thermodynamics. These two equations tell us that however
much work we do or extract to take our system, say our engine, from its initial
state 1 to its final state 2, there must be an exchange of heat corresponding
precisely to that amount of work so that the change of energy is just that
difference between the two states. Said another way, these equations tell us that
there is no way to create or destroy energy; we can only change its form and
where it is.
One very simple example illustrates this principle. The energy of a 20-pound
weight is higher if it is on top of a 50-foot pole than if it is on the ground at the
base of the pole. But the difference between the energies of that weight does not
depend in any way on how we brought it to the top of the pole. We could have
hoisted it up directly from the ground, or dropped it from a helicopter hovering
at an altitude of 100 feet; the difference in the energy of the ground-level state
and the state 50 feet higher in no way depends on how the states were produced.
This also tells us something more about how real machines operate. Think of
an engine such as the gasoline engine that drives a typical automobile. Such an
engine operates in cycles. Strictly, each cylinder goes through a cycle of taking
in fuel and air, igniting and burning the fuel, using the heat to expand the gas-air
mixture, which pushes the moveable piston of the cylinder. The moving piston
connects via a connecting rod to a crankshaft and the expansion in the cylinder,
pushing the piston, in turn pushes the crankshaft to make it rotate. That
crankshaft’s rotation eventually drives the wheels of the car. The important part
of that complex transformation for us at this point is simply that each cylinder of
the engine goes regularly through a cycle, returning to its initial state, ready to
take in more fuel and air. In the process of going through the cycle, the burning
of the gasoline in air releases energy stored in chemical bonds and turns it into
heat. The engine turns that heat into work by raising the pressure of the burned
gasoline-air mix enough to push the piston through the cylinder. At the end of
each cycle of each cylinder, the piston returns to its original position, having
converted chemical energy to heat, and then, transformed that heat into work.
In an ideal engine, with no friction or loss of heat to the surroundings, the
engine would return to its initial state and as much as possible of the chemical
energy that became heat would have become work, turning the wheels of the car.
(“As much as possible” is a very important qualifier here, which will be a central
topic of a later discussion.) Such an engine, first envisioned by the French
engineer Sadi Carnot, has the remarkable property that at any stage, it can move
equally easily in either direction, forward or backward. We call such an ideal
engine or process a reversible engine or reversible process. One extremely
important characteristic of reversible processes is that they can only operate
infinitely slowly. At any instant, the state of a reversible engine is
indistinguishable from—or equivalent to—a state of equilibrium, because it can
go with equal ease along any direction. Such reversible processes are ideal and
unreachable limits corresponding to the (unreachable) limits of the best possible
performance that any imaginable engine could do. However, we can describe
their properties and behavior precisely because they are the limits we can
conceive by extrapolating the behavior of real systems. They would operate
infinitely slowly, so obviously are only useful as conceptual limits, and in no
way as anything practical.
Of course real engines are never those ideal, perfect machines. They always
have some friction, and there is always some leakage of heat through the metal
walls of the engine to the outside. That the engine feels hot when it has been
running testifies to the loss of heat to the environment; that we can hear the
engine running is an indication that there is friction somewhere that is turning
some, maybe very little, of that energy that was heat into sound waves, generated
by friction. But we try to minimize those losses by using lubricating oil, for
example, to keep the friction as low as we can. We make real engines that are as
much like those ideal engines as we can, but of course require the real engines to
operate at real rates, not infinitely slowly, to deliver the work we want from
them.
One very important, inherent characteristic of what we call energy is the
remarkably broad scope of what it encompasses, as we already have begun to
see. Every activity, every process, even every thing there is has associated with it
an energy or energy change. The concept of energy includes, in its own way,
some aspect of virtually everything in human experience. In the next chapter, we
shall examine how this concept evolved and grew, as human experience and
understanding of nature grew. At this stage, we need only recognize that energy
is a property we can identify, often measure, and detect in an amazingly broad
variety of forms. Light, sound, electricity and magnetism, gravitation, any
motion whether constant or changing, the “glue” that holds atomic nuclei
together, even mass itself, are all examples of the manifestation of energy. And
now, there are even forms of energy that we do not yet understand, what we call
“dark matter” and “dark energy.” That the human mind has recognized the very
existence of such a universal manifestation of the natural world is itself a source
of amazement and wonder.

A Closer Look at Some Basic Concepts for the Next Big Step

First, we consider temperature. Strictly, the concept evolved as a descriptive

property only of systems in equilibrium. In practice today, it is also applicable in
many circumstances, but not all, to systems undergoing transitions from one
state to another. All that is required to do this is that the process is slow enough
to put in some kind of temperature-measuring device that will give us a number
for that place where we inserted our thermometer. Our intuitive sense of
temperature is a measure of an intensity, specifically of how much energy a
system contains in any part of itself, and intuitively, that form of energy that we
recognize as the degree of warmness or coolness. (We choose to use these words,
rather than “the degree or intensity of heat” because we will shortly return to
“heat” as a related but different concept.) A system in equilibrium has the same
temperature everywhere in that system. It doesn’t matter whether the system is
small or large; temperature is a property that is independent of the size of the
system, and hence is an intensive property. We shall say a bit more below about
properties and variables that are neither intensive nor extensive.
Heat is the form of energy that most often changes the temperature of a
system. Heat, as the term is used in the sciences, is a property associated with the
change of energy of something, with its change of state, not a property of the
state of the system. As we saw, properties or variables that describe changes—
gains or losses—are called process variables, in contrast to state variables. The
concept of heat went through a fascinating history that we examine in the third
chapter. In essence, there were two incompatible ideas of what heat really is.
One view saw heat as an actual fluid, which was given the name “caloric”; the
other view saw heat as a measure of the degree of intensity of motion of the
small particles that constitute matter. This latter concept emerged even before the
clear establishment of the atomic structure of matter, although the evolution of
the two concepts linked the ideas of heat and atomic structure. (Of course the
notion that matter is composed of atoms had its origin in ancient Greece, with
Democritus and Lucretius, but the concept remained controversial for centuries.)
Today, of course, we interpret heat and temperature in terms of the motions of
the atoms and molecules that are the basic constituents of matter. Temperature is
the specific measure of the average motion of each elemental component, and
more specifically, of the random motion of the atoms and molecules; heat is the
form of energy that directly changes that random motion of the atoms and
molecules. If, for example, a substance absorbs light of a specific wavelength,
then in most typical situations, that light energy is quickly converted from being
concentrated in the particular atomic or molecular structure that acts as an
antenna to absorb it, into energy distributed evenly, on average, throughout the
entire system, in all the kinds of motion and other ways that energy can exist in
the system. This is what happens, for example, when sunlight warms your skin;
the energy in the visible, ultraviolet, and infrared solar radiation strikes your
body, your skin absorbs that radiation and converts it to the motion of the atoms
that make up your skin and muscles that then feel warmed. We say that the
energy is equipartitioned, distributed evenly among all the places it can be, when
the system has come to equilibrium. The average amount of energy in any part
of a system at a specific temperature is the same as in any other connected part
of the system. We shall see, however, that it is the average that is constant when
the temperature is constant, and that the amount of energy in any small part of a
system actually fluctuates in a random fashion, but with a specific probability
distribution.
The other property that can express a change of energy, and hence a change of
the state of a system, is work. If a piston moves to push a crankshaft, then all the
atoms of that piston move together to achieve that push. If there is any resistance
to the pushing force, then energy must be exchanged between the piston and
whatever is resisting the push. That energy must come from the system of which
the piston is a part, and must go to that other system resisting the push. (Think of
the inertia and friction that must be overcome by whatever energy a source
supplies.) In this situation, as whenever work is done, energy is exchanged, but
in a very non-random way. Work is, in that sense, completely different from
heat, a complementary way for energy to pass out of or into a system. Heat is
energy exchanged in the most randomly distributed way possible; work is energy
exchanged in the least randomly distributed way possible. In real systems, of
course, there are always frictional and other “losses” that prevent the complete
conversion of energy from its initial form into the intended work. Nevertheless,
we can imagine ideal engines in which those losses disappear, and build our
theoretical ideas with those ideal examples as the natural limits of real systems,
as we make them better and better. The ideal reversible engine is that kind of
ideal system.
We describe the forms in which energy can be stored and transferred through
the concept of degrees of freedom. Motion of an object in a specific direction is
kinetic energy in a single degree of freedom, the degree corresponding to motion
in that direction. A tennis ball can move in three independent directions in space,
which we identify with three degrees of freedom, specifically three translational
degrees of freedom. If it were confined to stay on a plane, on the ground of the
tennis court, it would have only two degrees of freedom of translation. But the
tennis ball can spin, too; it can rotate about three different axes, so, being a
three-dimensional object, it also has three rotational degrees of freedom. A
typical molecule, composed of two or more atoms, can exhibit vibrations of
those atoms as well. The atoms can oscillate toward and away from one another,
still retaining the basic structure of the molecule they form. Hence there are
vibrational degrees of freedom. Molecules, like tennis balls, can rotate in space,
and thus have rotational degrees of freedom as well. Every free particle, every
atom, brings with it its own three translational degrees of freedom. (Here, we
neglect the possibility that such a free particle might also be capable of rotation.)
When these atoms stick together to make molecules or larger objects, the nature
of the degrees of freedom changes but the total number of degrees of freedom,
the number of ways or places where energy can go, remains the same—three
times the number of particles that make up the system.
A complex object such as a molecule made of n atoms has 3n degrees of
freedom total, but only three of those degrees are degrees of translational
freedom, describing the motion of its center of mass. This is because those n
atoms are bound together into a relatively stable object. Besides its three
translations, the molecule has three degrees of rotational freedom, and all the
rest, 3n - 6, are vibrational. As an example to illustrate this, think of two
hydrogen atoms and an oxygen atom. If they are all free, each atom has three
translational degrees of freedom, nine in all; if, on the other hand, they are bound
together as a water molecule, two hydrogens attached to the oxygen, this system
has only three translational degrees of freedom, corresponding to motion of its
center of mass in the x, y, and z directions; however, it has three rotational
degrees of freedom, rotation about each of those three directions, and three
vibrational degrees of freedom, a symmetric stretching motion of the O-H bonds,
an asymmetric stretching motion with one bond extending while the other
shrinks, and a bending vibration. Hence the bound system has the same number
of degrees of freedom as the three free particles, but of different kinds. We shall
use this concept in several ways in the following discussion.
Some state variables such as temperature and pressure are, as we just saw,
independent of the size of the system; these are called intensive variables. The
other variables most frequently encountered, such as mass, volume, energy, and
entropy, depend directly on the size of the system, the amount of stuff that
composes that system. These are called extensive variables. Together, these are
the variables we use to describe systems that, for heuristic purposes, we can
validly treat as being in states of equilibrium, whose state variables do not
change on time scales such that we can observe them. (There is at least one other
kind of variable, exemplified by gravitational energy, discussed below, which
depends on the product of the masses of two interacting bodies, and hence is
quadratic in its dependence on the amount of matter.)
Much of the thermodynamics we encounter in everyday life can be
encompassed entirely with intensive and extensive variables. However, if we
wish to use thermodynamics to describe the behavior of a cloud of galaxies, for
example, for which a large fraction of the energy is gravitational, we can’t use
just that simple pair of characteristics because gravitational energy depends on
the product of the masses of pairs of attracting bodies, such as the earth and the
sun. This means that energy associated with gravitation depends on the product
of two interacting masses, not simply on a single mass, as with an extensive
variable. Hence astronomers wanting to use thermodynamics to portray such
things as galaxies and solar systems have had to develop a form of the subject
called nonextensive thermodynamics to address their systems. Here, however, we
shall be able to stay almost entirely within the scope of conventional
thermodynamics and its two kinds of variables.

Summary

We have seen how the first law of thermodynamics can be expressed in words:
energy can never be created or destroyed but only changed in form and location.
We have also seen how this law can be expressed in a very simple equation,

ΔE = Q - W

which tells us that ΔE, the change in energy of any system, is precisely the heat
Q that the system takes in, minus the work W that the system does on something
outside itself. It says that the change of energy of the system, from an initial state
to some final state, depends only on the difference of the energies of those two
states and not on the pathway taken to go from one to the other. It says that if a
system traverses a closed loop, that brings it back to its initial state, that the
system has undergone no change in its energy. It is a law we believe is valid
everywhere in the universe, a law that describes every form that energy can take.
When we realize all the forms that we know energy can have—heat, work,
electromagnetic waves, gravitation, mass—this becomes a remarkable, even
awesome statement.
 TWO
Why We Can’t Go Back in Time
The Second and Third Laws

The second law of thermodynamics is very different from the first law. The first
law is about something that doesn’t change, specifically the total amount of
energy, or even the amount of energy in some specific part of the universe,
closed off from the rest of the world. The second law is about things that can’t
happen, which can be interpreted as distinguishing things that can happen from
those that can’t—which is a way of showing what the direction of time must be.
The first explicit formulations of the second law were three verbal statements
of the impossibility of certain kinds of processes. In 1850, the German physicist
and mathematician Rudolf Clausius put it this way: no engine, operating
cyclically, can move heat from a colder to a hotter body without doing work.
Refrigerators can’t operate spontaneously. Then, the next year, the British
scientist William Thomson, Lord Kelvin, stated it differently: no engine,
operating cyclically, can extract heat from a reservoir at a constant temperature
and convert that heat entirely into work. The third of these statements, by
Constantin Carathéodory in 1909, is, in a sense, more general and less
specifically related to work than its predecessors; it says that every equilibrium
state of a closed system has some arbitrarily close neighboring states that cannot
be reached by any spontaneous process (or by the reversible limit of a
spontaneous process) involving no transfer of heat. “You can’t get there from
here.” This, one can see, encompasses the two earlier statements and carries with
it the implication that one may be able to go from state A to state B by a
spontaneous or limiting reversible process, yet be unable to go from B back to A
by such a process. This is a very fundamental way to express the unidirectional
nature of the evolution of the universe in time. For example, if state A consists of
a warm object X connected to a cooler object Y by some means that can conduct
heat, then X will cool and Y will warm until X and Y are at the same
temperature—and that is our state B. We well recognize that the system
consisting of X, Y, and a connector between them will go spontaneously from
the state A in which the temperatures of X and Y are different to the state B, in
which X and Y are in thermal equilibrium, but that we will never see the system
go from thermal equilibrium to two different temperatures.
At this point, we must introduce a new, fundamental concept that becomes a
most important variable now, for our second law of thermodynamics. This
variable is very different from energy, and the natural law associated with it,
likewise, is very different from the first law. This new, fundamental variable is
entropy. Of all the fundamental concepts in thermodynamics, entropy is surely
the one least familiar to most people. The concept of entropy arose as the
concepts of heat and temperature evolved. More specifically, entropy emerged
out of an attempt to understand how much heat must be exchanged to change the
temperature of an object or a system by a chosen amount. What is the actual heat
exchanged, per degree of temperature change in the system? Then the natural
next question is, “What is the absolute minimum amount of heat, for example
the number of calories, that must be exchanged to change the system’s
temperature by, say, 1 degree Celsius?” The most efficient engine imaginable
would exchange only the minimum amount of heat at each step of its operation,
with none wasted to friction or loss through the system’s walls to its
environment.
Now, in contrast to those earlier statements, the most widely used expression
of the second law is in terms of entropy. Entropy, expressed as a measure of the
amount of heat actually exchanged per degree of temperature change, means that
it is very closely related to the property of any substance called its heat capacity,
or, if it is expressed as a property of some specified, basic unit of mass of a
material, as specific heat. Precisely defined, the heat capacity of a substance is
the amount of heat it must absorb to increase its temperature T by 1°.
Correspondingly, the specific heat of that substance is the amount of heat
required to raise the temperature of one unit of mass by 1°. Typically, both are
expressed in units of calories per degree C or K, where K refers to the Kelvin
temperature scale, named for Lord Kelvin. The units of measure in this scale,
which are called kelvins, not degrees, are the same size as those of the Celsius
scale, but zero is at the lowest physically meaningful temperature, a presumably
unattainable level at which point absolutely no more energy can be extracted
from the system; at 0 K, there is zero energy. This point, which is often referred
to as absolute zero, is just below -273° C; hence, 0° C is essentially 273 K. (We
shall return to the Kelvin scale when we consider the third law of
thermodynamics.) At this point, we need to look at heat capacities and specific
heats, in order to probe this approach to the concept of entropy.
The heat capacity of anything obviously depends on its size; a quart of water
has twice the heat capacity of a pint of water. To attribute the temperature
response of a substance without having to refer to the amount of matter, we use
the heat capacity per unit of mass of that substance, which is the specific heat of
the substance. Thus the specific heat of water is 1 calorie per degree C, per gram
or cubic centimeter of water. Specific heat is an intensive quantity (which does
not depend on the size of the object); heat capacity is its corresponding extensive
quantity, proportional to the mass of the object. Hence the heat capacity of 1
gram of water is the same as its specific heat, but the heat capacity of 5 grams of
water is 5 calories per degree C. In the next chapter, we shall discuss the role
played by specific heats and heat capacities in the evolution of the concepts of
thermodynamics. Typical specific heats range from about 1.2 calories per degree
C per gram for helium to much lower values, such as approximately 0.5 for ethyl
alcohol, only 0.1 for iron, and 0.03 for gold and lead. Metals require
significantly smaller amounts of heat to raise their temperatures.
One subtlety we can mention here is the dependence of the specific heat and
heat capacity on the constraints of the particular process one chooses to heat or
cool. In particular, there is a difference between heating a system at a constant
pressure and heating it while fixing its volume. If the pressure is constant, the
volume of the system can change as heat is added or withdrawn, which means
the process involves some work, so we must make a distinction between the heat
capacities and specific heats under these two conditions.
The entropy change, when a substance absorbs or releases energy as heat, is
symbolized as ΔS, where S is the symbol generally used for entropy, and Δ, as
before, is the symbol meaning “change of.” As we discussed, we use Δ to
indicate a significant or observable change, while the lowercase δ indicates a
very tiny or infinitesimal change—of anything. Then when a system undergoes a
change of state by absorbing a tiny bit of heat, δQ, its entropy change δS must be
equal to or greater than δQ/T. Formally, we say δS ≥ δQ/T. The entropy change
must be equal to or greater than the amount of heat exchanged, per degree of
temperature of the system. Thus, for a given exchange of heat, the entropy
change is typically greater at low temperatures than at higher ones. An input of a
little heat at low temperatures makes a bigger difference in the number of newly
accessible states than at higher temperatures, and it is just that number of states
the system can reach that determines the entropy.
That is one quantitative way to state the second law of thermodynamics, in
terms of an inequality, rather than an equation of equality, such as the statement
of the first law. But that doesn’t yet tell us what entropy is, in a useful, precise
way. That only tells us that it is a property that is somehow bounded on one side,
that its changes can’t be less than some lower limit related to the capacity of a
system to absorb or release energy as heat. And that it depends inversely on the
temperature at which that heat exchange occurs and hence somehow on the
number of ways the system can exist. But from this inequality, we can get a first
sense of what entropy tells us. If a substance absorbs an amount δQ of heat and
all of that heat goes into raising the temperature of the substance according to
that substance’s heat capacity, then the process would have an entropy change δS
precisely equal to δQ/T. If some of the heat were to escape by leaking to the
surroundings, then the entropy change would be greater than δQ/T. Hence we
can already see that entropy somehow reflects or measures something wasteful
or unintended or disordered.
A simple example illustrates how temperature influences entropy. For a gas
with a constant heat capacity, the entropy increases with the logarithm of the
absolute temperature T. Hence the entropy doubles with each tenfold increase in
the absolute temperature; it doubles when the temperature goes from 1 K to 10
K. But it only doubles again when the temperature rises from 100 K to 1000 K!
The lower the temperature, the more a given rise in temperature (in degrees)
affects and thereby increases the system’s entropy. Hot systems are inherently
more disordered and thereby have higher entropy than cold systems, so a small
increase in the temperature of a hot and already disordered system doesn’t make
it much more disordered.
Thus far, we have used the word “entropy,” and begun to describe a little of
its properties, but we have not yet said what entropy is. To do that, we will begin
to bridge between the macroscopic world of our everyday observations and the
microscopic world of atoms and molecules. At our macroscopic level, we
describe a state of a system in terms of the familiar macroscopic properties,
typically temperature, pressure, density, and mass or volume. In fact, any
equilibrium state of a macroscopic system can be specified uniquely by
assigning specific values to only a subset of these variables; that is enough to
determine all the other state variables. That is because the state variables of a
substance are linked in a way expressed by an equation, the equation of state,
unique for each substance. These equations can be simple for simplified models,
as we shall see, or they can be very complex if they are put to use to get precise
information about real systems, such as the steam in a steam turbine.
In the following discussion, we shall designate a state specified by those state
variables as a macrostate. The number of those variables that are independent—
that is, can change independently—is fixed by what is perhaps the simplest
fundamental equation in all of science, the phase rule, introduced by a great
American scientist, long a professor at Yale, J. Willard Gibbs, between 1875 and
1878. This rule relates three things. The first, the one we want to know, is the
number of degrees of freedom, f, which is the number of things one can change
and still keep the same kind of system. This is directly and simply determined by
the second quantity, the number of substances or components c in the system,
and the third, the number of phases p, such as liquid, gas, or solid, of a specific
structure that coexist in equilibrium. These are linked by the relation

f = c - p + 2.

Not even a multiplication! It’s just addition and subtraction. We will later go
much deeper into this profound equation. For now, we need only say that every
phase of any macroscopic amount of a substance has its own relation, its
equation of state, that fixes what values all the other variables have if we specify
the values of just any two of them.
Typically, there is a different equation of state for the solid, the liquid, and the
gaseous form of any chosen substance. (Some solid materials may have any of
several forms, depending on the conditions of temperature and pressure. Each
form has its own equation of state.) The simplest of all these equations is
probably the equation of state for the ideal gas, a sort of limiting case to describe
the behavior of gases in a very convenient way that is often a very good
approximation to reality. This equation relates the pressure p (not to be confused
with the number of phases p), the volume V, and the temperature T based on the
absolute Kelvin scale, where the lowest point is known as absolute zero. The
relation involves one natural constant, usually called “the gas constant” and
symbolized as R, and one measure of the amount of material, effectively the
number of atoms or molecules constituting the gas, counted not one at a time or
even by the thousands, but by a much larger number, the number of moles.
A mole is approximately 6 × 1023 things, and is essentially an expression of a
conversion from one mass scale to another. In the atomic mass scale, a hydrogen
atom has a mass of 1—that is, 1 atomic mass unit. Collecting 6 × 1023 hydrogen
atoms, we would have 1 gram. Hence we can think of that number, 6 × 1023, as
the conversion factor between atomic mass units and grams, just as 2.2 is the
number of pounds in a kilogram. The number 6 × 1023 is known as Avogadro’s
number, and is symbolized as NA. The number of moles of the gas is represented
as n. To get a feeling for the amount of material in one mole, think of water,
H2O, which has a molecular weight of 18 atomic mass units, so one mole of
water weighs 18 grams. But 18 grams of water occupies 18 cubic centimeters, or
about 1.1 cubic inches. Hence 18 cubic centimeters of water contains 6 × 1023
molecules, one mole of water.
So now we can write the ideal gas equation of state:

pV = nRT.

Most other equations of state are far more complicated. The equation of state for
steam, used regularly by engineers concerned with such things as steam turbines,
occupies many pages. But every state of every substance has its own equation of
state. From these equations, we can infer the values of any macroscopic variable
we wish from knowledge of any two and, if the variable we want is extensive, of
the amount of material. We shall return to this subject in later chapters and
derive this equation. (If we wanted to count the particles individually, instead of
a mole at a time, we could write the equation as pV = NkT, where we use N to
indicate the number of atoms or molecules and k is known as the Boltzmann
constant, so the gas constant R = NAk, Avogadro’s number of little Boltzmann
constants.)
At this point we can begin to explore what entropy is. We easily recognize
that every macroscopic state of any system, the kind of state specified by the
variables of the appropriate equation of state, can be realized in a vast number of
ways in terms of the locations and velocities of the component atoms or
molecules. All those individual conditions in which the positions and velocities
of all the component particles are specified are called microstates. Microstates of
gases and liquids change constantly, as the component atoms or molecules move
and collide with one another and with the walls of their container. In fact, when
we observe a macrostate, whatever process we use invariably requires enough
time that the system passes through many microstates, so we really observe a
time average of those microstates when we identify a macrostate. The molecules
of oxygen and nitrogen in the air in our room are of course in constant—and
constantly changing—motion, hence constantly changing their microstate.
However, we think of the air in a room as being still, at a constant temperature
and pressure, and thus in a nonchanging macrostate.
In a microstate, any two identical particles can be exchanged, one for the
other. We would call the two states, with and without that exchange, two
microstates. Nevertheless we would identify both as representative of the same
macroscopic state. The entropy of a macroscopic state or macrostate is the
measure of the number of different microstates that we identify as corresponding
to that macrostate. Put more crudely, we can say that entropy is a measure of
how many ways all the component atoms could be, that we would say are all the
same for us. For example, the microstates of the air molecules in a room are all
the possible positions and velocities of all those molecules that are consistent
with the current temperature and pressure of the room. Obviously the numbers of
those microstates are so vast that, instead of using the actual number of
microstates accessible to a system, we use the logarithm of that number.
Logarithms are very convenient for any situation in which we must deal with
very large numbers. The logarithm of a number is the exponent, the power to
which some reference number must be raised to produce the given number. The
reference number is called the base. The three most common bases are 10, 2, and
the “natural number” e, approximately 2.718. Logarithms to the base 10 are
written “log,” so 2 is log(100) because 102 = 100. Logarithms to the base e are
usually written as ln, meaning “natural logarithm.” Logarithms to the base 2 are
often written log2( ) so log2(8) = 3 because 23 = 8. We shall discuss this in more
detail in the context of entropy in the next chapter.
Defined precisely, the entropy of any macrostate is the natural logarithm, the
logarithm to the base e, of the number of microstates that we identify with that
macrostate. Of course the microstates are constantly changing on the time scale
of atomic motions, yet we observe only the one constant macrostate when we
observe a system in thermodynamic equilibrium. Virtually every macrostate we
encounter is some kind of time average over many, many microstates through
which our observed system passes during our observation. The oxygen and
nitrogen molecules in a room move constantly and rapidly—but randomly—
while we sit in what we consider still air at a constant temperature and pressure.
Logarithms are pure numbers, not units of any property. However, strictly for
practical use, we express the entropy in physical units; usually in units of energy
per degree of temperature, the same units that express heat capacity. These may
be either energy per mole per degree, in which case we multiply the natural
logarithm by R, or energy per molecule (or atom, if the system is composed of
independent atoms), in which case we use the much smaller constant factor, the
Boltzmann constant, symbolized k, the gas constant divided by Avogadro’s
number, R/N. It is in these units that the entropy was introduced implicitly in the
statement of the second law of thermodynamics.
Now we can express the second law in other ways. One is this: All systems in
nature evolve by going from states of lower entropy to states of higher entropy—
that is, from macrostates of fewer microstates to macrostates of more possible
microstates—unless some work is done in the process of changing the state.
Another way to say this is that systems evolve naturally from states of lower
probability to states of higher probability. A gas expands when its volume
increases because there are more ways the molecules of that gas can be in
macroscopic equilibrium in the larger volume than in the smaller. To go to a state
with fewer microstates, we must do work; we must put in energy.
Another statement of the second law, less general than the above, is this:
Systems in contact that are initially at different temperatures but are allowed to
exchange energy as heat will eventually come to some common temperature,
determined only by the initial temperatures and heat capacities of each of the
systems. The composite system relaxes to a common equilibrium by going to the
state of highest entropy available to it. That is the state in which the systems
share a common temperature. All states in which the systems have different
temperatures are at lower entropy.
Here we have crossed a bit from the traditionally macroscopic approach to
thermodynamics to the statistically based microscopic approach. While
thermodynamics developed originally as a macroscopic view of heat, work, and
energy, the introduction of an approach based on statistical mechanics, the
statistical analysis of the behavior of the mechanics of a complex system,
provided a whole new level of insight and a perspective on the relationship
between the traditional approach and our new understanding of the atomistic
nature of matter. Statistical mechanics is our first bridge between the macro-level
and the micro-level for describing natural phenomena. And it is in the second
law that the statistical, microscopic approach gives us the greatest new insight.
The world simply moves from the less probable state where it is now to the next,
more probable state, the new state which has more available microstates, more
ways of being—that is, has a higher entropy—than the present state which it is
now leaving. If we want some subsystem, some machine for example, to do
otherwise, we must provide energy and do work to make that happen.
Historically, the way entropy was first introduced was, in fact, based on the
amount of heat exchanged, per degree of temperature. Specifically, the entropy
change is expressed as a limit, based on the ideal, reversible process. If there are
just two components, one at a higher temperature T1 and the other at a lower
temperature T2, then the heat lost from the high-temperature component, Q1, and
that gained by the low-temperature component, Q2, satisfy the relation that
Q1/T1 + Q2/T2 = 0 in that ideal limit. Obviously in a natural flow of heat, Q1
must be negative and Q2 positive. The ratio of heat exchanged to the temperature
at which that heat is gained or lost is the lower bound on the entropy change in
this purely thermal process. If the process is real and not an ideal, reversible one,
then the total entropy change must be greater than the sum of the ratios of heat
exchanged per degree and the sum of heats per degree of temperature must be
greater than zero. This macroscopic approach does not offer a way to evaluate
the actual entropy change; it only sets a lower bound to its value. Another way to
look at that relation is to rearrange the equation to the form Q1/Q2 = -T1/T2. That
is, the ratio of heat changes is given by the ratio of the initial temperatures; since
T1 is the higher temperature, the heat lost from that component is greater than
the heat gained by the colder component, when the two components come to
equilibrium at a common temperature between T1 and T2.
In the following chapter we shall investigate the nature of entropy in more
detail, as we explore how the concept developed, first at the macroscopic level,
and then in terms of behavior at the microscopic level. The essence of the latter,
which perhaps gives us the deepest and fullest understanding of the concept, is
the intuitively plausible idea that if a system can evolve spontaneously from a
macrostate with a smaller number of microstates to one with a larger number of
microstates, it will do just that. We can think of that statement of spontaneous
behavior, obvious as it seems when expressed in those terms, as one statement of
the second law of thermodynamics.
 Figure 1. Walter Nernst. (Courtesy of the Smithsonian Institution Libraries)

We can conclude this chapter with a statement of the third law of

thermodynamics, presented by the German chemist Walter Nernst in 1912 (Fig.
1). This law states that there is an absolute lower bound to temperature, which
we call “absolute zero,” and that this bound can never actually be reached in a
finite number of steps. In practice, experimental advances have enabled us to
reach temperatures in the range of levels as low as nanokelvin, or 10-9 K.
Substances behave very, very differently under those conditions than at the
temperatures with which we are familiar. Some, for example, lose all of their
electrical resistance, becoming what are called “superconductors.” Even more
dramatic changes can occur for substances such as helium atoms, which have
nuclei that are composed of two protons and two neutrons. At such low
temperatures, these atoms can all go into a common quantum state, forming what
is called a “Bose-Einstein condensate.” Nevertheless, the third law says that
absolute zero of temperature remains infinitely far away, at least for macroscopic
systems, those we encounter in everyday life.
The concept of an absolute zero of temperature is closely linked to the
concept of energy. For a system to reach that absolute zero, it would have to lose
all of its energy. But to achieve that, it would have to be in contact with some
other system that could accept that energy. It is possible to extract all of the
internal energy from an individual atom or molecule, to bring it to its lowest
internal state in which it has no more energy in vibration or rotation, but to bring
a molecule to a dead stop, with no translational motion at all, in an unbounded
volume, did not seem possible in the past. If the atom is confined, effectively in
some sort of box or trap, then it is possible in principle to bring that atom to its
lowest possible energy state in that box. That would correspond to bringing a
microsystem to its absolute zero of temperature. Modern methods of
experimenting with one or a few atoms at extremely low temperatures might
make it possible to use light to trap an atom in a tiny “box” and remove all of its
translational as well as internal energy. If this can be done, it will be a
demonstration of a gap or distinction between natural scientific laws valid at the
micro level and those applicable for macro systems. It might be possible to
determine how small a system must be in order that, in practice, all of its energy
can be removed; that is, at present, an open question. (However, that introduces a
rather philosophical question as to whether “temperature” has any meaning for a
single atom.) We shall return to this topic in Chapter 4, when we discuss the
thermodynamics of systems that require quantum mechanics for their
description.
 THREE
How Did Classical Thermodynamics Come to Exist?

One particularly fascinating aspect of the history of thermodynamics is the

stimulus for its origins. In the twentieth century and perhaps even a little before,
we have experienced a flow of concepts from basic science into applications.
This has certainly been the way nuclear energy and nuclear bombs evolved from
implications in the theory of relativity. Likewise, lasers and all of the ways we
use them grew out of basic quantum mechanics. Transistors emerged from basic
solid-state physics. All of these examples (and others) suggest that ideas and
concepts naturally and normally flow from the basic, “pure” science to “applied”
science and technology, to the creation of new kinds of devices. But in reality,
the flow between stimulus and application can go in both directions, and
thermodynamics is an archetypal example of the applied stimulating the basic.
In sharp contrast to the “basic-to-applied” model, the origins of
thermodynamics lay historically in a very practical problem. Suppose I am
operating a mine and, to keep it operating, I must provide pumps to extract the
water that leaks in; the water would flood the mine if it is not removed. I will
fuel the pumps with coal. But I would like to know the smallest amount of coal I
need to burn in order to keep the mine free enough of water to be workable. This
is precisely the problem that stimulated the thinking that led to the science of
thermodynamics. In the late seventeenth century, the tin mines of Cornwall in
southern England were typical of those that needed almost constant pumping to
keep the water levels low enough to allow miners to dig ore. Here we see how
the challenge of a most practical and very specific question has led to one of the
most general and fundamental of all sciences—the basic arising from the
applied. Even at this early stage of our development, we get an insight into a
very general characteristic of science: virtually any question we might ask about
anything in the natural world may lead to new levels of understanding into how
nature functions and how we can make use of that new understanding. When a
new idea is conceived, or a new kind of question is raised, we have no
conception of its future implications and applications.
The history of the concepts and, eventually, the laws of thermodynamics can
be said to have begun in England with Francis Bacon (1561–1626), then in
France with René Descartes (1596–1650), again in England with Robert Boyle
(1627–1691), or even earlier in Italy with Galileo Galilei (1564–1642). These
were among the early investigators who, independently, identified heat as a
physical property, specifically one they associated with motion of the atoms of
which they assumed matter is composed—even though nobody, at that time, had
any idea what those atoms were or what any of their properties would be. Their
contributions to this topic, however, were qualitative concepts; the formulation
of a more precise treatment of heat, and the connection between heat and work,
came a little later, through the thoughts and experiments of several key figures
and through a fascinating period of controversy.
One very important step in that development came from the study of the
relation between heat and the behavior of gases. Likewise, some crucial
advances led to the invention and development of the steam engine. The French
scientific instrument inventor Guillaume Amontons (1663–1705) showed that
heating a gas increases its pressure and leads to its expansion. More specifically,
he was apparently the first to recognize that all gases undergo the same
expansion when they receive the same amount of heat. Furthermore, the pressure
increases smoothly, in direct proportion to an increasing temperature, so that the
gas pressure can be used as a measure of temperature—that is, as a thermometer.
That behavior, the precise connection between temperature and pressure, later
took the more exact form of a law, now known in most of the world as Gay-
Lussac’s law, or Charles’s law in English-speaking countries. (As we saw in the
previous chapter, we now incorporate the relationship of pressure, temperature,
volume, and the amount of a substance into that substance’s equation of state.)
Amontons also recognized that the temperature of water surprisingly does not
change when it boils, and that its boiling temperature is always the same, hence
providing a fixed point for a scale of temperature. British scientist Edmund
Halley had noticed this previously but did not realize its significance, in terms of
its role in determining a fixed point on a temperature scale. Amontons also
proposed a kind of steam engine, using heated and expanding air to push water
to one side in a hollow wheel, creating an imbalance and hence a rotation.
However, the first practical engine actually built and used was that of Thomas
Newcomen (1663–1729) in England, in 1712.

Steam Engines and Efficiency

The Newcomen engine operated in a simple way (Fig. 2). A wood fire supplied
heat to convert water to steam in a large cylindrical boiler equipped with a piston
or in an external boiler from which steam flowed into the cylinder with its
moveable piston. The steam pushed the piston, which, in turn, moved the arm of
a pump, which would lift water, typically out of a flooded mine. The hot cylinder
would be cooled with water, enough to condense the steam, and the piston would
return to its position near the bottom of the cylinder. Then the whole cycle would
start again, with heating the cylinder, and repeat itself as long as its work was
needed. Each stroke of the piston required heating and then cooling the entire
cylinder. The first installed Newcomen engine drove pumps that drew water
from coal mines on the estate of Lord Dudley, and soon these engines became
the most widely used means to pump water from mines in England, Scotland,
Sweden, and central Europe. Occasionally, others tried to introduce different
kinds of heat engines, but at the time none could compete with Newcomen’s.
One of his, called “Fairbottom Bobs,” is preserved at the Henry Ford Museum in
Dearborn, Michigan (Fig. 3). A functioning model of the Newcomen engine still
operates at the Black Country Museum in Birmingham, England.

Figure 2. Schematic diagram of a Newcomen engine. Steam generated in the cylinder pushes the piston up,
which in turn moves the lever arm at the top. Then water cools the steam and cylinder so the piston moves
back down, pulling the lever arm down. (Courtesy of Wikimedia)

Figure 3. The Newcomen engine “Fairbottom Bobs,” a real, working example of this early steam engine.
(Courtesy of Wikimedia)

Chronologically, now we momentarily leave steam engines and come to the

introduction of the important concepts of heat capacity and latent heat. Joseph
Black (1728–1799) was the key figure in clarifying these (Fig. 4). He found, for
example, that when equal amounts of water at two different temperatures were
brought together, they came to equilibrium at the mean temperature of the two.
However when a pound of gold at 190 degrees was put together with a pound of
water at 50 degrees, the result was a final temperature of only 55 degrees. Hence
Black concluded that water had a much greater capacity to hold heat than did
gold. Thus, generalizing, we realize that the capacity to store heat is specific to
each substance. It is not a universal property like the uniform expansion of gases
under any given increase in temperature. Likewise, Black, like Amontons before
him, recognized that although only a little heat would raise the temperature of
liquid water under almost all conditions, if the water was just at the boiling
point, one could add more and more heat and the temperature would not change.
Instead, the water would turn to steam. Each of those forms, ice, liquid water,
and steam, we denote as a phase. We now associate this added heat, which Black
called the “latent heat,” with a change of phase of the substance. Thus, we
learned that putting heat into something could affect it in different ways, by
raising its temperature or by changing its form from one phase to another. The
temperature rises as heat is introduced when only a single phase is present, but
remains constant if two phases are present (Fig. 5).

Figure 4. Joseph Black. (Courtesy of the U.S. National Library of Medicine)

 Figure 5. Latent heat diagram. (Courtesy of Barbara Schoeberl, Animated Earth LLC)

Here we introduce a brief aside, on what constitutes a phase. There is one and
only one gas phase. Likewise, there is only one ordinary liquid phase for most
substances. Each structural form of a solid, however, is a phase of its own. Many
solids change their form if they are subjected to high pressures. We identify each
of those forms as a unique phase. Furthermore, some substances, especially
those whose elements have shapes far from spherical, can exhibit both the
ordinary liquid form with those elements oriented and distributed randomly (but
dense, of course) and at least one other form, under suitable conditions, in which
those nonspherical elements maintain some ordered orientation. We identify two
or more different liquid phases for such substances. Hence phase is a
characterization of the kind of spatial relation of the component elements of a
system.
The next major advance in steam engines came half a century after
Newcomen, and was a major breakthrough. James Watt (1736–1813) was a
young instrument maker and scientist at Glasgow University, where Joseph
Black was a professor. Watt speculated at least as early as 1760 about creating a
better engine, but his breakthrough came in 1763, when he was repairing a
miniature model of a Newcomen engine. He recognized that the model’s
cylinder cooled much more than the cylinders of full-sized counterparts, when
cooling water condensed the steam. This led him to investigate just how much
steam was actually generated on each cycle, compared with the amount of steam
needed just to fill the volume of the expanded cylinder. Watt found that the
actual amount of heat used, and hence of steam generated, for each cycle was
several times what would generate just enough steam to fill the cylinder. From
this, he inferred, correctly, that the excess steam was simply reheating the
cylinder. He then realized that trying to improve the performance of the
Newcomen engine posed a dilemma: How could one simultaneously keep the
cylinder hot, so that it would not require all that reheating on each cycle, and still
achieve an effective vacuum by condensing the steam to liquid water cold
enough to have a low vapor pressure? At that point, his genius led him to the
idea of having a second cylinder, or chamber of some sort, that would be cool
enough to condense the steam, but that would only become accessible to the
steam after the piston had been pushed to maximize the volume of the cylinder.
That second chamber can be merely a small volume that can be opened to allow
steam to enter from the main cylinder, in which that steam can be cooled and
condensed to the much smaller volume of liquid water. This was the genesis of
the external condenser, which essentially revolutionized steam engines and
stimulated a whole new level of interest in improving their performance. The
small external condenser could be kept cool, while the main cylinder could
remain hot. The key to its use is simply the valve that stays closed except when
it’s time to condense the steam and reduce the volume of the main chamber.
(And, of course, the condenser has a drain to remove the water that condenses
there.)
Watt, who was deeply concerned with making steam engines as economically
efficient as possible, also recognized that the pressure of the steam on the piston
did not have to remain at its initial high level. If the pressure was high enough
initially, it would be possible to shut off the supply of steam and the steam in the
cylinder would push the piston to the further limit of its stroke, with the pressure
dropping all the way but still remaining high enough to keep the piston moving
and expanding the chamber. At that extreme point of expansion, Watt realized
that the still-hot steam would be at a pressure still barely above atmospheric, so
that a valve could open to allow the steam that caused the excess pressure to
condense and then flow back to the boiler. Only then would another valve open,
this one to the external condenser, into which the remaining steam would flow,
emptying the cylinder so the piston would return to its original position. Watt
recognized that by letting the steam expand and cool almost to the surrounding
temperature, one could obtain the maximum work that the steam could provide.
It would be a way to obtain as much of the heat energy in the steam as
practically possible (Fig. 6).
This combination of simple steps emerged from Watt’s remarkable insight
into how the real Newcomen engine operated, and, more important, what kinds
of changes would make a steam engine much more efficient than the original
model. Four related ideas—keeping the cylinder hot, using an amount of steam
as close as possible to the minimum required to drive the piston through its
cycle, letting it cool as it expanded, and then condensing the remaining steam
externally, outside the cylinder—enabled Watt to make an altogether far more
efficient and more economical engine than its predecessors.

Figure 6. The steam engine of James Watt. Instead of cooling the entire large cylinder on each cycle, only
the small chamber, C, receives cooling water, but C opens to the main cylinder B at the end of the
expansion of chamber B, so the steam in B condenses in C, B remains hot and new water can be introduced
into B and converted into steam, driving the piston P. (Courtesy of Wikimedia)
 Another very important contribution emerged from the firm of Boulton and
Watt, which made and oversaw the operations of those efficient steam engines.
This was the graphical means to measure the amount of work done by the engine
on each cycle, which became known as the “indicator diagram.” This diagram,
still constructed in its original and traditional form, represents each branch of the
cycle as a curve in a graph whose axes are pressure (on the vertical axis) and
volume (on the horizontal axis). John Southern, working with Boulton and Watt,
invented a simple device that literally had the steam engine draw its own
indicator diagram. The device consisted of a rolling table linked to the shaft
holding the piston, so that the table moved back and forth as the piston moved.
This motion revealed the changes in volume of the cylinder. The table held a
sheet of paper. Connected to the cylinder was a pressure-sensitive arm that held a
pencil, which moved across the paper, perpendicular to the back-and-forth
motion of the table, with its tip on the paper, so that the pencil inscribed a curve
on the paper as it and the table moved. Thus, with each cycle, the table moved
and the pencil drew a closed curve of pressure as a function of piston position—
that is, of the volume of the chamber. We shall address this in the next chapter,
where we see that the area enclosed by the closed curve measures the work done
by the engine on each cycle. Boulton and Watt incorporated these devices into
their engines and thus were able to assess the performance of each one. The
devices were held in locked boxes that only the employees of Boulton and Watt
could open, keeping the nature of these devices secret, until someone in
Australia opened one and revealed what they were and what they measured.
While Watt did make remarkable advances in the performance of steam
engines, an even better steam engine than his appeared and actually became the
dominant form of engine in the Cornish mines and elsewhere. Advances by
Richard Trevithick, Jonathan Hornblower, and Arthur Woolf specifically used
steam at as high a temperature and pressure as possible to start pushing the
piston and then continued the expansion until the steam had nearly cooled.
Consistent with the later findings of Sadi Carnot, these engines proved to be
even more efficient than those made by Boulton and Watt and were widely
adopted.

What Is Heat?

During the years when the steam engine was being developed, the question arose
regularly of just what heat is. Some of the properties of heat emerged from
observations and measurements. The amount of heat required to convert a given
amount of water to steam, for example, was the same as the amount one could
recover by converting that amount of steam back to water at its original
temperature. Melting a given sample required exactly as much heat as would be
delivered when that sample again turned solid. The concept that heat is
conserved emerged from such observations (but, as we shall see, that concept
was inadequate and hence incorrect). Furthermore, people began to recognize
that heat could be transferred in at least three different ways. It can go from one
place to another much like light, via radiation, for example warming by the sun.
It can also be transferred by the flow of a warm substance such as hot air or hot
water, which we call “convection.” Or it may be transferred by conduction, as it
passes through a material that itself does not appear to change or move, as it
does through metals, for example, but not through straw or wood. Yet these
properties were not sufficient to tell what heat is. They only tell us something
about what it does. The essential nature of heat remained a mystery,
notwithstanding the manifold ways it was used for practical purposes.
The recognition that motion, and what we now call kinetic energy, can be
converted to heat was the crucial observation from the famous cannon-boring
experiment of 1798 by Count Rumford, born Benjamin Thompson in America
(Fig. 7). He showed that the friction from boring out the metal from a cylinder
could generate enough heat to boil the water in which the borer and cylinder
were immersed. His finding laid the groundwork for the subsequent realization
that mechanical work can be converted to heat and then, with heat engines, that
the conversion can go in the opposite direction. Thus people came to realize that
heat itself need not be conserved—but something more fundamental and general
is indeed conserved, namely energy, as expressed in the first law of
thermodynamics.
Two concepts, mutually exclusive and thus competitive, evolved regarding
the nature of heat. One identified heat as the degree of motion of the (still
conjectured) particles that composed materials, including solids, liquids, and
gases. The other identified heat as a material substance, often as a mutually
repelling matter that surrounds every material particle. The former, the kinetic
model, actually emerged first, but the material or “caloric” concept overtook the
kinetic model, especially in the late eighteenth century. Two eminent and
distinguished French scientists, Antoine Lavoisier and Pierre-Simon Laplace,
published Mémoir sur la Chaleur in 1783, in which they presented both theories,
without suggesting that one was superior to the other, but observing that both
were consistent with the conservation of heat and with the different modes by
which it can move. It was Lavoisier who introduced the term “caloric” in 1789,
suggesting that perhaps he personally favored the fluid model. However, based
on his friction experiments, Rumford himself rejected the idea that heat was a
fluid, a concept then considered plausible by many people, obviously including
Lavoisier. (Lavoisier was executed during the French Revolution; his widow, a
brilliant woman herself, later became the wife of Rumford.)

Figure 7. Benjamin Thompson, Count Rumford; engraving by T. Müller. (Courtesy of the Smithsonian
Institution Libraries)

According to the concepts of that time growing from the work of Joseph
Black, the content of heat in any body of material is a simple function of the heat
capacity of the material, or of the specific heat and the amount of material, since
“specific” here refers to the heat capacity of a specific amount of the material,
such as one gram. There was a belief for a time that the specific heat of any
given substance is constant, but that was proven wrong. Heat capacities and
specific heats definitely depend on conditions of temperature and pressure, for
example. But every substance has its own specific heat, which we can think of as
a property we can represent by a curve in a graph with temperature as the
horizontal axis and the value of the specific heat as the quantity on the vertical
axis. But we would need one such curve for every pressure, so we should think
of the specific heat as represented by a surface in a space in which one axis in
the bottom plane is temperature and the other is pressure, and the vertical axis is
the specific heat. But constant or variable as the heat capacity might be, it was a
property that could be interpreted in terms of either of the concepts of what heat
is, motion or caloric fluid. Heat capacity could be interpreted in terms of the
intensity of atomic motion or of the amount of a physical fluid. This property,
important as it is for knowing how real materials behave and for explaining the
conservation of heat, gives us no new insight into the nature of heat. It is simply
useful as a property characterizing each substance, in terms of how it contains
heat and responds to its input or outflow. Nonetheless, it will turn out to provide
insight into how complex the particles are that make up each substance.
An important property of heat capacities, particularly of gases, is the way
they depend on the conditions under which heat is exchanged. Specifically, we
can add heat to a given volume of gas, holding the volume of the gas constant,
or, alternatively, we can hold the pressure of the gas constant. If we hold the
pressure constant, then the gas expands to a larger volume; if we hold the
volume constant, the pressure exerted by the gas increases, as Amontons
showed. We now understand why this difference exists. If the pressure is held
constant, the gas, by expanding, moves the walls of its container, doing work in
that process as it increases its temperature. If the volume is kept constant, the
system increases its temperature without doing any work. We shall explore this
more quantitatively shortly, but at this stage, we merely need to recognize that
the extra heat required to raise the gas temperature at constant pressure is
responsible for doing the work of expansion, an amount not needed for constant-
volume heating. As a consequence, we distinguish heat capacities at constant
volume from those at constant pressure. (These are even slightly different for
solids and liquids, because they do change their volumes and densities a bit with
changes in temperature—but much less than gases do.)
Another related aspect of the behavior of gases is the way they behave when
they expand under different conditions. Recognizing that there are such
differences was the initial step toward the discovery of the ideal gas law, the
simplest of all equations of state. If a gas expands against a force, such as the
pressure of the atmosphere or a piston linked to a driveshaft, and there is no heat
source to maintain its temperature, that gas cools as it expands. It loses heat by
doing work against that resisting force. However if a gas expands into a vacuum,
its temperature remains essentially unchanged. John Dalton (1766–1844, Fig. 8),
who is best remembered for his exposition of the atomic theory of matter,
showed that all gases expand equally when they undergo the same increase in
temperature. This finding, ironically, reinforced the belief by Dalton and many
contemporaries that heat is a pervasive fluid, known as “caloric,” which
surrounds every atom, and that it is the increase of caloric with temperature that
is responsible for the uniform expansion. This property of expansion was made
more precise by Dalton’s French contemporary Joseph-Louis Gay-Lussac
(1778–1850, Fig. 9), who studied and compared the expansion of oxygen,
nitrogen, hydrogen, carbon dioxide, and air. He showed that when the
temperature of a specific volume of each of these gases increases from what we
now call 0° Celsius, the freezing point of water, to 100° Celsius, the boiling
point, they all undergo the same increase in volume. This discovery led to the
relation that the change in pressure p is directly proportional to the change in
temperature, or Δp = constant × ΔT. This relation is now known in many parts of
the world as Gay-Lussac’s law, but because of a historical anomaly it is called
Charles’s law in the English-speaking world. Here we have the first precise
relation that will be incorporated into the ideal gas law.
 Figure 8. John Dalton. (Courtesy of the Library of Congress)

This pressure-volume relation immediately suggests that there should be a

limiting lowest-possible temperature at which the pressure of any gas would be
zero. As we saw in Chapter 2, we now recognize this as the third law of
thermodynamics, the existence—and unattainability—of a fixed, finite lower
limit to what we call “temperature,” which we define as “absolute zero.” Dalton
and others of his contemporaries attempted to determine what that lowest
possible temperature is, but their estimates turned out to be far off from the
actual value, which is approximately -273° Celsius, known now as 0 K, or
absolute zero.
 Figure 9. Joseph-Louis Gay-Lussac. (Courtesy of Wikimedia)

The response of gases to changes of temperature, as revealed through their

heat capacities, ultimately led to a deepened understanding of the nature of gases
and, eventually, of the atomic nature of matter. In particular, it was found by
Gay-Lussac and then much refined by F. Delaroche and J. E. Bérard that, under
ordinary room-temperature conditions, equal volumes of all gases at the same
pressure have equal heat capacities, while the heat capacities of equal masses of
different gases differ, one from another. Specifically, their heat capacities depend
inversely on the densities of the gases. This finding eventually led to the
realization, based on the atomic theory of matter, that equal volumes of gases, at
the same temperature and pressure, must contain the same number of atomic
particles (strictly, of molecules), but that the weights of those individual particles
depend on the specific substance.

Sadi Carnot and the Origin of Thermodynamics

The origins of thermodynamics as a science—a self-contained intellectual
framework to link observation, logical analysis, and a capability to describe and
predict observable phenomena with quantitative accuracy—truly lie with the
contributions of the engineer Sadi Carnot (Fig. 10). His father, Lazare Carnot, an
engineer himself, was also an important figure during the mid-nineteenth century
in promoting the understanding of what made machines efficient or inefficient.
He was a member of a circle of French colleagues that included J. B. Biot, the
mathematics professor J. N. P. Hachette, and especially Joseph Fourier, who
articulated general properties of heat in terms emphasizing (and distinguishing)
heat capacity, heat conductivity, and heat exchange with an environment. Hence
Sadi Carnot, as a young man, was aware of the growing concern about
understanding heat and heat engines.

Figure 10. Sadi Carnot as a young man, about eighteen years old. (Courtesy of Wikimedia)

After service as a military engineer during the Napoleonic wars, in 1824 he

produced his magnum opus, Réflexions sur la puissance motrice du feu, or
Reflections on the Motive Power of Heat. In this volume, a crucial conceptual
pillar for building ideas in thermodynamics emerged: the ideal, reversible
engine. This is an engine with no losses of heat via friction or heat leaks to the
external world, an engine that, at every instant, could be going in either direction
and hence operates infinitely slowly. This concept enabled Carnot to determine
the natural limit of the best possible performance that any engine can attain.
“Best possible” here means the maximum amount of useful work obtainable from
any specified amount of heat. We now call this ratio, the amount of useful work
per amount of heat, the “efficiency” of a heat-driven machine, and denote it by
the Greek letter η. Generalizing, we use the term to mean the amount of useful
work per amount of energy put into the machine, in whatever form that energy
may be. The concept of energy, however, was not yet part of the vocabulary for
Sadi Carnot, and certainly the idea of using different kinds of energy to drive
machines to do work lay years ahead.
As an aside, it is important to recognize that the meaning of “best possible”
depends on the criterion we choose for our evaluation. Carnot used efficiency,
which is the amount of useful work per amount of heat put in. Another, different
criterion is power, the rate at which work is delivered; a machine optimized for
efficiency will, in general, not be optimized for the power it delivers, and vice
versa. One can imagine a variety of criteria of performance, each of which
would have its own optimum.
As we discussed in Chapter 1, to carry out Carnot’s analysis of the maximum
possible efficiency of a heat-driven machine, he first invented the hypothetical
engine that lent itself to just that determination of optimum performance. Then
he went on to show that no other ideal engine could have an efficiency different
from that of his model system, the system we now call the “Carnot cycle.” This
ideal engine has no friction and no exchange of heat that does not contribute to
the delivery of useful work, meaning there are no heat losses through the walls
of the system to the outside world. The engine operates in four steps that, when
completed, bring it back to its original state.
The first step begins when the working fluid, such as air or steam, is at its
highest temperature, call it TH, and is in contact with a heat source at exactly that
temperature. The fluid expands, pushing a piston to do work, and absorbing
more heat from the hot heat source so the fluid’s temperature remains constant.
At the end of that step, the high-temperature source disconnects and the fluid
expands further to reach a predetermined maximum volume, but, since it is
doing work and taking in no heat, the fluid cools during this second step to a
lower temperature TL. In the third step, the now cool fluid undergoes
compression, and is in contact with a cold reservoir at the same temperature as
the fluid, TL. The compression would heat the fluid but, being in contact with the
cold reservoir, the temperature remains constant at its low value, TL. Heat flows
from the fluid to the cold reservoir in this step. The fourth step is another
compression, but now out of contact with any external heat source or sink, so the
compression process heats the fluid, until it reaches the high temperature TH and
the initial volume that began the cycle. The first and third steps are said to be
“isothermal,” meaning the temperature of the system remains constant; the
second and fourth steps, in which the system is isolated from any outside
environment, are called “adiabatic,” the name for a process in which no heat
passes to or from the system. (The term originates from a Greek word meaning
“impassable”; it was first introduced by William Macquorn Rankine in 1866, and
then used by Maxwell five years later.) As we shall see, there are many other
possible cycles, but this particular one serves especially well to reveal its
efficiency. The Carnot cycle can be diagrammed as a simple closed loop in a
plane with x- and y-axes representing volume and pressure, respectively (Fig.
11). The top and bottom branches are carried out at constant temperature, when
the system is in contact with a thermostat; the temperature changes during the
other two branches when the system is isolated.
Carnot recognized that work could be generated from heat only if heat is
allowed to flow from a high temperature to a lower one; no work could result if
there were not two different temperatures. Using this fact together with his
model of the ideal heat engine, he discovered a fundamental limit to the
performance of all heat engines. He realized that if an amount of heat we can call
QH is taken in at the high temperature TH, and an amount of heat QL is deposited
at the low temperature TL, then the amount of work W would be equal to the
difference between these two, QH - QL. This is because, with this ideal engine,
there is no other place the heat can go. Consequently the efficiency, work per
unit of heat taken in, is η = W/QH, or (QH - QL)/QH. But the heat QH taken in is
proportional to the temperature TH; specifically, it is the heat capacity C or heat
exchanged per degree of temperature, multiplied by that temperature: QH = CTH.
Likewise, the heat exchanged at the lower temperature is QL = CTL, if we
assume that the heat capacity is the same at both temperatures. Therefore the
efficiency of this ideal engine depends only on the temperatures between which
it operates:
 Figure 11. Indicator diagram for the Carnot cycle; the four branches of the cycle operate successively
between the higher temperature T1 and the lower temperature T2. The pressure changes continuously with
the volume during each of these steps. (Courtesy of Barbara Schoeberl, Animated Earth LLC)

This remarkably simple expression tells us that the best possible conversion of
heat into work, using the cycle devised by Carnot, depends only on the two
temperatures between which the system operates. Moreover if we think of the
low temperature as something close to room temperature and the high
temperature as the one we can control, this expression tells us to make the high
temperature TH as high as possible if we want to make the efficiency as high as
possible. This was, of course, the basis for Trevithick’s efficient steam engines.
One important characteristic to recognize about Carnot’s ideal engine is that
in order to have no heat losses to friction, the engine must operate very, very
slowly—infinitely slowly, to be precise. The engine must operate so slowly that
at any instant, it is impossible to tell whether it is moving forward or backward.
We call such an engine reversible. Yes, it is unrealistic to think of using an ideal,
reversible engine to carry out any real process, yet the concept serves as a limit,
however unattainable, to real processes. It enabled Carnot to find that ultimate
limit of efficiency for his engine. But it was more: introducing the idea of a
reversible engine as the hypothetical limit of real engines was one of the novel
concepts that Carnot introduced to revolutionize the way we use
thermodynamics.
Then Carnot took one more major step, often overlooked in discussions of his
contributions. He began with the realization that it is impossible to create a
perpetual motion machine. With this as a basic premise, he showed that all ideal
machines must have that same limit of efficiency. The reasoning is remarkably
straightforward. Imagine an ideal engine of any kind operating between the same
two temperatures as the Carnot engine, but suppose it has an efficiency greater
than that of the Carnot engine. Then it can run forward, driving the Carnot
engine backward, so that it delivers heat QH to the high-temperature reservoir,
having extracted it from the cold reservoir. Then that heat can be used to drive
the more efficient engine, which in turn pushes the Carnot engine to raise
another QH to the hot reservoir. This means that the two engines could go on
forever, together making a perpetual motion machine, taking heat from the cold
reservoir at TL and raising it to the hot reservoir at TH. Carnot recognized that
this was completely inconsistent with all our experience and therefore must be
impossible. Hence the other ideal engine can be no more efficient than the
Carnot cycle. The same argument tells us that the other ideal engine cannot be
less efficient than Carnot’s, because if it were, we could drive it backward with
the Carnot engine and, in that way, raise heat from a low temperature to a high
temperature without limit. Hence all ideal heat engines must have the same
limiting efficiency, given by equation (1) above. This reasoning, with its
amazing generality, is the culmination of Carnot’s contribution and is effectively
the origin of thermodynamics as a precise science.
Despite its deep innovation and key role in shaping the subject, Carnot’s work
was virtually unnoticed for a decade. Its first real recognition came in 1834,
when Émile Clapeyron published a paper utilizing Carnot’s concepts,
particularly by representing Carnot’s hypothetical engine via a form of its
indicator diagram, the plot of the cycle of the engine as a function of the pressure
and volume for each of the four steps.

What Are Heat and Energy?

The nature of heat was a central issue in the evolution of the concepts that
became thermodynamics. That heat can be carried by motion of a warm fluid,
whether liquid or gas, was long recognized, especially through the work of
Count Rumford at the end of the eighteenth century. Recognized as well was the
capability of some materials, notably metals, to transmit heat efficiently, while
other materials, such as straw, could not. Another significant step came with the
realization that heat can also be transmitted as radiation, which is clearly
different from conduction through metals or from convection, the way fluids
flow to transmit heat. It was a major step to recognize that the radiation from the
sun that carries heat obeys the same rules of behavior as visible light, and that,
consequently, radiant heat must be essentially the same as light, a wave
phenomenon. This identification was one of the major contributions of the
French physicist and mathematician A. M. Ampére (1775–1836, Fig. 12). He
showed that radiant heat differed from light only in the frequencies of the two
kinds of radiation. Thus, people came to recognize that heat can pass
spontaneously in different forms from one object to another—and that this
passage always occurs from a warmer body to a colder one, never in the reverse
direction.
 Figure 12. André-Marie Ampère. (Courtesy of the Smithsonian Institution Libraries)

Some insight into the capacity to do work came with the identification by G.
W. Leibnitz (Fig. 13) of the product of mass times the square of velocity, mv2, as
what he and others called vis viva, his measure of force. But not long after, both
G. G. de Coriolis and J. V. Poncelet showed that it is mv2/2 that is the relevant
quantity. Meanwhile, others, Descartes and the “Cartesians” (in opposition to the
“Leibnitzians”), argued that mass times velocity, mv, should be the measure of
capability to do work and that, in effect, this quantity is conserved. At that time,
the first half of the nineteenth century, it seemed impossible that both could be
valid and useful concepts. Today, we recognize mv2/2 as the kinetic energy of an
object and mv as its momentum, both of them valid, useful characteristics of a
body’s motion, and have forgotten altogether that at one time people believed
only one or the other could be a valid “measure of motion.”
 Figure 13. Gottfried Wilhelm Leibnitz. (Courtesy of the Herzog Anton Ulrich-Museum)

The fundamental question of what heat really is remained a long-standing

dispute over many years. There were those two opposing views: one that
originated in the eighteenth century said that heat is the motion of the particles
that make up matter; the other held that heat is a fluid, called “caloric,”
something incompressible and indestructible. Of course the former, the
association of heat with kinetic energy, won out, but we still use some of the
language of the fluid model when we talk of how “heat flows from hot to cold.”
An early demonstration by Sir Humphry Davy, that rubbing two pieces of ice
against each other could melt them, was invoked to support the kinetic theory,
but the controversy continued well after that, at least partially because there were
other interpretations of the Davy experiment that suggested it was not
necessarily friction that melted the ice.
Thus, bit by bit, the different forms that heat transfer can take, and the
interconvertibility of heat and mechanical work, became an acceptable, unifying
set of ideas that laid the foundation for the emergence of the concept of energy.
 That concept arose, even before it received its name, from an insight of J. R.
Mayer in Heilbronn, Germany, in 1842, when he realized that energy is a
conserved quantity. Mayer referred to the “indestructability of force,” rather
than, with all its possible transformations, what we now call energy, but the
concept was indeed the same. One of the keys to his insight came from the
recognition of the difference between the heat capacities of a gas held at constant
volume and that at constant pressure. The gas at constant volume simply absorbs
any heat put into it, so all that heat goes to increasing the temperature of the gas.
The gas at constant pressure expands when heat is put into it, so that it not only
experiences an increase in temperature but also does work in expanding, for
example by pushing a piston. The result is that the heat capacity of the gas at
constant pressure is larger, by the amount required to do the work of expanding,
than that at constant volume. That difference is essentially a simple universal
constant added to the constant-volume specific heat, the heat capacity per unit of
gas, typically one mole. Thus if Cv is the specific heat of a gas at constant
volume and Cp is its specific heat at constant pressure, then Cp = Cv + R (where,
as previously, R is the “gas constant”), which we realize is the energy required to
expand a gas when it absorbs enough heat to increase its temperature by 1° at
constant pressure. This was a key step in the recognition of work and heat as
forms of something common to both of them, and then to radiation and other
manifestations of what we now call energy. Mayer also demonstrated in 1843
that he could warm water by agitating it, supporting the kinetic model for heat.
Another person who played a key early role in developing the concept of
energy and its conservation was James Prescott Joule (Fig. 14), professor of
physics at the University of Manchester, one of the great centers of science and
engineering throughout the nineteenth century. It was Joule who showed, in
1843, precisely how electric currents generate heat. By using falling weights to
operate the electric generators that produced those currents, he was able to
determine the quantitative equivalence between mechanical work and heat. He
established this equivalence in other ways, including demonstrating the heating
that can be achieved when a gas is compressed adiabatically, that is, in a manner
allowing no exchange of heat with the environment during the compression
process. He also confirmed that if a gas, air in his experiments, expands into a
vacuum, it does no work and no heat is lost. In 1847 he also showed, like Mayer,
that mechanical work could be converted directly into heat, such as by simply
stirring or shaking a liquid. This was strong supporting evidence for the idea that
heat is motion of the elementary particles composing a system, and thus a blow
to the idea that it is a fluid called caloric. His work stirred the attention of at least
two important scientific figures, Hermann von Helmholtz in Germany and
William Thomson (Lord Kelvin) in Britain; this happened at the time Joule
described his latest, most improved device for measuring the mechanical
equivalent of heat at a meeting of the British Association in Oxford in 1847. It
was then that people first began to accept the conservation of energy, a concept
very well articulated by Joule and strongly accepted by Kelvin. It was a major
step, to recognize that something more general than heat must be conserved,
because different forms of that more general property can be interconverted.

Figure 14. James Prescott Joule. (Courtesy of Wikimedia)

William Thomson, who became Lord Kelvin of Largs in 1892 (Fig. 15), and
his older brother James played very important roles in the next steps in
developing the science of energy. Working first in Glasgow, a center at least as
important for the field as Manchester, they focused first on the efficiency of
engines of various kinds. Both steam and heated air were the driving materials in
the heat engines of that time, which by midcentury had replaced water-powered
pumps. William devised a temperature scale, taking the freezing point and
boiling point of water as the fixed points, with regular steps or degrees between
them. His scale was essentially identical to the one developed by Anders Celsius
in 1742, a century earlier. Thomson introduced the term “thermodynamic” in a
crucial paper of 1849, in which he analyzed the work of Sadi Carnot—having
finally found and read Carnot’s nearly forgotten book. He connected that
analysis to the experiments by Clapeyron and by Victor Regnault (Fig. 16),
measuring the ratio of the work done to compress a gas to the heat released by
that compression. Specifically, these experiments were meant to determine a
quantity fundamental to quantification of Carnot’s analysis, the maximum
amount of work that could be done by a temperature change of 1°.

Figure 15. William Thomson, Baron Kelvin. (Courtesy of the Smithsonian Institution Libraries)

Thomson’s analysis of those experiments effectively demonstrated that there

should be an absolute scale of temperature, with a lowest value at -273° C. This
was a much more precise determination than the inference made by Amontons a
century earlier, that air would lose all its elasticity at a temperature of about
-270° C. Although Thomson recognized both that there was a natural “fixed
point” and what its value was, he did not himself introduce the Kelvin
temperature scale. With its zero point at -273° C, now recognized as the lowest
temperature attainable in principle, the Kelvin scale has a true natural fixed
point, so it is an absolute scale of temperature. This is in contrast to the arbitrary
choices of the Celsius scale or the Fahrenheit scale, which place the freezing
point of water at 0° C and 32° F, respectively.

Figure 16. Victor Regnault. (Courtesy of the Smithsonian Institution Libraries)

The emergence of energy as an encompassing concept was largely the result

of work by Kelvin and especially by his German contemporary, Hermann von
Helmholtz. In 1847, Helmholtz published a paper with a title rather freely
translated as “On the Conservation of Energy,” which uses Joule’s results to infer
precisely that idea. He made his inference from two premises: one, that there are
no perpetual motion machines, and the other, that all interactions involve the
attractive and repulsive forces between the elementary particles that matter
comprises. Incidentally, he used the then established expression vis viva, and
also “Kraft,” strictly translated as “force,” rather than “energy” for the conserved
quantity. The distinction between force and energy was not yet clear at that time.
Kelvin, then still William Thomson, discovered Helmholtz’s work in translation
in 1852 and used it to unify the mechanical, thermal, and electrical
manifestations of energy, effectively establishing the conservation of energy as a
fundamental principle in itself. In his own paper of 1852 built on Helmholtz’s
work, he stated explicitly that energy is conserved, using that word. His Scottish
contemporary Macquorn Rankine, another important figure in the development
of thermodynamics, was the person who introduced the explicit expression “the
law of the conservation of energy.” Rankine was also the first to draw a
distinction between potential energy and what he called “actual energy” (which
we call “kinetic energy”). It was with his usage that the word “energy” became
fully established over the various previous wordings. It was quickly adopted by
the Thomson brothers, and by the Germans including Helmholtz and another
important figure, Rudolf Clausius. It was Rankine who, in a paper in 1857 and a
book in 1859, first used the term “principles of thermodynamics” and cited “the
first and second laws of thermodynamics.” Although that citation incorporated
work from an early paper by Clausius, at that time the second law did not yet
have the precision it was given by Clausius in the following years.
It was Rudolf Clausius, a German physicist, who first enunciated the second
law of thermodynamics. He was a professor in Zürich when, in 1854, he
published his first statement of the law: that heat can never flow spontaneously
from a colder to a warmer body, and, by implication, that work must be done to
make that happen. In 1865, he introduced explicitly the concept of “entropy,” as
well as the word, giving it a precise mathematical definition. It was then that he
expressed the first and second laws in essentially the words we use today: The
energy of the universe is constant (the first law), and the entropy of the universe
tends to a maximum (the second law). Until his statement, the concept of heat
flowing from a cold body to a warmer one was apparently consistent with the
conservation of energy, yet it is obvious from all of our experience that this
doesn’t happen. Hence it was necessary to enunciate a second fundamental
principle, which is precisely the second law, to be an integral part of the basis of
the new science of thermodynamics.
The precise statement of that law, mathematically, is in terms of an inequality,
rather than as an equation. If a small amount of heat, δQ, is added to a body
whose temperature is T, then the entropy δS of that body increases, satisfying the
inequality relation δS ≥ δQ/T. The equality constitutes a lower bound for the
change of entropy. The concept of entropy, also called “disgregation” by
Clausius, was, at that time, still imprecise by today’s standards, but it was
recognized to be a measure of disorder, in some sense. Only later, when
thermodynamics was linked with the microscopically based statistical
mechanics, largely by Ludwig Boltzmann of Germany, and then more precisely
and elaborately by an American, J. Willard Gibbs (Fig. 17), did entropy acquire
a truly precise, quantitative meaning, which we shall examine briefly later. Here,
we need only recognize that (a) entropy is a quantity measured in units of heat
per degree of temperature, such as calories per kelvin, and (b) that there is a
lower bound to the amount it increases if a system acquires heat. In fact, of
course, entropy can increase even if no heat is injected into a system. A gas,
initially confined to a small volume, expands spontaneously into an adjacent,
previously evacuated larger chamber, with no input of heat. A drop of ink that
falls into a glass of water eventually diffuses to become uniformly distributed in
the water, with no input of heat. A raw egg, dropped, breaks into an irreversible
state. All of these examples illustrate the natural increase of entropy in the
universe. For all three, δQ is zero, so we can say that δS > 0 for them. If a
system is cooled by some refrigeration process (so work is done), then δQ is
negative, and the entropy change may also be negative, but if it is, it must be
closer to zero than δQ/T. The equality sign can be appropriate only for ideal,
reversible processes, never for real processes that involve heat leaks, friction, or
other “imperfections.”
 Figure 17. Josiah Willard Gibbs. (Courtesy of the Beinecke Rare Book and Manuscript Library, Yale
University)

Entropy and Statistics: Macro and Micro Perspectives

In the mid-nineteenth century, a serious issue developed from the apparent

incompatibility of Newtonian mechanics and the evolving second law of
thermodynamics. Newton’s laws of motion implied that all mechanical processes
must be reversible, so that any process that can move “forward” can equally well
move “backward,” so that both directions, being compatible with the laws of
motion, should be equally likely. On the other hand, the second law of
thermodynamics overtly gave a specific direction in time to the evolution of the
systems it describes. The laws of motion seem to imply that if a gas, a collection
of molecules, in a volume V, is allowed to expand into a larger volume, say 2V,
that it is perfectly reasonable to expect that, at some time, those molecules would
all return to the original volume V. But the second law of thermodynamics says
that we’ll never see it happen.
Here we encounter a fundamental problem of reconciling two apparently
valid but incompatible descriptions of how matter behaves. One, based on the
laws and equations of motion of individual bodies, looks at matter from an
individual, microscopic point of view. The other, based on thermodynamics,
looks at matter in the larger sense, from a macroscopic point of view. In some
ways, the problem of reconciling these two viewpoints continues to arise in the
natural sciences. This problem led to two conflicting views of the second law
during the late nineteenth century: is the second law strictly true, with entropy
always increasing, or is it valid only in a probabilistic sense? This could be seen
as a question of the compatibility of Newtonian mechanics, for which all
motions are equally valid with time running forward or backward, and the
second law, with its inherent direction of time. However, the problem of
reconciling the second law with Newtonian mechanics did achieve a satisfactory
resolution through the recognition of the importance of statistics, and the
creation of what we now call statistical mechanics, or statistical
thermodynamics. And of course, the validity of the second law does rest in the
nature of statistics and statistically governed behavior. In particular, statistics
tells us that the more elements there are that constitute the system, the less likely
are recognizable fluctuations from the most probable microstates. The second
law is, strictly, valid only in a probabilistic sense, but for systems of macroscopic
size, the probabilities weigh so heavily in one time direction that there is
essentially no likelihood that we would ever observe a macroscopic system
evolving spontaneously from a higher to a lower entropy.
We shall describe how this approach emerged historically, but for the
moment, it suffices to say that, yes, the gas molecules could perfectly well go
back from occupying volume 2V to being entirely in the initial volume V, and
the laws of mechanics virtually assure that at some time, that would happen. But
we can examine this in a bit more detail. Having all the molecules return to V is
a very, very unlikely event for any large assembly of molecules. For two or ten
molecules, it is quite likely that we could, in some reasonably short time,
observe all of them returning from 2V to V, however briefly. But if we were
dealing with 1,000 molecules, such a return would be quite improbable, and if
we consider a sample, for example, of all the air molecules in a volume of 1
cubic meter, and allowing them to expand to 2 cubic meters, the probability of
their returning to the original 1 cubic meter is so infinitesimal that we could
expect to wait many, many times the age of the universe before we would see
that happen. And even then, that condition would persist for only an
infinitesimal time. We’d have to be unbelievably lucky to catch it in our
observations. To get a feel for the number of molecules in a typical macroscopic
system, recall that we typically count atoms and molecules on this scale using
units we call moles. The number of particles in a mole is the number of atomic
mass units (with the mass of a hydrogen atom equal to 1) in a gram. That
number, Avogadro’s number, is approximately 6 × 1023. Hence, for example, one
mole of carbon atoms, atomic weight 12 atomic mass units, is just 12 grams, an
amount you can easily hold in your open hand.
In other words, the macroscopic approach of thermodynamics has at its roots
the assumption that the atoms and molecules that compose matter obey statistics
of large numbers that describe random behavior. Macroscopic science depends
inherently on the properties of systems of very large numbers of component
particles. That means that what we observe of a system at any time is determined
by what is the most probable way that system’s constituent particles can be,
given how the system was prepared. The key here is that the more constituents
that the system contains, the more the most probable condition and tiny
fluctuations from that condition dominate over all others. Having essentially
uniform density of air throughout the 2-cubic-meter volume can be achieved in
so many more ways than having measurable fluctuations that deviate from
uniform density that we simply never see those potentially measurable
fluctuations. Yes, Newtonian mechanics and the requirement that the molecules
move randomly does assure that those fluctuations can, and even will, occur—in
principle. But observably large fluctuations are so rare for macroscopic systems
that we would never observe them. We can’t wait that long, and they wouldn’t
last long enough to be clearly distinguishable, either.
There is another aspect to the reconciliation of the micro and macro
approaches to describing the world. Newtonian mechanics is indeed formally
reversible, so that if we specify precisely the positions and velocities of all the
particles of a system at some instant, the system can go forward, and at any later
time, if we reverse the velocities, the system will return to its initial state. In
reality, however, we can never describe those initial positions and velocities
precisely, meaning that we can specify them only to some number of significant
figures. Beyond that limit, we have no knowledge. But then, as the real system
evolves, with the particles interacting, colliding and bouncing off one another,
our knowledge of the positions and velocities becomes less and less precise.
Each collision increases our uncertainty of the microstate of the system,
ultimately but in a finite time, to a point that stopping the system and reversing
its direction would have at most an infinitesimal probability of returning to that
initial state, even to just the imperfectly specified state that we knew it to have at
the initial instant. This decrease in our knowledge of the system reveals another
aspect of entropy, namely its connection with information—or lack thereof.
Let us look a bit deeper into the precise, quantitative concept of entropy. As
we discussed in Chapter 2, we again need to distinguish macrostates from
microstates. A macrostate of a system is one describable by thermodynamic
variables, such as temperature and pressure, if it is in equilibrium. If it is not in
true equilibrium, then perhaps it may be described by those thermodynamic
variables plus additional variables specifying the bulk flows and any other
nonstationary properties, such as the mean velocity of the water in a river. We
can, after all, speak of a fluid that cools as it flows through a pipe, assigning a
temperature to all the positions along the pipe; such a system is not in
equilibrium but is still describable by thermodynamic variables, augmented with
others such as the flow velocity. In contrast, a microstate is one in which the
positions and velocities of all the elementary particles that make up the system
are specified. Obviously we would not care to describe, for example, the air in a
room, even at equilibrium, in terms of its microstate. The important point here is
that, for a macroscopic system, the number of microstates that are consistent
with its macrostate is vast, so enormous that we would not wish to deal with
them. However, the number of microstates corresponding to any chosen
macrostate is important; a macrostate that can be achieved by some moderately
large number of microstates is still different from one that can be achieved by,
say, an even much larger number of microstates. In the former, if it does not have
too many elementary components, we may be able to observe fluctuations from
the most probable; in the latter, the probabilities of observable fluctuations are
simply far too small for us ever to see them. The extent of the fluctuations in a
random system of N particles, such as the air in a room, is proportional to
which grows much more slowly than N itself. Hence the larger the system, the
narrower is the range of fluctuations and the more difficult they are to observe.
(One interesting challenge, with current capabilities to observe small systems, is
determining the approximate maximum size system for which we might observe
those fluctuations.)
To deal with this situation of large numbers, we use not the number of
microstates that can yield a given macrostate, but rather the logarithm of that
number, the exponent, normally taken to be the exponent of the natural number
e, or Euler’s number, which is approximately 2.718—the power to which e must
be taken to reach the number of microstates. (The number e is the limit of the
infinite sum 1 + 1/1 + 1/(1 · 2) + 1/(1 · 2 · 3) + . . . ) If we call the number of
microstates corresponding to a chosen macrostate N, then the quantity we use is
called the natural logarithm of N, written ln(N). Hence exp[ln(N)] = eln(N) = N.
Thus we use ln(N) as the basis of the microscopically based definition of the
entropy of that state. Readers may be more familiar with logarithms based on
powers of 10, which we write as log(N). Thus log(100) is 2, since 102 = 100.
Logarithms are pure, dimensionless numbers. In normal usage, we give the
entropy its numerical scale and dimension of energy per degree of temperature,
by writing the entropy S as either k ln(N), counting individual atoms, or R ln(n),
counting a fixed large number of atoms at a time, specifically Avogadro’s
number. Here k is the Boltzmann constant, approximately 1.38 × 10-23 joules per
kelvin, or 1.38 × 10-16 ergs per kelvin, and R is the gas constant, just the
Boltzmann constant multiplied by approximately 6 × 1023, the number of
particles in a mole (the number of individual hydrogen atoms that would weigh 1
gram). Avogadro’s number is the conversion factor that tells us the number of
atomic weight units in 1 gram (since the hydrogen atom weighs essentially 1
atomic weight unit). Working with macroscopic systems, we typically count
atoms or molecules by the number of moles they constitute.
Now we can tie the statistical concept of entropy to the second law: systems
changing their state spontaneously always move from a state of lower entropy to
a state of higher entropy—in other words, they always move to a macrostate
with a larger number of microstates. The more ways a macrostate can be
achieved in terms of microstates, the more probable it is that it will be found,
and the greater its entropy will be. The precise relation, first enunciated by
Boltzmann in 1877, for the entropy S of a macroscopic system is simply S = k ln
W, where W is the number of microstates that correspond to the macrostate of
interest.
There is one other powerful kind of tool that thermodynamics provides. We
previously mentioned thermodynamic potentials, the natural limits of
performance of processes. These are variables that balance the tendency for
systems to go to states of low energy (and typically low entropy) and to states of
high entropy (and likewise, typically high energy). The specific form of each
depends on the conditions under which it applies. For example, if the
temperature and pressure of the process are the controlled variables under which
a system is evolving, the appropriate thermodynamic potential is called the
Gibbs free energy, which takes the form G = E - TS, so, since it is useful under
conditions of constant temperature T, its changes for such conditions are
expressed as ΔG = ΔE - TΔS.
An application of this particular potential gives us the ratio of the amounts of
two interconvertible forms of matter, such as two chemical forms of a substance,
say A and B. We call that ratio, which depends on the temperature and pressure,
an equilibrium constant, Keq. Explicitly, the ratio of the amounts [A] and [B],
when the two are present in equilibrium, is given by the relation Keq = [A]/[B] =
exp(ΔG/kT), where ΔG = G(A) - G(B) and the G’s, the Gibbs free energies per
molecule of A and of B are in atomic-level units. (If we use molar-level units for
the amounts of A and B, we must replace the Boltzmann constant k with the gas
constant R.)
It is sometimes useful to express the thermodynamic potentials in terms of
their values per particle, instead of in terms of the total amounts of the
substances; if we count by atoms or molecules, rather than by moles, we write G
= Nμ, or, for a change, ΔG = N Δμ. The free energy per atom, μ, is called the
chemical potential. The thermodynamic potentials, which include energy itself,
the Gibbs free energy and other parallel quantities, are powerful analytic tools
for evaluating the real and potential performance of real engines and processes.
They enable precise comparisons of actual and ideal operations, in order to help
identify steps and stages with the greatest opportunity for technological
improvement. We shall return to the Gibbs free energy and the expression for it
in the next chapter.
Now we can examine how statistics came to play a central role in the natural
sciences, and specifically how it provided a link from the macroscopic approach
of thermodynamics and the microscopic approach of mechanics, whether
Newtonian or the later quantum variety—which brings us to our next chapter.
 FOUR
How Do We Use (and Might We Use) Thermodynamics?

Thermodynamics is, as we have seen, a basic science stimulated by the very

practical issue of making steam engines operate efficiently. Carnot realized from
the expression he derived for efficiency that it is desirable to make the
temperature of the high-temperature heat as high as possible, and the
temperature of the cold reservoir as low as possible. This naturally played a role
in how steam engines were designed, and continues to be a basic guideline. But
thermodynamics has given us many other insights into ways we can improve
technologies that derive useful work from heat.
A class of very straightforward applications of thermodynamics consists of
the cooling processes we use, the most obvious being refrigeration and air
conditioning. These achieve the desired lowering of temperature by evaporating
a volatile substance; when a condensed liquid converts to vapor, this process
requires the evaporating material to absorb energy from its surroundings and
hence to cool those surroundings. But we expect refrigerators to operate for long
periods, not just once, so we must re-condense that material back to liquid, a
process that requires work. Therefore, refrigerators operate by going through
cycles much like the one we saw in the indicator diagram for the Carnot cycle in
the previous chapter. Instead of going clockwise, however, the direction that
generates work by extracting heat from a hot reservoir and depositing the energy
that wasn’t converted to work into the cold reservoir, a refrigerator runs
counterclockwise around its cycle. The refrigerator removes heat from the cold
reservoir, first by evaporation, and then undergoing an adiabatic expansion, a
process that cools the vapor but allows no energy exchange with the
surroundings during the process; it then deposits heat at the higher temperature
when the vapor is compressed and liquefied, and of course does work in that step
—rather than producing work. The area within the cycle is precisely the work we
must supply to go around the cycle one time—if the system has no friction or
heat leaks, real imperfections that of course require some extra work. Because
the cycle follows a counterclockwise path, the area within the cycle has a
negative sign, indicating that the area gives us the amount of work we must
supply for each cycle—in contrast to the clockwise paths we saw previously,
whose enclosed area carried a positive sign, revealing the amount of work
provided to us on each cycle. Naturally there must be some way for the heat
extracted in the cooling process to leave; we normally dissipate that heat into the
surroundings, into the room where the refrigerator operates, then into the air
outside. Otherwise the refrigerator would heat its immediate surroundings!
We can think of the refrigeration process as a means to maintain a low
entropy in whatever it is that we are cooling. So long as that food or whatever it
is remains at a temperature below the room temperature of the surroundings, its
entropy is lower than what it would be if it were allowed to come to thermal
equilibrium at room temperature.
Of course one of the most widely used applications of thermodynamics, in the
sense of converting one form of energy into another, is the generation of
electricity, the conversion of any of a variety of forms of energy into that one
specific form. We typically use energy as heat, typically in the form of hot
steam, as the immediate form of energy to drive generators, so we convert that
heat into mechanical energy, which, in turn, we use to produce electrical energy.
But the heat comes, in most of our power stations, from combustion of coal, oil,
or gas, which means the primary source from which we eventually obtain
electric energy is actually chemical energy, stored in the fuel. We are learning,
however, to convert the kinetic energy of wind and even of ocean water into the
mechanical energy of generators, and to convert radiation energy in sunlight
directly into electrical energy. We even know now how to convert energy stored
in atomic nuclei into heat in a controlled way, then into mechanical energy and
ultimately into electrical energy—it’s of course called “nuclear energy.”
One approach whose application continues to grow is called “combined heat
and power” or CHP. It also goes by the name “cogeneration.” This makes use of
the obvious fact that heat itself has many uses, so that the heat produced when
we generate electricity can be put to use, rather than just discarded into the
environment. After the very hot steam has given energy to the mechanical
generator, it is still hot, albeit not as hot as it was. The energy remaining in that
steam can be used, for example, to warm buildings. This is probably the most
widely applied form of combined heat and power. More than 250 universities,
including Yale, now use CHP to generate their electricity and heat their
structures. Typical overall efficiencies of separate generating and heating
systems are about 45%, while combined heat and power systems can have
overall efficiencies as high as 80%. (Here, efficiency is the amount of energy put
to deliberate use, whether as heat or electricity, per energy input.)
The substantive information contained in thermodynamics is a constant guide
to how to get the maximum benefit from whatever energy conversion process we
are using, whether it is conversion of heat to mechanical energy that we use
directly, or then transform that mechanical energy to electrical energy, or convert
chemical energy to heat or directly to electrical energy, as in a battery. We
learned from Carnot that in converting heat to mechanical energy, the high-
temperature step of a cycle should be carried out at as high a temperature as
possible. The simple comparison of real and ideal machines tells us to minimize
any subsidiary processes, such as friction and heat leaks, so we learn to make
efficient lubricants and thermal insulators. (Sometimes we inadvertently create
new problems when we try to minimize those losses. An example is the wide use
people made of asbestos as a heat insulator—for which it is indeed very good.
However it was some time before the harmful health effects of inhaling asbestos
were recognized, and when that happened, other materials replaced asbestos in
many applications.)
Another closely related application from thermodynamics is in the design of
what is called an open-cycle turbine. In such a device, a compressor takes in air
mixed with fuel, typically natural gas (methane, specifically), and the mixture
burns, generating heat and warming the gas mixture, thereby increasing its
pressure. At that high pressure, the hot gas drives the turbine wheel of an electric
generator. The more heat the hot gas contains, the more efficient is the turbine.
Hence such devices are designed to capture heat left in the burned gas after it
leaves the turbine; they are built to transfer heat from the gas exiting the turbine
to gas that has not yet undergone combustion. In a sense, this is a sort of
combined heat and power system, rather different from the one discussed above.
A growing application of improved efficiency in producing a useful form of
energy is in conversion of electrical energy into visible light. Before the
invention of the electric light, people produced their light directly from chemical
reactions that release energy as heat and light—in flames, naturally. Candles
convert chemical energy to light and heat by enabling oxygen to combine with
substances that contain carbon, such as wax. Gas lamps used methane, “natural
gas,” as the combustible material but, like the candle, generated light from a
chemical reaction that releases considerable energy.
The incandescent lamp produces light in an altogether different way, by
passing electricity through a metal filament, typically tungsten, because it is a
substance that, as a thin piece of wire, can withstand heat for long periods, even
when raised to temperatures at which it radiates visible light—so long as no
oxygen is present. The incandescent light bulb therefore has its tungsten filament
sealed inside a transparent glass bulb, where there is either a vacuum or an inert
gas atmosphere. This device was an enormous advance and became the principal
source of “artificial light” throughout the world.
Yet more energy-efficient ways to generate visible light from electrical energy
eventually appeared. One is the fluorescent lamp, in which an electrical
discharge operating continuously in a gas keeps exciting a gas filling the lamp;
the excited atoms or molecules collide with fluorescent material on the walls of
the lamp, and transfer their excitation energy to the fluorescent material, which
then emits visible light.
Another technology, now widely used, is the “light-emitting diode,” or LED,
a solid, typically quite small, that has a portion in which electrons are available
to move if a voltage is applied to “push” them and a portion where there are
empty sites where electrons can go, “fall in,” and give off visible light as the way
they lose what is then their excess energy. Both the fluorescent bulb and the LED
produce a given amount of light from smaller amounts of input electrical energy
than the incandescent lamp requires. They achieve their higher efficiency by
converting the excitation energy directly into light, either from the atoms struck
by electrons in the electric discharge of the fluorescent lamp or from the
electrons excited by an applied voltage to move from the “donor” side to the
“acceptor” side of the LED, where their excitation energy becomes light energy.
In contrast, the incandescent lamp gives off visible light when the filament is
heated to such a high temperature that it radiates a whole spectrum, visible light
as well as infrared and other electromagnetic radiation over a broad range of
wavelengths and frequencies. The fluorescent and LED lamps emit radiation
only of specific frequencies, corresponding to the precise amounts of energy
released when electrons lose their excess energy or the fluorescent material
excited by the electrons changes from a specific high-energy state to a more
stable one at lower energy. So it may be a serious challenge to create a
fluorescent lamp or an LED that has some specific, desired color, particularly if
that color doesn’t coincide with a known emission frequency, such as the yellow
sodium line so apparent in many street lights.
This brings us to an explanation that comes directly from thermodynamics, of
just why hot systems emit light. Because radiation is a form of energy that most
materials can absorb or emit spontaneously, if an object is not somehow isolated
from its environment, it will exchange energy as electromagnetic radiation with
that environment. If the object is warmer than the environment, there is a net
flow of radiation out of the object; if the environment is warmer, that flow is
inward. The radiation from a typical heated object consists of a broad spectrum
of wavelengths, whose shape and especially whose peak intensity depend on the
temperature. We refer to such an object as a “black body,” in contrast to
something that emits its radiation at a few specific wavelengths, such as a
mercury vapor lamp. If the “black body” is fairly warm, that peak is likely to be
in the infrared part of the spectrum, where the wavelengths are longer than those
of visible light, or, for a barely warm object, even in the microwave region,
where wavelengths can be centimeters or meters long. If the object is hot
enough, however, its most intense radiation can be in the visible region of the
spectrum, where the wavelengths are between 400 and 700 nanometers, from
blue to red. (One nanometer is one billionth of a meter, written 10-9 meter.) The
hotter the emitting system is, the shorter—and more energetic—is the peak
intensity of the radiation it emits. A hot emitting object that looks blue or green
must be hotter than one that looks red. If an object is hot enough, its peak
radiation can be in the (invisible) ultraviolet region, with a wavelength shorter
than 400 nanometers, or even in the X-ray wavelength region, from 0.01 to 10
nanometers.

Figure 18. Max Planck. (Courtesy of the Smithsonian Institution Libraries)

 An interesting aside is worth noting at this point. The classical theory of
thermodynamics of radiation included a prediction that was obviously wrong.
Specifically, that classical theory implied that the radiation from any object
would become more and more intense at shorter and shorter wavelengths, so that
the object would emit an infinite amount of radiation at arbitrarily short
wavelengths. This was obviously wrong, but what was the explanation? It came
with the work of Max Planck (Fig. 18), who proposed that energy has to come in
discrete units, packages called “quanta,” and that the energy in a single quantum
is a fixed amount, a function of wavelength λ (or frequency ν), given by a
constant, universally written as h, times the frequency, so a single quantum or
minimum-size package of radiation carries an amount of energy hν. This means
that radiation energy of high frequency and short wavelength—and high energy
—can come only in very energetic, “big” packages. If a system has only some
finite amount of energy, it can’t produce a quantum that would have more than
that amount. That condition thereby limits the amount of short-wavelength
radiation any object, however hot, can radiate. This acts as a natural limiting
factor that keeps finite the amount of radiation anything can emit.
Planck’s explanation of the limitation on high-frequency radiation led to the
concept of quantization of energy, which in turn stimulated the entire quantum
theory of matter and radiation. These developments demonstrate how our
understanding of nature and the world around us is always incomplete and
evolving, as we recognize limitations in the explanations we have and use, and
as we find new ways to explore natural phenomena. There are, even at the time
of this writing, unexplained and poorly understood but well-observed
phenomena that call for new, as yet uncreated science: the source of much of the
gravitation around galaxies, which we call “dark matter,” and the source of the
accelerating expansion of the universe as revealed by the rate at which distant
galaxies move away from us, which we call “dark energy.”
 FIVE
How Has Thermodynamics Evolved?

In the context of the steam engine, we first encountered the indicator diagram, a
graph showing pressure as a function of the volume for a typical steam engine.
This is a closed loop, exemplified by the indicator diagram for the idealized
Carnot engine that we saw in Figure 11. There are many kinds of real engines,
with many different cycle pathways; in reality, no operating engine actually
follows a Carnot cycle. Traditional automobile engines, burning gasoline and air
when their mixture is ignited by a spark, follow what is called an “Otto cycle.”
Diesel engines follow a different cycle called a “Diesel cycle.” These differ
simply in the particular pathway each follows. The important point for us here is
to recognize something that Boulton and Watt and their engineer and later
partner John Southern recognized, and that still remains an important diagnostic
application of thermodynamics: that the area within the closed figure of the
indicator diagram is the actual work done in a single cycle.
We see this because (a) the work done to produce a small change of volume
dV against a pressure p is just the product of these two, pdV; (b) if we follow
one curve of the indicator diagram as the arrows indicate, say the highest branch
of the loop, by adding all the small amounts of work that are the result of going
from the smallest to the largest volume on that branch of the loop—that is,
adding up all the small pdV values—we would obtain the total work done when
the system moves along that curve.
We call that process of adding all the tiny increments “integration,” and we
symbolize it as ∫ pdV, which we call the integral. Each increment, each product,
is just the area under the curve for that tiny interval dV. Hence the integral from
beginning to end of that top branch is the area under that branch! That area is
just the work done by the system as it expands along that top curve, as the
volume increases. We can go on to the next branch, and see what work is done
there. Then, when we come to the long, lower branch, a compression step, the
volume diminishes with each dV so work is done on the system. The process of
integration along each of the lower branches involves volume changes that
reduce the size of the chamber, so there, the dV values are negative, work is
done on the system, and hence the areas under the lower branches are negative.
Thus, when we add the integrals, the areas beneath all four branches, those from
the top two are positive, but those from the bottom two are negative, so the sum
of all four is just the difference between the two sets, the area within the closed
loop. This is precisely why finding the area of the loop of the indicator diagram
is a straightforward way of determining the work done on each cycle by an
engine, any cyclic engine.
We can construct the diagram for the ideal performance of any cyclic engine
we can devise. Then we can let the machine draw its own indicator diagram, as
Boulton and Watt did; we can compare it to the ideal diagram for the same
ranges of volume and pressure. The comparison of the areas of the two loops
immediately tells us how near the actual performance comes to that of the ideal
engine.
An example is the indicator diagram for an ideal Otto cycle (Fig. 19). This
cycle is actually more complex than the Carnot cycle, because it is a “four-
stroke” cycle. In the Carnot cycle, one complete cycle has the piston moving out
and back, in what we call two “strokes.” In contrast, in the Otto cycle, the piston
moves back and forth twice in each full cycle, doing useful work only in one
expansion step. That is the step represented by the highest curve in the diagram,
the step in which the gasoline burns in air, forcing the piston out as the gases in
the cylinder expand. At the end of that stroke, the gases, still at fairly high
pressure, escape through a now open valve, so the second stroke consists of a
reduction of pressure while the volume remains constant. Then the piston
returns, at constant pressure, to its position corresponding to the minimum
volume of the cylinder (to the left in the diagram), driving the rest of the burned
gases out. Next, the exhaust valve closes and the intake valve opens, fresh air
enters the cylinder and the piston moves out, still at constant pressure. These two
steps correspond to the back-and-forth horizontal line in the diagram. Then the
piston again moves back to compress the air in the cylinder, preparing for the
introduction of gasoline. That step is the one represented by the curve rising
from right to left at the lower side of the loop in the diagram. At the end of that
step, gasoline enters the cylinder, either by simple flow through a now open
valve or through a jet. Then the spark goes off, igniting the gasoline-air mixture.
The introduction of the gasoline and the firing of the spark correspond to the
vertical upward line in the diagram, the step that occurs so suddenly in the ideal
engine that the volume does not change. That brings the engine back to the
“power stroke” from which the automobile derives its motion.
 Figure 19. Indicator diagram for the ideal Otto cycle that drives the gasoline-powered automobile. Unlike
the earlier diagram for the Carnot cycle, this diagram uses pressure and volume for the axes, rather than
temperature and volume. On each full cycle, the system moves twice between the lowest and highest
volume; in the upper loop, the cycle generates work equal to the area within that loop; the lower “back-and-
forth” line is merely emptying and refilling the cylinder. (Courtesy of Barbara Schoeberl, Animated Earth
LLC)

Thus a practical diagnostic of the performance of a real engine is the

comparison of the area in the cycle of the ideal engine’s indicator diagram with
the corresponding area in the cycle of the real engine’s indicator diagram. We
learn from this comparison how near to the ideal limiting performance our real
engine actually comes. In practice, we know the area within the cycle for the
ideal engine, and we can measure the work done, per cycle, by the real engine by
a variety of means, in order to make the comparison and hence evaluate the
performance of our real engine. This gives us a diagnostic tool to identify
potential ways to improve the performance of an engine. For example, we learn
from Carnot that the highest temperature in the cycle should be as high as
possible, so we design internal combustion engines to withstand very high
temperatures and to lose as little of that heat through the engine’s shell as
possible. We could, if we find a way to devise the appropriate mechanical link,
capture as much of that high-temperature heat as possible by allowing the piston
to move fast while the burned fuel is at its hottest; that represents the kind of
intellectual challenge that we encounter as we try to use thermodynamics to
improve performance of real processes.

Linking Thermodynamics to the Microscopic Structure of Matter

A major step in the evolution of the science of thermodynamics was precisely

the establishment of a bridge between the macroscopic description of matter
characteristic of the traditional approach via properties of “human-size” systems
and the microscopic view, rooted in mechanics, which treats matter as composed
of elementary particles—atoms and molecules, typically. This was triggered in
large part by the apparent incompatibility of the reversibility of Newtonian
mechanical systems with the irreversibility demanded by the second law.
Scientists in Germany, England, and the United States developed that bridge,
called statistical thermodynamics, a name introduced by J. Willard Gibbs. The
essence of this subject is the incorporation of the statistical behavior of large
assemblages into the micro-oriented, mechanical description of the individual
elementary constituents—atoms or molecules—as they interact with one another.
A second very important later advance that had a major influence on
thermodynamics was the introduction of quantum mechanics. While it is
predominantly concerned with microscopic behavior of matter and light,
quantum mechanics has some very strong implications for thermodynamics, via
the allowable statistical properties of the microscopic particles, especially at low
temperatures. We shall examine those implications in this chapter.
James Clerk Maxwell (Fig. 20) was a believer in the idea that matter was
made of small particles that could move about and interact with one another. In
gases, these particles should be relatively free. In the period in which Maxwell
first worked, the mid-nineteenth century, this view was still controversial. Many
people then believed that matter has a smooth, continuous basic structure, and it
was not until late in that century that the atomistic concept became widely
accepted. One of the reasons for rejecting the atomistic model was the apparent
incompatibility of classical mechanics with the second law of thermodynamics
and the obvious irreversibility of nature. In the 1860s, Maxwell made a major
contribution when he took a first step to address this problem by introducing
statistics to describe an atomistic gas. Specifically, he developed a function that
describes the distribution of energies, and hence of velocities, of the moving
particles comprising a gas, when they are in equilibrium at a fixed total energy
as, for example, in a closed, insulated box. The velocities of individual particles
could vary constantly, with each collision between particles or of particles with a
wall. However the distribution of those velocities would remain essentially
constant. There could be fluctuations in the distribution, of course, but these
would be small and very transient.

Figure 20. James Clerk Maxwell. (Courtesy of the Smithsonian Institution Libraries)

Maxwell’s contribution was the “Maxwell distribution,” or, as it is now

called, the “Maxwell-Boltzmann distribution.” Ludwig Boltzmann also derived
that distribution and explored its implications extensively. This is still a mainstay
in describing the most probable distribution of the energies of the molecules or
atoms of any gas in equilibrium at moderate or high temperatures. Maxwell
developed an expression for the probability that a molecule with mass m in a gas
at temperature T has a speed between v and v + dv. This is an exponential; the
probability is of the form

Prob (v) ∽ e-K.E.(v)/kT or, written alternatively, exp[K.E.(v)/kT],

where the kinetic energy K.E.(v) is mv2/2 and e is the “natural number,”
approximately 2.718. (As we saw previously, the quantity in the exponent is
called a “natural logarithm”; were we to write the expression as a power of 10,
we would multiply that natural logarithm by the power to which 10 must be
raised to get the value of e, approximately 0.435.) But the number of ways the
speed v could be directed (in other words, the number of ways the velocity could
be aimed) is proportional to the area of a sphere with radius v, so the probability
of finding v must also be directly proportional to v2. Without deriving the full
expression, we can write the probability distribution so that the total probability,
summed over all directions and magnitudes of the speed, is exactly 1:

This is about the most complicated equation we shall encounter. This

probability distribution rises from essentially zero for atoms or molecules with
speeds close to zero, up to a peak, the most probable speed, which occurs at a
speed vmost prob of Thus, the most probable speed increases with the
square root of the temperature, and particles of small mass, such as hydrogen
molecules or helium atoms, move much faster than heavier ones, such as
nitrogen molecules. The nitrogen molecules have masses 14 times larger than
those of hydrogen molecules, so the hydrogen molecules move roughly 3½ times
faster than the nitrogen molecules. The probability distribution also tells us that
there will be some, not many, particles moving at very high speeds, even at low
temperatures. These will be rare, but they can be found nonetheless (Fig. 21).
Implicit in Maxwell’s concept but not contained at all in the expression for
the probability distribution is the idea that there are fluctuations, both of the
individual particle speeds and of the instantaneous average of their distribution.
If the particles collide with the walls of their confining chamber, which they
must, and with one another, which we now know they do (but Maxwell could
only conjecture since he didn’t know their size or shape), then the actual
distribution of speeds changes constantly. However, the magnitude of the
fluctuations of any system from its most probable distribution has a very
interesting dependence on the number of particles in the system. If there are N
individual particles in our system, then the probability of a deviation of the
average speed or energy from its most probable value varies as and hence the
fractional probability varies as Thus, if N is only 9, that fraction is 1/3,
but if N is 1,000,000, the fraction is only 1/1000. And when we deal with
macroscopic amounts of gases, we are working not with a million molecules but
with roughly 1024 of them, so the chance of seeing a fluctuation away from the
most probable distribution is about 1 in 1,000,000,000,000, hardly something we
can hope to see.

Figure 21. The shapes of the Maxwell-Boltzmann distribution of particle speeds at three temperatures. The
peak moves to higher temperature and the area spreads to higher speeds, but the area under the curves is the
same for all temperatures because the amount of matter is the same for all three of them. (Courtesy of
Barbara Schoeberl, Animated Earth LLC)

We can be more precise about the deviations from the most probable by using
the concept of standard deviation from the mean, especially for the most
common and widely encountered form of distribution called a “Gaussian
distribution,” a symmetric, bell-shaped curve. A band within that curve that is
one standard deviation wide from the most probable (and mean) value of that
distribution either above or below that most probable value includes about 34%
of the total population of the distribution, so about 68% of the total population
lies within less than one standard deviation above or below the most probable
value. Put another way, only about 1/3 of all the values of the distribution fall
more than one standard deviation away from the mean, half of that third in the
low tail and half in the high. And the deviation to which the above paragraph
refers is specifically labeled as the standard deviation.
Maxwell arrived at the expression for the distribution of speeds by using
plausible and intuitively acceptable ideas about the randomness of the particle
velocities and their collisions. He did not find it by any set of logical steps or
mathematical derivations. Carrying out that derivation mathematically was the
first major contribution to this subject made by Ludwig Boltzmann. In 1872,
Boltzmann started with the ideas that (a) the particles of a gas must be moving
randomly because their container does not move, (b) their speeds, more strictly
their velocities (which indicate both speed and direction), are randomly
distributed at every instant, and (c) the particles collide with one another and,
thereby, change speeds and directions with every collision, and the collisions
happen very often. Using these completely plausible assumptions, he was able to
derive, from a combination of physics, mathematics, and statistics, a distribution
of velocities (and hence of speeds) for a gas in a state of equilibrium at a fixed
temperature. And that distribution was exactly the one that Maxwell had
produced intuitively! That distribution, the one given in the equation above and
shown schematically in the accompanying diagram, is now known as the
Maxwell-Boltzmann distribution, and is one of the fundamental links between
the macroscopic approach of traditional thermodynamics and the microscopic
approach based on the mechanics describing the behavior of matter at the atomic
scale.
Boltzmann went further in linking thermodynamics to other aspects of
science. He explained the basis of the connection, found experimentally by his
mentor Josef Stefan, between the electromagnetic radiation emitted by a warm
body and the temperature of that body; this relation, that the total rate of
emission of electromagnetic energy is directly proportional to the fourth power
of the body’s temperature, is known as the Stefan-Boltzmann law. Moreover,
Boltzmann showed that radiation exerted a pressure, just as a gas does. As we
have seen, a warm object gives off radiation of very long wavelength, radio
waves; a still warmer object loses energy by emitting microwaves; we know that
very hot objects, such as flames and filaments in incandescent lamps, emit
visible light. Still hotter objects, such as our sun, emit visible, ultraviolet, and
even X-ray radiation.
One more very important contribution Boltzmann made to thermodynamics
was a further application of statistics. He showed that if any collection of atoms
with any arbitrary initial distribution of velocities is left in an isolated
environment to evolve naturally, it will reach the Maxwell-Boltzmann
equilibrium distribution. He also introduced a function he labeled “H” that
measured how far that evolution to equilibrium has gone; specifically, he proved
that function achieves its most negative value when the system comes to
equilibrium. After Boltzmann first introduced this concept, it was called the H-
function. The initial negative sign, however, was later reversed, so that the proof,
applied to the redefined statistical H-function now called entropy, shows that
function evolves to a maximum. And the proof was a microscopically based,
statistically based demonstration that a system’s entropy takes on a maximum
value when it is in equilibrium. The H-function was precisely the micro-based
equivalent and counterpart of the macro-based entropy, as developed by
Clausius, but with the opposite sign.
The next major contribution to the linking of thermodynamics and mechanics
came from J. Willard Gibbs. Largely unrecognized in the United States until
European scientists read his works and realized their import, Gibbs established
the field of statistical thermodynamics on a new, deeper foundation. His
approach to applying statistics was based on the then novel concept of
ensembles, hypothetical large collections of replicas of the system one is trying
to describe. For example, if one wants to understand the behavior of a system in
equilibrium at a constant temperature, presumably by its contact with a
thermostat, then one imagines a very large collection of such systems, all with
the same macroscopic constraints of constant temperature and presumably
volume and pressure, but each system with its own microscopic assignments of
molecular or atomic properties that would be consistent with those macroscopic
constraints. Thus the individual systems are all different at the microscopic level,
but all identical at the level of the macroscopic, thermodynamic variables. Then
the properties one infers for the real system are the averages of those properties
over all the systems of the ensemble. Constant-temperature ensembles, also
called “canonical ensembles” or “Gibbsian ensembles,” are not the only useful
ones; others, such as constant-energy ensembles, called “microcanonical
ensembles,” are used to describe the behavior of isolated systems, for example.
The energies of the systems in the constant-temperature ensemble are not all the
same; they are distributed in a manner analogous to the Maxwell-Boltzmann
distribution of velocities in a single constant-temperature system. Other kinds of
ensembles also come into play when one introduces other constraints, such as
constant volume or constant pressure.
A very specific product of Gibbs’s work was the famous phase rule, often
called the Gibbs phase rule, which we discussed in Chapter 2. We shall return to
this later, when we investigate the boundaries of applicability of
thermodynamics. It was Gibbs who introduced the term “statistical mechanics”
and the concepts of chemical potential (free energy per unit of mass) and of
phase space, the six-dimensional “space” of the three spatial position coordinates
and the three momentum coordinates.

Thermodynamics and the New Picture of Matter: Quantum

Mechanics

A revolution in our concept of the nature of matter—at the microscopic level—

began in 1900 when Max Planck, in explaining the distribution of spontaneous
radiation from any object at a fixed temperature, proposed that energy as
radiation should come in discrete units, “quanta,” rather than in a continuous
stream. The previous ideas about radiant energy were clearly incorrect, because
they predicted that any object would radiate ever increasing amounts of energy
of ever smaller wavelengths, implying that the energy being radiated would be
infinite, and predominantly of infinitesimal wavelength. This was obviously
impossible and wrong, and came to be known as “the ultraviolet catastrophe,”
but it was not clear, until Planck, how to correct it. Requiring that radiation
comes in discrete quanta, and that the energy in each quantum be directly
proportional to the frequency of the radiation (and hence inversely proportional
to its wavelength) resolved that discrepancy, so that the amount of radiation from
any object diminishes with increasing frequency, beyond some frequency where
the radiation is a maximum. Obviously, since an object glowing dull red is not as
hot as one glowing bright yellow, the maximum frequency of the emitted
radiation, as well as its total amount, increases with the object’s temperature.
A major advance came soon after, when, in 1905, Einstein showed that
radiation, with the form of discrete quanta, explained the photoelectric effect, by
which electrons are ejected from a substance when it is struck by light of
sufficiently short wavelength. It was for this discovery that he received the
Nobel Prize—and he later said that, of the three major discoveries he published
that year (special relativity, the explanation of Brownian motion, and the origin
of the photoelectric effect), explaining the photoelectric effect was the one that
truly deserved the prize. It was in that analysis that Einstein showed that excited
atoms or molecules could be induced to emit radiation and reduce their energy if
they were subjected to the stimulus of incident radiation of precisely the same
frequency as that which they would emit, a process, now called “stimulated
emission,” which is the basis for lasers.
 Then, the next step came when Niels Bohr devised a model to explain the
radiation in the form of spectral lines coming from hydrogen atoms. He
proposed that the one negatively charged electron, bound to a much heavier,
positively charged proton, could only exist in certain specific states, each with a
very specific energy, and that a light quantum corresponding to a specific
spectral wavelength is emitted when an electron falls from a specific state of
higher energy to one of lower energy. Likewise, when a quantum of that specific
energy is absorbed by an electron in its lower-energy state, that electron is lifted
to the state of higher energy. The Bohr model has a lowest possible energy for
the bound electron, and an infinite set of higher-energy bound levels that
converge at just the energy required to set the electron free of the proton. Above
that level, the electron can absorb or emit energy of any wavelength; below that
limit, only quanta of energies corresponding to the difference between two
discrete levels can be absorbed or emitted. Extending the Bohr model to atoms
with more than one electron proved to be difficult, until the discovery of the
much more elaborate, sophisticated subject of “quantum mechanics,” by the
German physicists Werner Heisenberg and Irwin Schrödinger in 1924, and its
further elaboration by many others. Now we use quantum mechanics to describe
virtually all phenomena at the atomic level; the mechanics of Newton remains
valid and useful at the macroscopic level of real billiard balls and all our
everyday experience, but it fails badly at the atomic level, where quantum
mechanics is the valid description.
What does quantum mechanics have to do with thermodynamics? After all,
quantum mechanics is relevant at the atomic level, while thermodynamics is
appropriate for macroscopic systems. One context in which quantum mechanics
becomes especially relevant is that of systems at low temperatures. There, one of
the most important manifestations comes from the striking property that
quantum mechanics reveals regarding the fundamental nature of simple particles.
Specifically, quantum mechanics shows us that there are two kinds of particles,
called bosons (after the Indian physicist S. N. Bose) and fermions (after the
Italian-American physicist Enrico Fermi). Electrons are fermions; helium atoms
with two neutrons and two protons in their nuclei are bosons. The distinction
relevant here between these two is that there is no limit to the number of
identical bosons that can be in any chosen state, whereas only one fermion can
occupy a specific state. When one fermion is in a state, it is closed to all other
identical fermions. This means that if we cool a system of bosons, which has a
lowest state and states of higher energy, withdrawing energy from that system by
lowering its temperature more and more, the individual bosons can all drop
down and, as the system loses energy, more and more of the bosons can fall into
the state of lowest energy. The fermions, in contrast, must stack up, since a state,
once occupied by one fermion, is closed to any others. This means that there is a
qualitative difference between the nature and behavior of a system of bosons and
a system of fermions, a difference that reveals itself if the energies of these
systems are so low that most or all of the occupied energy states are their very
low ones.
Because of their discrete, quantized levels, atoms, molecules, and electrons
can exhibit properties very different at extremely low temperatures from those of
matter at room temperature. Bosons assembled in a trap, for example, can be
cooled to temperatures at which most of the particles are actually in their lowest
energy state; a system in such a condition is said to have undergone Bose or
Bose-Einstein condensation. Fermions, likewise, can be cooled to a condition in
which many or most of the particles are in the lowest allowed states in the stack
of energy levels.
There is an interesting implication from this behavior regarding the third law
of thermodynamics. Normally, we can distinguish systems of constant
temperature—called “isothermal” systems—from systems of constant energy,
called “isolated” or “isoergic” systems. We also distinguish the Gibbsian
ensembles of systems at a common constant temperature—“canonical
ensembles”—from ensembles of systems all at a constant common energy, or
“microcanonical ensembles.” Recall that the third law of thermodynamics says
that there is an absolute lower bound to temperature, which we call 0 kelvin, or 0
K. Moreover, that law states that it is impossible to bring a system to a
temperature of 0 K. However, if we could bring an assembly of atoms to its
lowest possible quantum state, we would indeed bring it to a state corresponding
precisely to 0 K. Whether that can be done in a way that the particles are trapped
in a “box,” a confining region of space and are all in the lowest possible energy
states allowed by the confines of the box, is an unsolved problem, but if it can be
done, and that may be possible with methods now available, then that system
would be at 0 K, in apparent violation of the third law. However, even if it is
possible to produce such an ultracold system, we recognize that this could be
done only for a system composed of a relatively small number of particles,
certainly not of a system of very tiny but macroscopic size, say of 1015 atoms. In
other words, we may expect to see a violation of one of the laws of
thermodynamics if we look at sufficiently small systems. Again, as with entropy
always increasing, we could see transient fluctuations violating the second law if
we look at systems so small that those fluctuations are frequent and large enough
to be detected. This simply emphasizes that thermodynamics is truly a science of
macroscopic systems, one that depends in several ways on the properties of large
numbers of component particles.
The concept of a real system attaining a temperature of 0 K introduces
another sort of paradox that distinguishes small systems from those large enough
to be described accurately by thermodynamics. Traditionally, the two most
widely used concepts of Gibbsian ensembles are the canonical or constant-
temperature ensemble, and the microcanonical or constant-energy ensemble.
Constant-temperature systems show very different behavior in many ways from
constant-energy systems, so we naturally recognize that the canonical and
microcanonical ensembles we use to describe them are, in general, quite
different. However, if we think of an ensemble of systems that have all been
brought to their lowest quantum state, those systems have a constant energy and
hence are described by a microcanonical ensemble—yet they are all at a
common temperature of 0 K, and hence are described by a canonical ensemble.
Thus we come to the apparent paradox of a special situation in which the
canonical and microcanonical ensembles are equivalent. Once again, we arrive at
a situation that doesn’t fit the traditional rules of classical thermodynamics, but
only for systems so small that we could actually bring them to their lowest
quantum state.
We thus recognize that, valid as it is for macroscopic systems,
thermodynamics is an inappropriate tool to describe very small systems, systems
composed of only a few atoms or molecules. Modern technologies give us tools
to study such small systems, something quite inconceivable in the nineteenth or
even much of the twentieth century. This new capability opens a new class of
questions, questions that are just now being explored. Specifically, we can now
ask, for any given phenomenon whose thermodynamic description is valid for
macroscopic systems but fails for very small systems, “What is the approximate
size of the largest system for which we could observe failure of or deviations
from the thermodynamics-based predicted behavior?” This brings us to our next
chapter.
 SIX
How Can We Go Beyond the Traditional Scope of
Thermodynamics?

We have already seen that there are some situations in which general properties
of macroscopic systems, well described by thermodynamics, lose their
applicability when we try to apply those ideas to small systems. One example we
have seen is the possible violation of the third law, for collections of atoms or
molecules so few in number that those particles might all be brought to their
lowest quantum state.
Another is the Gibbs phase rule; recall that the number of degrees of freedom
f depends on the number of components c or independent substances, and on the
number of phases p in equilibrium together: f = c - p + 2, so that water and ice,
two phases of a single substance, can have only one degree of freedom. Hence at
a pressure of 1 atmosphere, there is only one temperature, 0° C, at which these
two phases can be in equilibrium. The “phase diagram” for water illustrates the
regions of temperature and pressure in which solid, liquid, and gaseous water
can exist as stable forms (Fig. 22). Phases can coexist at the boundaries between
regions. Thus, if we change the pressure, the temperature at which water and ice
can coexist changes. This rule describes the behavior of coexisting phases of
macroscopic systems, and seems to be a universally true condition—for macro
systems.
Figure 22. The phase diagram for water. In the white region, ice, solid water, is stable. In the striped region,
liquid water is stable. In the dotted region, the stable form is water vapor. Ice and liquid water can coexist in
equilibrium under conditions along the boundary separating the white and striped regions. Liquid water and
vapor can coexist in equilibrium along the curve separating the striped and dotted regions. The three forms
or phases can coexist together under only one temperature-pressure pair, labeled as the “triple point.”
(Courtesy of Barbara Schoeberl, Animated Earth LLC)

If we look at the behavior of small clusters, however, say 10, 20, or 50 atoms,
we find, both in experiments and computer simulations, that solid and liquid
forms of these tiny beasts may coexist in observable amounts within observable
ranges of temperature and pressure! Their coexistence is not restricted to a single
temperature for any given pressure. This reveals that small clusters of atoms and
molecules simply do not obey the Gibbs phase rule. But if we look carefully at
the fundamental thermodynamic foundations of equilibrium, this apparent
anomaly becomes quite easy to understand, and is not actually a violation of
fundamental thermodynamics.
If two forms of a substance, A and B, are in equilibrium, as we have seen, the
ratio of their amounts [A] and [B] is the temperature-dependent equilibrium
constant Keq = [A]/[B] = exp(ΔG/kT) = exp(NΔμ/kT). We want to use that final
form because we’ll examine what this expression tells us both when N has a
value corresponding to a macroscopic system and when N corresponds to just a
few atoms or molecules. Suppose that the free energy difference between two
phases, per atom, Δμ/kT, is very tiny, say 10-10, so that kT, effectively the mean
thermal energy per particle, is 1010 times larger than that difference in free
energies per atom or per molecule. And suppose we are dealing with a small but
macroscopic system, only 1020 particles, less than a thousandth of a mole,
roughly a grain of sand. In this case, the equilibrium constant, the ratio of the
two forms of the substance, is either exp(1010) or exp(10-10), either an
enormously large or infinitesimally small number, depending on which of the
two phases is favored by having the lower free energy. This tells us that even just
this very tiny bit away from the conditions of exact equality of the free energies
and chemical potentials of the two forms, the minority species would be present
in such infinitesimal amounts that it would be totally undetectable.
It is precisely this character that makes phase changes of ordinary substances
sharp, effectively like sudden, abrupt changes with temperature at any fixed
pressure. In principle, phase transitions are smooth and continuous, but for all
practical purposes we see them as sharp changes, so that we can treat them as if
they were discontinuous—for macroscopic systems. This is why we say water
has a sharp, specific melting temperature and a sharp, specific boiling
temperature at any chosen pressure.
Now let us apply the same kind of analysis to small clusters. Suppose our
system consists of only 10 atoms. That is, suppose N = 10, not 1020. In that case,
Keq can easily be close to 1 over a range of temperatures and pressures. For
example, if Δμ/kT is ±0.1, meaning the free energy per atom of the favored form
is 10 times lower than that of the unfavored, then Keq = exp(±0.1 × 10) =
exp(±1) or ca. 2.3 or 1/2.3. This means that it would be easy to observe the less-
favored form in equilibrium with the more favored when the free energies per
atom of the two forms differ by a factor of 10—which is a clear violation of the
Gibbs phase rule. Hence this apparent anomaly is entirely consistent with
fundamental thermodynamics. We see that the Gibbs phase rule is a rule about
macroscopic matter, not about atomic-size systems. We can go further, if we
know about how large the fraction of a minority species would be detectable
experimentally. We can find how large the size a cluster of atoms would be that
would just allow us to detect, say, a liquid at a temperature below its formal
freezing point, for example 0° C for water, and also what the lower limit of
temperature would be of the band of coexisting liquid and solid. We learn, for
example, that our experiments could reveal solid and liquid clusters of argon
atoms coexisting over a few degrees Celsius, if the clusters consist of 75 or
fewer atoms. But if the clusters were as large as 100 atoms, their coexistence
range of temperature would be too narrow for us to detect anything but a sharp
transition with the tools we have today.

Thermodynamics and Equilibrium

Thermodynamics, as we have examined it, concerns itself with macroscopic

systems, with the equilibrium states of those systems, with idealized changes
among those equilibrium states, and, by treating those idealized changes as
natural limits, with the natural bounds on performance of real systems that
execute real changes among states. The ideal, reversible heat engine, whether it
follows a Carnot cycle or some other cyclic path, must have an efficiency, the
amount of work produced per amount of heat taken from a hot source, partly
using that heat energy to produce work and part to deposit in a cold sink. That is
precisely what Carnot taught us: the maximum possible efficiency is 1 - (TL/TH)
where TL and TH are the temperatures of the cold sink and the hot source,
respectively. The infinitesimal steps that the ideal engine takes to go through its
cycle are really intended to be steps from one equilibrium condition to the next.
The very nature of the idealized reversible cycle is one of a succession of
hypothetical equilibrium states; that is the essence of reversibility. We can never
tell which way the system came to its present state or which way it will go next,
just by looking at the system and not at its history. Real engines are of course
never in equilibrium when they are operating; we describe them in a useful,
approximate way by attributing a pressure, a volume, and a temperature to our
engine at every instant along its path, but strictly, it is constantly changing its
conditions and is never in equilibrium while it operates. We make a fairly
accurate, useful approximation when we attribute a single pressure, volume, and
temperature to the real engine at every instant, but strictly, we know that it is an
approximation, a fiction. But it’s accurate enough to be very useful.
That traditional thermodynamics is about states in equilibrium and the
changes associated with passages between these states has long been accepted.
In effect, there were tacit limits to the domain of applicability and validity of
thermodynamics; the systems for which traditional thermodynamics can be used
should always be close to equilibrium. Well into the twentieth century, however,
those limits were challenged, and the subject of thermodynamics began to be
extended to systems that are not in equilibrium. We shall not go at all extensively
into these extensions here, but merely give a brief overview of what they do and
how they can be used.
The first step, a very major one, was the introduction of flows as variables.
Clearly, describing any system not in equilibrium must require additional
variables for its characterization. Lars Onsager, a professor of chemistry at Yale,
formulated a way to describe systems near but not in equilibrium by finding
relations among the various flows, the rates at which mass of each specific kind,
and heat, and whatever other variables undergo changes in the system. A system
out of equilibrium is more complex than one in equilibrium, so we need to use
more information to describe it and its complexity. The rates of flow provide just
that additional information in the formulation developed by Onsager. These
additional variables augment the traditional variables of thermodynamics. Of
course systems that are not in equilibrium typically have different values of
temperature, for example, at different places within them. They often have
different relative amounts of different constituent substances as well. But in the
context in which Onsager’s approach applies, we can associate values of the
traditional thermodynamic variables with each point, each location, within the
system. We call that condition “local thermal equilibrium,” often abbreviated
LTE.
With that as a stimulus, other investigators opened new ways to extend
thermodynamics to treat systems operating at real, nonzero rates. For example,
one productive direction has been the extension of the concept of
thermodynamic potentials to processes operating in real time. As we discussed
earlier, traditional thermodynamic potentials are the functions whose changes
give the absolute natural limits on performance, typically the minimum energy
required to carry out a specific process with an ideal, reversible machine, one
that operates infinitely slowly. Each set of constraints such as constancy of
pressure or volume or temperature, or combination of any two of these, calls for
its own thermodynamic potential. The extension to real-time processes requires
more information, of course. Specifically, the analogues of traditional
thermodynamic potentials for real-time processes require knowledge of whatever
totally unavoidable loss processes must be accepted and hence included in the
way the process is specified. For example, if a process involving a moving
piston is to be carried out in a specified time, it is likely that the friction of the
piston against its container’s walls would be an unavoidable loss of energy,
something that would have to be included in the appropriate thermodynamic
potential for the process. These extended limits on performance are naturally
called “finite-time potentials.”
 Another aspect of the extension of thermodynamics to finite-time processes is
the recognition that, other than efficiency—work done, per total energy put into
the process—there is at least one other quantity that one may wish to optimize,
namely the power of the process, the amount of useful energy delivered, per unit
of time. This was first investigated by two Canadians, F. L. Curzon and B.
Ahlborn; they compared the optimization of a heat engine for maximum power
with that for maximum efficiency. In contrast to the Carnot expression for
maximum efficiency, 1 − TL/TH, its efficiency when a machine delivers
maximum power is given by with the square root of the temperature
ratio replacing the simple ratio of temperatures. Of course the pathway that
yields maximum power is very different from the maximum-efficiency path, the
infinitely slow reversible path, since power requires that the energy be delivered
at a real, nonzero rate. Other kinds of thermodynamic optimizations can be
useful for real, finite-time processes as well, such as minimization of entropy
production. That choice uses a measure of “best way” different from both the
maximum efficiency and the maximum power criteria. Each of these leads to a
different “best” pathway to achieve the desired performance. But all fall well
within the spirit of thermodynamics, its reliance on macroscopic properties and
variables, and its goal of optimizing performance, by whatever criterion.
There is still another important step one must carry out in order to design and
construct real processes aimed toward achieving or approaching the “best
possible” performance, whatever criterion of “best” one chooses. That is finding
the pathway, typically of a cyclic or continuous flow process, that comes as close
as possible to meeting the chosen criterion of “best” performance. This is
achieved by a mathematical approach known as “optimal control theory.”
Finding extremal pathways, such as longest or shortest, began in the 1870s with
Johann Bernoulli and other mathematicians, and continued as an active topic,
perhaps culminating in the work of the blind mathematician Lev Pontryagin and
other Russians in the 1950s and ’60s. The formal mathematical methods they
introduced can now be supplemented or even replaced by computational
simulations, enabling one to design process pathways that bring the process as
close as the available technology allows to whichever optimal path one wishes.
Sometimes thermodynamics opens realizations that something can be treated
as a controllable variable that was not previously recognized as such.
Distillation, for example, one of the most widely used industrial heat-driven
processes, common in petroleum refining and whisky making, is traditionally
employed the same way as it is in undergraduate laboratory courses, with the
heat put in entirely at the bottom of the distillation column, where the material to
be distilled lies, while the column above that container is cooled by a steady
flow, typically of cold water. The most volatile component of the original
mixture rises to the top of the column more readily than any less volatile
component, but only by going through many steps of condensation and re-
evaporation all along the height of the column. It turns out that much waste heat
goes into that re-evaporation. By treating the temperature profile of the column
above the source at the bottom as a control variable, one can minimize the heat
input required to achieve the desired separation. This is an example of the
application of optimal control to achieve maximum efficiency in distillation.
One can carry out analogous optimizations to maximize power or to minimize
entropy production, for example, for other processes. A key step and often an
intellectual challenge is identifying and choosing the right control variable, the
variable whose values we can choose; once selected, we can go through a
mathematical analysis to find the pathway for those values that yields the best
performance, based on the criterion we have chosen to optimize. The
temperature profile of the distillation column is just one example, specific to that
situation; each process we may wish to optimize will have its own control
variable or variables. Of course it is important to choose as one’s control variable
something that can truly be controlled in practice!
In its present state, the extension of thermodynamics to nonequilibrium
systems is a useful tool for describing systems for which traditional variables—
temperature, volume or density, pressure—can be applied, supplemented by
others such as flow rates. However, if a system is so far from equilibrium that
those traditional variables are changing at a rate so fast that they are not useful,
as in combustion, for example, the methods of thermodynamics, even as they
have been developed and extended, are not applicable. For those phenomena, we
must find other approaches, some of which are well understood—for example,
much of traditional kinetics of chemical reactions—and others, including several
aspects of reaction kinetics, that literally lie on the frontier of current research.
 SEVEN
What Can Thermodynamics Teach Us About Science
More Generally?

Thermodynamics as a unified science evolved initially from the stimulus of a

very practical effort to make heat engines more efficient, to get as much work
from heat as we can. The challenge to mine operators was to spend as little
money as possible for the fuel required to run the steam engines that pumped
water out of their mines. Watt’s engine and Carnot’s analyses came from that
search. Only after their contributions and those of their contemporaries and
immediate successors benefited the mining community did the basic science
emerge. The solution to this engineering problem provided the understanding of
the nature of heat and the recognition of the interconvertibility of light, heat,
mechanical work, and then electricity. With these understandings, of course,
came the explicit idea of energy. We have seen how it was this interconvertibility
that eventually led to the unifying concepts of energy and entropy, perhaps
painfully, as the central foundations of a powerful science. Here is a paragon of
the basic, general, and unifying arising from the stimulus of specific applied
problems.
This is an interesting contrast to a widely recognized and accepted paradigm
of the twentieth and twenty-first centuries, that the applied follows as a
consequence of the basic. That model has many examples to demonstrate its
validity. What we learn from the evolution of thermodynamics, however, is that
concepts and their consequences can flow in either direction, the basic out of
problems of application, or all sorts of applications from recognition of
potentialities exposed by new basic knowledge. The laser is a consequence of
Einstein’s analysis of radiation, and of the necessity that light of a precise
frequency can stimulate an appropriate energized particle to emit still more light
of that frequency—a case of basic science leading us to what has become an
extremely important application. The determination of the helical, sequential
structure of DNA is another example of a discovery of a “pure” basic character
eventually yielding new, practical methods to carry out genetic control. So here
is a way that thermodynamics can broaden our perspective on how scientific
advances come about; the basic can lead to the applied, or, with
thermodynamics, the applied can lead to the basic.

Variables and Ways of Looking at Nature

Classical and quantum mechanics approach the description of nature in terms of

the behavior of individual elements, whether atoms or tennis balls or planets.
The traditional variables we use for these in both approaches are the ones
describing the position and motion of those elements, how they change and how
they influence those properties of each other. This is what we have called the
“microscopic approach,” for which we have adopted variables of position,
velocity, acceleration, momentum, force, and kinetic and potential energy. But
that approach is of course not restricted to truly microscopic things. It is equally
valid for describing individual objects such as billiard balls and baseballs. It is
applicable to atoms and baseballs because we characterize them using properties
such as position and velocity. Thermodynamics and statistical mechanics take a
very different pathway to describe nature. They use properties we associate with
everyday objects and aspects of our environment as the variables telling us about
the world we inhabit. Temperature and pressure are properties of complex
systems, typically composed of many, many of the elements that mechanics
treats individually. Temperature and pressure have no meaning for individual
elements; momentum and velocity have no meaning for a complex system,
particularly when it is in equilibrium, unless we treat the entire system as an
element, such as a tennis ball.
It is amusing, in a way, to recognize that the tennis ball can be considered as a
single element whose complex molecular structure we totally neglect, for which
mechanics provides the appropriate description, or as a complex of molecules
with a temperature and an internal pressure, whose velocity and momentum we
neglect, when we use the appropriate thermodynamic, macroscopic description.
When treated as a single object, we can ascribe to it a mass, an instantaneous
position of the center of that mass, a velocity and an acceleration of the center of
that mass, and still other characteristics such as the compressibility of the tennis
ball and the instantaneous distortion from spherical shape when the ball strikes a
surface. On the other hand, we could, in principle, describe the tennis ball as
composed of a vast number of molecules bound together to form a stable but
somewhat elastic object. But in practice, we cannot play this two-faced game of
macro and micro descriptions with individual molecules or with the steam
driving an engine; we must use the microscopic approach for the former and the
macroscopic, thermodynamic approach for the latter. Of course, if we go
exploring into the nature of the particles of which atoms are made, we come to a
new level of description. Protons and neutrons themselves consist of particles,
called quarks and gluons. Is there another level, still deeper, of which these are
made? We have no evidence now of such, but we cannot predict what people
might find in the future. The history of science tells us that we continue to invent
new tools to probe ever deeper into the structure of matter; where this journey
will take us in the future, we cannot predict.
Statistics—as statistical mechanics and statistical thermodynamics—provides
a bridge to link these two very different modes. By recognizing that we cannot
imagine describing the instantaneous velocities and positions of all the air
molecules in a room, but that they nevertheless do obey the laws of mechanics,
quantum or classical, and then introducing the extremely critical concept of
probability, we can ask, “What are the most probable conditions, in terms of
microscopic variables, of those many air molecules, if we specify the
macroscopic variables that describe what we observe?” And then we must ask,
“What is the probability that we could observe a measurable deviation from
those most probable conditions? How probable is the most probable, compared
with other possible conditions?” As we saw in Chapter 4, the behavior of
systems of large numbers assures that we have virtually no possibility of
observing a significant deviation from the most probable condition for a typical
macroscopic system; we will never see all the air molecules moving, even very,
very briefly, to one side of the room, leaving none on the other side. But if we
consider, say, only ten air molecules in a box of one cubic centimeter, it is quite
likely that we could sometimes find all ten on one side of that box. The more
elementary objects that the system comprises, the more unlikely and difficult it is
to observe deviations from the most probable macrostate of that system.
This way of using statistics and probability tells us that our macroscopic
description is indeed valid, that the macro variables we use are indeed the
appropriate ones for the macro world in which we live. Because deviations from
the most probable conditions have probabilities that depend on the square root of
the number of elements that compose the macro system, and because square
roots of very large numbers are very much smaller than those large numbers, the
deviations become too improbable for us to expect to observe them—ever.
There is another way to use that information, however. There are some kinds
of behavior of macroscopic systems that we know and understand and accept as
universally correct, but which become inaccurate and inappropriate for small
systems, for which fluctuations from the most probable can be readily observed.
A striking example that we have seen is the case of the Gibbs phase rule, which
says that while we can vary the temperature and pressure of a liquid, say a glass
of water, as we wish (within limits, of course), if we insist on having ice in the
water, with the two forms in equilibrium, there is only one temperature, 0° C, at
one atmosphere pressure, at which we can maintain those two phases together. If
we change the pressure, then the temperature has to change in order for the two
phases to stay in equilibrium. If we want an even stricter condition, in which the
ice, the liquid water, and water vapor are all three in equilibrium, there is only
one temperature and pressure at which that can happen. This is the content of
that phase rule, f = c - p + 2. The condition for different phases to be in
equilibrium is simple to state: they must have exactly equal free energies, per
particle or per mole, or, in other words, exactly equal chemical potentials.
If we look at the properties of very small systems, for example a cluster of
only 20 or 50 atoms, we see something very different from what the phase rule
tells us. We have seen that solid and liquid forms of these small but complex
particles can coexist within a range of temperatures and pressures. Moreover,
more than two phases of these clusters can coexist within a finite range of
conditions. At the point at which the free energies of the solid and liquid forms
are equal, the probabilities of finding solid or liquid are exactly equal. If the free
energies of the two are different, however, the phase with the lower free energy
is the more probable, but the other, the “less favored phase,” is present as a
minority component of the total system—in observable amounts.
How can this be? How can small systems violate the presumably universal
Gibbs phase rule? The answer is straightforward and lies in the properties of
systems of large numbers. The ratio of the amounts of two forms of a substance
that can coexist in equilibrium is determined by the exponential of the free
energy difference between the two forms. The exponent, that free energy
difference, must of course be in dimensionless units; we cannot take e or 10 to a
power of a dimensioned quantity. The free energy must be expressed in units of
the energy associated with the temperature of the system, which, as we have
seen, we can write as kT if the free energy difference is expressed in a per-atom
basis or as RT if in a per-mole basis. If, for example, substance A and substance
B could interconvert, and at a specific temperature and pressure, their free
energy difference per molecule, divided by kT, is equal to the natural logarithm
of 2, then the equilibrium ratio of the amount of A to the amount of B would be
elog2, just 2:1. However, if our system is a small but definitely macroscopic one,
say of 1020 particles, roughly the size of a grain of sand, and the free energies of
the two phases differ by a very tiny amount, say 10-10 in units of kT, the ratio of
the two phases is exp{±1010}, either an enormous number or an extremely tiny
number. This tells us that although, in principle, the two phases could coexist in
unequal amounts under conditions where they have different free energies, those
coexistence ranges would be far too narrow for us ever to observe them. What is
formally a continuous change from solid to liquid, for example, occurs over a
range of temperatures (at constant pressure) so narrow that it appears to be a
sudden, discontinuous change by any means that we have to observe such a
change. Only at the point of exact equality can we see the two phases in
equilibrium—in practice.
If, however, we look at small systems, the free energy differences between the
solid and liquid phases of a 20- or 50-atom cluster can be small enough that the
exponential of that difference can be near but different from 1, meaning that the
unfavored phase, the one with the higher free energy, can easily be present as a
minority species in observable amounts. In short, the phase rule is absolutely
valid for macroscopic systems, because of the properties of large numbers, but it
loses its validity for systems of small numbers of component particles because
the large-number dominance doesn’t hold for those systems. In fact, it is even
possible to estimate the largest size of systems for which this violation of the
phase rule would actually be observable. Clusters of a hundred or so metal atoms
can exhibit coexisting solid and liquid phases over an observable range of
temperatures, with the relative amounts of the two phases governed by the
difference in the free energies of those phases. However, although we could
observe coexisting solid and liquid clusters of 75 argon atoms within a range of
temperatures, such a range of coexistence would be unobservably narrow for a
cluster of 100 argon atoms.
More broadly, this illustration is meant to show that there can be macroscopic
phenomena that are fully and accurately described by our macroscopic scientific
methods, which lose their validity for small systems, simply because that
validity is a consequence of the behavior of large numbers. We can generalize
the conclusion of the previous paragraph by pointing out that, for any property
described accurately by our macroscopic approach but which fails for small
enough systems, it is possible at least in principle to estimate the size of systems
that fall at the boundary, the lower size limit for validity of the macro
description, below which we must use some more microscopic approach. In the
case of the failure of the phase rule, we can, in fact, continue to use some aspects
of the macroscopic approach because we can continue to associate a free energy
with the atomic clusters of each phase, and to use the exponential of that
difference to tell us the relative amounts of each that we can expect to observe.
However, we may also want to ask, and this is at the frontier of the subject, what
fluctuations from the most probable distribution we can expect to observe. In a
sense, we can define a boundary size for a property well described at the macro
level that, for small systems, does not conform to that description; we can ask,
and sometimes estimate, the largest size system for which we would be able to
observe behavior deviating from the macro description. We can ask, for
example, what would be the largest number of molecules in a box of given size,
for which we could actually observe, briefly of course, all the molecules on one
side of the box. But finding such “boundary sizes” for specific phenomena is just
now at the frontier of science.
This tells us something more general about science. We find concepts,
sometimes from everyday experience, such as temperature, pressure, and heat,
that we can learn how to relate to one another. But we also seek to give some
kind of deeper meaning to those concepts so we may propose ideas, notions we
might not (yet) be able to observe or confirm, such as the proposal that matter is
composed of atoms, or that protons are composed of quarks and gluons, or that
heat consists of a fluid we call caloric. There was a time, well into the nineteenth
century, when the existence of atoms was only a conjecture, and a very
controversial one—but when Maxwell and Boltzmann showed how one could
derive many observable properties from the assumptions of the existence,
motion, and collisions of atoms, people became much more willing to believe in
their existence. And when Einstein explained Brownian motion, the random,
fluctuating movement of small particles visible under a microscope, in terms of
the collisions of those particles with molecules, the acceptance of atoms and
molecules became virtually universal.
This is an example of how we developed tools to test proposed explanations.
In this case, eventually we were able to go so far as to verify the atomic
hypothesis by actually seeing the proposed objects, and, in fact, not only to see
but even to manipulate atoms. With tools to work at the scale of ten billionths of
a centimeter, we now are able to move and place individual atoms. But for many
years, we lived with the unverified belief (unverified at least in terms of actually
observing them) that atoms are the basic constituents of the matter we see all the
time. Of course the observation in 1827 by Robert Brown that tiny particles,
pollen grains, observable under a microscope, jiggled and wiggled in random
ways we call “Brownian motion” was especially strong evidence in support of
the idea of atoms; something had to be hitting those particles we observed to
make them move randomly, in small steps. It was Einstein who, in 1905, showed
that random collisions of molecules with those observable pollen grain particles
accounts very well for those jiggles. Now, on a finer scale of size, we see strong
evidence that protons are not elementary particles but are made from smaller
things, the things we call “quarks” that are held together by things we call
“gluons.” But we have not seen quarks or gluons, not yet. In contrast, we believe
that electrons are not made of smaller things, but are actually elementary
particles.
Hence, just as we have seen how thermodynamics came to be linked via
statistical mechanics to atomic theory, other sciences evolve as we propose, test,
and develop new tools to probe new concepts. Some of those tools are
experimental, the results of new tools or new ways to use what we already have;
others are new concepts, new ways to ask questions or to envision and
hypothesize how things might be happening, concepts that lead to elaborate
theories that can only achieve validation, ultimately, by experiments or
observations. The gravitational force exerted by the mysterious dark matter is an
example of a major puzzling challenge arising from observations. We see the
consequences of that gravitational force, but we have not yet seen what is
responsible for producing that force, so we don’t yet know what its source is.
Observing results consistent with or predictable from a theory only says that the
theory may be correct, of course, and the more experiments and observations
consistent with that theory, the more we are ready to accept and believe it. On
the other hand, one experimental result or observation whose validity we accept
that contradicts the predictions of the theory is enough to disprove the
correctness of the theory. In science, it is often far easier to show that an idea is
wrong than that it is correct. A single (but confirmable) example that contradicts
a theory or an idea is enough to disprove it.
Nonetheless we can often go far with concepts and theories that turn out not
to be universally valid, but that have clear, justifiable validity for some
situations. This is exactly the case with thermodynamics; it is a valid science for
systems composed of very many elementary components, but some of its “rules”
and predictions lose their validity when we deal with systems of only a very few
components. (Of course some aspects we think of as coming from
thermodynamics do remain valid for all systems, such as the conservation of
energy.) The same kind of limitation is true of the mechanics of Newton; it is
quite valid and useful for describing baseballs or planets, but loses its validity at
the very small scale of atoms, where we must use quantum mechanics.
Nonetheless, in both of these examples, we now understand how the large-scale
science, whether thermodynamics or Newtonian mechanics, evolves in a self-
consistent way, from the small-scale science, how statistical mechanics explains
the way thermodynamics becomes valid for many-body systems and how
quantum phenomena become unobservable as we move from the atomic scale to
the scale of human beings.
Sometimes we can demonstrate that a proposed concept is wrong; that is what
happened with the caloric theory of heat, and that process of negation was the
main pathway for the evolution of science. Negation has always been one of
science’s strongest tools; it is usually far easier to show that a wrong idea is
wrong than that a correct idea is indeed right. This is obviously so because one
“wrong” is enough to eliminate the idea. But there are occasional opportunities
in which we can actually demonstrate that a proposed concept is right, as in the
case of observing individual atoms. This only became possible in very recent
years, yet we accepted their existence for well over a century. Often, we live and
work with a concept that is useful and consistent with all we can observe but that
we cannot prove is correct. That, for many years, was the situation with the
concept that matter consists of atoms. And that condition led to the realization
that much of science evolves by depending on our ability to show that an idea or
concept is wrong and inconsistent with what we can observe, while we must live
with concepts that are indeed consistent with all we can observe but that we are
unable to prove to be correct. And sometimes we eventually do develop means
to test those ideas in a positive way, not just in the negative way of seeing
whether they are wrong, or inconsistent with what we observe.

How Science Evolves

Sciences and scientific “laws” can never be thought of as permanent, inviolate

statements about nature. We saw how our concept of the nature of heat emerged
from the controversy over whether heat is a fluid, “caloric,” or the result of the
motion of the elementary particles in matter. We saw the difficulties and tensions
associated with the emergence of the concept of energy, and we now can look
back to see how that concept has changed and expanded as we have learned
more and more about it. Until Einstein’s theory of relativity, energy and mass
were “known” to be two totally different manifestations of nature, but when the
relation that showed that mass is indeed a form of energy, related to energy itself
by the famous, simple expression E = mc2, we had to expand our concept of the
various forms that energy could take. “Atomic energy” is, literally, energy
associated with the conversion of mass into other forms of energy. In a nuclear
reactor, relatively unstable nuclei of high mass, typically of the uranium isotope
U235, break apart, and the sum of the masses of the fragments is slightly smaller
than the mass of the original uranium. That loss of mass reappears typically as
heat, which we use to generate electricity.
The concept of energy is undergoing another substantive expansion just as
this is being written. We now know that there is something pervasive in the
universe that exerts gravitational force, something whose nature we do not yet
understand but whose presence is now well established, something we call “dark
matter.” Because it does exert observable, even very large gravitational effects, it
must have some of the properties of mass and hence must be another form of
energy. Its properties, revealed by its gravitational effects on galaxies, tell us
much about where it is, but we still know very little about what it is. Beyond the
puzzle of dark matter, and even more puzzling and less understood, is that there
must be another form of energy—something, whatever it turns out to be, that is
responsible for the accelerating expansion of the observable universe. That
“something” goes by the name “dark energy,” which merely acknowledges that
it must be included with all the other manifestations of nature that we call
“energy.” So far as we know now, the one common property that we can
associate with the many forms that energy can take is that of its conservation.
And that is a truly amazing thing, perhaps one of the very few things about
nature that we are inclined to accept, at least in this age, as a true, inviolable
property of nature. A tantalizing question arises here, which the reader might
ponder: is energy a discovery or an invention of the human mind?
We have seen Newton’s mechanics, long thought to be totally fundamental,
transformed into a useful description of what we experience in our everyday
lives, but, nevertheless, a consequence of the more fundamental quantum
mechanics. We are seeing thermodynamics transformed from a universal
doctrine into a useful representation valid for large enough systems. We must be
prepared, always, to learn that concepts that are useful at some scale or under
some circumstances may be revealed to have more limited applicability than we
originally thought—or even that those concepts, however plausible they might
have been, were simply wrong, as with the concept of “caloric” as a fluid
thought to constitute heat, and how that idea was discarded when people realized
that heat is a consequence of random motion at the atomic level. A somewhat
analogous idea was a widely accepted notion that fire was caused by a substance
called “phlogiston,” and that it was the release of phlogiston that was literally the
flame. That concept, of course, was discarded when we came to recognize that
burning in air is a consequence of a chemical reaction in which the combustible
material, be it wood, paper, coal, or whatever else, combines with oxygen. But it
was not until people found that in burning, materials gained weight, rather than
losing it as the phlogiston theory predicted, that the true nature of combustion
became known.
Let us take a broader, more distant view now, of science, of what it does, what
it is, and how it evolves. Science consists of at least two kinds of knowledge,
what we learn from observation and experiment and what we learn from creating
interpretations of those experiments, the intellectual constructs we call theories.
We can recognize, by looking at the history of thermodynamics, how our tools of
observation have evolved, improved, become ever more powerful and
sophisticated, from following the temperature associated with cannon-boring to
the observation of Brownian motion in tiny particles, to the astronomical
observations that reveal the presence of dark matter. And with the changing tools
of observation, we have had to revise and reconstruct the theories we use to
interpret these observations. Another vivid illustration is the history of the
atomic theory of matter. As we saw, a kind of atomic theory arose in ancient
Greece, through the ideas of Democritus and Lucretius, but the concept
remained controversial, opposed to a concept that matter was infinitely divisible,
until Einstein explained the random Brownian motion of colloidal particles as
the result of random collisions of those particles with tinier particles that were
still not visible then. But now, with the remarkable advances in the tools we can
use, we can actually see and manipulate individual atoms. The atomic theory of
matter is indisputable because we have unassailable evidence for the existence of
these objects.
We can think of the evolution of each science in terms of the levels of
understanding through which it passes with time, as our tools of observation
develop and become more powerful, and our theories advance to enable us to
understand those observations. Thus thermodynamics, and all other areas of
science, are never static, locked combinations of observation and interpretation;
they are constantly changing, becoming ever broader in the sense of
encompassing new kinds of experience. Newton’s physics remains valid in the
domain for which it was intended, the mechanics of everyday experience.
Quantum mechanics, in a sense, displaced it but actually left it valid and in a
sense deepened in its proper domain, while enabling us to understand
phenomena at the molecular level where the Newtonian picture fails to account
for the observations.
An analogous pattern is apparent in thermodynamics, where we see the
evolving, broadening nature arising not only from new kinds of observations but
also from asking new questions. We can trace its history by following the
questions that stimulated conceptual advances. Perhaps the first key question
was the basis of Carnot’s work, “How can we determine the best possible
(meaning most efficient) performance of a heat-driven machine?” But there were
other deep questions that followed: “What is energy and what are its inherent
characteristics?” “What gives the directionality to evolution in time?” Then,
more recently, “How can we extend the tools of traditional thermodynamics,
based on equilibrium and idealized systems and processes, to more realistic
situations, systems not in equilibrium, and systems operating at nonzero rates?”
And “How can we extend thermodynamics to systems such as galaxies, whose
energy depends on the products of masses of interacting bodies?” And still
further, “How can we understand why some very powerful, general concepts of
thermodynamics lose their validity when we try to apply those concepts to very
small systems?” Thermodynamics is, in a sense, a science of things made of very
many elements, not a means to treat and understand tiny things consisting of five
or ten or a hundred atoms. We can expect thermodynamics, and all our other
sciences, to continue to evolve, to explain newly observed phenomena and to
answer new, ever more challenging questions. There probably will be no end to
the evolving nature of science; it will continue to grow, to be ever more
encompassing, to enable us to understand ever more about the universe, but we
can see no end to this process.
Thus we see, partly through the evolution of thermodynamics, that our
concepts of nature and what we think of as the substantive knowledge contained
in a science is never firm and fixed. There have even been situations in which
the concept of conservation of energy has been challenged. Until now, it has
withstood those challenges, but we cannot say with total certainty that there are
no limits to its validity. Science will continue to evolve as long as we can find
new ways to explore the universe and whatever we find in it.
 Index

Absolute zero, 46, 68

Adiabatic, 73, 81–82, 100
Ahlborn, B., 139
Amontons, Guillaume, 50, 51, 65, 85
Ampére, A. M., 77, 78
Angular momentum, 6–7
Asbestos, 103
Atomic energy, 157
Atomic theory, 66
Atomistic model, 115
Atoms, 24
Avogadro’s number, 38, 42, 92, 96

Bacon, Francis, 49
Baseball, 7
Basic science to applied science, and vice versa, 48–49, 143–144
Bérard, J. E., 69
Bernoulli, Johann, 140
Biot, J. B., 69
Black, Joseph, 53–55, 64
Black body, 106
Bohr, Niels, 125
Bohr model, 125–126
Boltzmann, Ludwig, 88, 96, 116, 120, 121–122, 153
Boltzmann constant, 39, 42, 96, 97
Bose, S. N., 126
Bose-Einstein condensate, 46, 127
Bosons, 126
Boulton and Watt steam engines, 59–60, 109
Boyle, Robert, 49
Brown, Robert, 153
Brownian motion, 125, 153, 159–160

Caloric, 23, 63, 66, 82

Candles, 104
Canonical ensembles, 123
Carathéodory, Constantin, 31
Carnot, Sadi, 11, 21, 69, 70, 71, 75–77, 83, 99, 102, 160
Carnot cycle, 72–73, 74, 76, 100, 109, 111, 112, 135
Cartesians, 78
Celsius, Anders, 8, 66, 83
Celsius (centigrade) temperature scale, 8, 33, 66–67
Charles’s law, 50, 67
Chemical bonds, 13
Chemical potential, 98, 123, 134, 149
CHP (combined heat and power), 102
Clapeyron, Émile, 77, 83
Classical theory of thermodynamics of radiation, 107
Clausius, Rudolf, 30, 87, 88, 122
Clusters, 133, 134, 135
Coal, 13
Coal mines, 52
Cogeneration, 102
Combined heat and power, 102
Components, 37
Composition, 5
Conduction, 62
Conservation of energy, 15, 83, 86
Constant pressure, 65, 80
Constant volume, 65, 80
Control variable, 141
Convection, 62
Cooling processes, 99
Coriolis, G. G. de, 78
Cubic centimeters, 9
Cubic inches, 9
Curzon, F. L., 139
Cycle, 20
Dalton, John, 66, 67, 68
Dark energy, 108, 157–158
Dark matter, 108, 154, 157, 159
Davy, Sir Humphry, 80
Degrees of freedom, 26, 37
Delaroche, F., 69
ΔE, 18
δp, 18
δx, 17
Democritus, 24, 159
Density, 8
Descartes, René, 49
Diesel cycle, 109
Dimension, 11, 12
Distillation, 140

e, 41, 95, 117

Efficiency, 71, 73
Einstein, Albert, 124–125, 144, 153, 159–160
Electric charge, 12
Electric generator, 103
Electromagnetic radiation, 106
Energy, 4, 7, 9, 10–11, 12, 13–14, 15, 17, 18, 19, 22, 80, 86, 157
Engine, 10
Ensembles, 122
Entropy, 5, 32, 34, 36, 40, 43, 87, 88–89, 90, 96
Equation of state, 36, 37
Equilibrium, 1–4, 14, 15, 21, 22–23, 28, 31, 36
Equilibrium constant, 97, 133
Equipartition, 24
Equivalence of work and heat, 81
Erg, 11
Euler’s number (e), 41, 95, 117
Evolution, 31
Exhaust valve, 112
Extensive variables, 8, 15
External condenser, 57

Fahrenheit temperature scale, 8

Fairbottom Bobs, 52
Fermi, Enrico, 126
Fermions, 126
Finite-time potentials, 138
First law of thermodynamics, 4, 6, 14, 19, 29, 87
Fluctuations, 95, 119
Fluorescent lamp, 104, 105
Fourier, Joseph, 69
Free energy difference, 133
Friction, 10, 11, 21

Galilei, Galileo, 50
Gas constant, 38, 39, 42, 81, 96
Gas lamps, 104
Gasoline engine, 20, 109, 111–113
Gas phase, 37–38, 50, 55, 56, 65–66, 132
Gaussian distribution, 119
Gay-Lussac, Joseph-Louis, 50, 66, 68
Gay-Lussac’s law, 50, 67
Generation of electricity, 101
Generator, 101
Gibbs, J. Willard, 37, 88, 89, 114, 122, 123
Gibbs free energy, 97, 98
Gibbs phase rule, 37, 123, 131, 133, 134–135, 148, 149
Gluons, 154
Gravity, 9

Hachette, J. N. P., 69
Halley, Edmund, 51
Heat, 7, 9, 10, 16, 20, 23, 50, 55, 61–69
Heat capacity, 32–34, 35, 36, 53, 64, 74, 81
Heisenberg, Werner, 126
Helmholtz, Hermann von, 83, 86
H-function, 122
History of thermodynamics, 48, 49
Hornblower, Jonathan, 61
Hydrogen atoms, 27, 38, 96, 125

Ideal engine, 21, 135–136

Ideal gas, 38, 39
Ideal gas law, 66
Ideal reversible engine, 70, 73
Ideal systems, 11
Incandescent lamp, 10, 104
Indicator diagram, 60, 109, 110, 113
Inequality, 34, 35
Information theory, 5
Intake valve, 112
Integration, 110
Intensive variables, 8
Isoergic, 128
Isothermal, 72, 128
Joule, James Prescott, 81–83, 86
Joules, 11, 96

Kelvin, William Thomson, Lord, 31, 33, 83–85, 86

Kelvin temperature scale, 33, 38, 85
Kinetic energy, 13

Laplace, Pierre-Simon, 63
Lasers, 48, 125, 144
Latent heat, 53, 55
Lavoisier, Antoine, 63–64
Laws of motion, 6
Leakage, 21
Leibnitz, G. W., 78, 79
Leibnitzians, 78
Light-emitting diode (LED), 104–105
Liquid phase, 37–38, 40, 55, 56, 132, 135, 148, 149, 150–151
Local thermal equilibrium, 137
Logarithm, 41, 95
Lubricating oil, 21
Lucretius, 24, 159

Macroscopic systems, 2, 36, 91, 92, 93, 94, 96, 130, 131, 132, 135
Macroscopic variables, 6, 7, 17, 36, 39, 147–148
Macrostates, 6, 37, 40, 41, 42, 45, 94–95, 96
Mass, 5
Maximum power, 141
Maxwell, James Clerk, 73, 115–116, 119, 120, 153
Maxwell-Boltzmann distribution, 116–119, 121–122, 123
Maxwell distribution, 116
Mayer, J. R., 80, 81, 82
Methane, 103, 104
Microcanonical ensembles, 123, 128, 129
Microscopic variables, 6–7, 16–17, 145, 147
Microstates, 6, 40–41, 42, 43, 45, 91, 93, 94–95, 96
Microwaves, 121
Mixtures, 5
Mole, 38, 92
Momentum, 6
Motive power of heat, 70

Nanometer, 106
Natural gas, 13, 103, 104
Negation, 155
Nernst, Walter, 45
Newcomen, Thomas, 51
Newtonian mechanics, 89, 90, 93, 114, 126, 155
Nonequilibrium systems, 141
Nonextensive thermodynamics, 28
Nuclear energy, 102

Onsager, Lars, 137

Optimal control theory, 140, 141
Otto cycle, 109, 111

Perpetual motion, 75
Phase changes, 134
Phase rule, 37, 123, 131, 133, 134–135, 148, 149
Phases, 37
Phase space, 123
Phlogiston, 158–159
Photoelectric effect, 125
Piston, 10, 11, 20, 25, 51–52, 58–59, 60, 111–113, 138
Planck, Max, 107–108, 124
Poncelet, J. V., 78
Pontryagin, Lev, 140
Position, 7
Potential energy, 86
Power, 71
Power stroke, 113
Pressure, 5, 6, 8, 50
Probability, 147
Probability distribution, 118
Process variables, 12, 16

Quantity of heat (Q), 9, 16, 17, 18–19, 29

Quantum energy, 107–108, 124
Quantum mechanics, 48, 114–115, 124, 126, 145, 155, 158, 160
Quantum theory, 6, 108
Quarks, 154

Radiation, 61, 77, 101, 105–106, 107–108

Random motion, 24
Rankine, William Macquorn, 73, 86, 87
Refrigeration, 99–101
Regnault, Victor, 83, 85
Reversible engine, or process, 21, 26, 70–71, 75, 89–90, 93, 114, 135, 136, 138
Rotational freedom, 27
Rumford, Benjamin Thompson, Count, 62, 63–64, 77

Schrödinger, Irwin, 126

Sea level, 9
Second law of thermodynamics, 5, 30, 32, 34, 42, 87, 91
Southern, John, 60, 109
Specific heat, 32, 33, 34, 64
Spectrum, 105, 106
Spontaneous behavior, 45
Standard deviation, 119
State variables, 12, 15
Statistical mechanics, 43, 88, 90–91, 123, 145, 147, 154, 155
Statistical thermodynamics, 114, 122, 147
Steam, 14
Steam engine, 51
Stefan, Josef, 121
Stimulated emission, 125
Sunlight, 101
Superconductors, 46

Temperature (T), 2, 3, 5, 6, 7–8, 9, 12, 15, 22–23

Thermodynamic potentials, 5, 97, 138
Thermodynamics, 87
first law of, 4, 6, 14, 19, 29, 87
second law of, 5, 30, 32, 34, 42, 87, 91
third law of, 5, 45, 68, 128
violation of laws of, 128, 131
Thermometer, 8
Thermostat, 15, 73
Third law of thermodynamics, 5, 45, 68, 128
Thompson, Benjamin (Count Rumford), 62, 63–64, 77
Thomson, James, 83
Thomson, William (Lord Kelvin), 31, 33, 83–85, 86
Tin mines, 49
Transistors, 48
Translational freedom, 26, 27
Trevithick, Richard, 61, 75
Tungsten filament, 104
Turbine, 14, 103
Units, 12
Universe, 15

Variables, 6
Velocity, 6, 12
Very small systems, 130
Vibrational degrees of freedom, 27
Volume (V), 5, 9, 15, 18
Waste, 11
Watt, James, 56
Wind, 101
Woolf, Arthur, 61
Work (W), 9–11, 12, 16, 17, 18–19, 20, 25, 29, 65, 71, 73, 100, 110, 112