The Second Law and Molecular Behavior

At the present time we are familiar enough with molecules to formulate the Second
Law entirely in relation to an intuitive perception of their behavior. It is easy to see that the
Second Law, as expressed in terms of heat flow in section 11, could be violated with some
cooperation from molecules.
Consider two systems each consisting of a fixed quantity of gas. At the boundary of
each system are rigid, impenetrable, and well insulated walls except for a metal plate made
of a good thermal conductor and located between the systems as shown in Figure 1. The gas
in one system is at a low temperature T1 and in the other at a higher temperature T2. In
terms of molecular behavior the

temperature difference is produced by different distributions of molecules among the
velocities in the molecular states of each system. The number of different velocities,
however, is so large that the high temperature system contains some molecules with lower

speeds than some molecules in the low temperature system and vice versa.
Now suppose we prepare some instructions for the molecules in each system as
shown on the signs in Figure 1. When following these instructions, only the high speed
molecules in the low temperature gas collide with molecules on the surface of the plate, give
up energy to them, and thus create on this surface a higher temperature than in the gas.
The boundary between the gas and the plate then has a temperature difference across it and,
according to our definition, the energy thus transported is heat. In the plate this becomes
thermal energy which is conducted through it because the molecules on its other surface are
at a lower temperature as a consequence of their energy exchanges only with the low speed
molecules in the adjacent high temperature gas system. This constitutes likewise a heat flow
into this system. The overall result is then the continuous unaided transfer of heat from a
low temperature region to one at a higher temperature, clearly the wrong direction and a
Second Law violation. The Second Law therefore, is related to the fact that in completely
isolated systems molecules will never of their own accord obey any sort of instructions such
as these.
                          Microstates in Isolated Systems

To explain why molecules always behave as though instructions of this type are
completely ignored, imagine that we have a fantastic camera capable of making a
multidimensional picture which could show at any instant where all the ultimate particles
in the system are located and reveal every type of motion taking place, indicating its
location, speed, and direction. Every type of distinguishably different action at any moment,
the vibration, twisting, or stretching within molecules as well as their translational and
rotational movements, would be identified in this manner for every molecule in the system.
This picture would thus be a photograph of what we have defined as an instantaneous
microstate of the system.
Now, instead of the instructions on the signs in Figure I, suppose we ask the
molecules to do everything they can do by themselves in a rigid walled and isolated
container where no external arranging or directing operations are possible. We will say,
"Molecules, please begin now and arrange yourselves in a sequence of poses for pictures
which will show every possible microstate which can exist in your system under the
restrictions imposed by your own nature and the conditions of isolation in the container".
If we expressed these restrictions as a list of rules to be followed in assigning molecules to various positions and motions, the list would appear as follows:
1. In distributing yourselves among the various positions and motions for each microstate
picture, do not violate any energy conservation laws. Consequently, because you are in
an isolated system the sum of all your individual translational, vibrational, and
rotational kinetic energies plus all your intermolecular potential energies must always
be the same and equal to the fixed total internal energy of the system.
2. Likewise, do not violate any mass conservation laws. There are to be no chemical
reactions among you so that the total number of individual molecules assigned must
always remain the same.
3. All of you must, of course, remain at all times within the container so the total volume
in which you distribute yourselves must be constant.
4. Do not violate any laws of physics applicable to your particular molecular species. You
must remember that no two of you can have all of your microstate position and motion
characteristics exactly the same otherwise you would have to occupy the same space at
the same time. Furthermore, do not be concerned that there might not be enough
different microstates available for each of you to have a different one. Although you are
numerous, the number of different possible position and motion values is even more
numerous, so that there will never be enough of you to fill all of them and many
possible values will be left unoccupied by a molecule in each picture for which you pose.

The Total Energy Transfer

Because thermodynamic systems are conventionally defined so that no bulk
quantities of matter are transported across their boundaries by stream flow, no energy
crosses the system boundary in the form of internal energy carried by a flowing fluid. With
the system defined in this way the only energy to cross its boundary because of the flow
process is that of work measured by the product of the pressure external to a fixed mass
system in a stream conduit and the volume change it induces in this system. In the case of
diffusion mass transport, as discussed in section 15, the system does not have a fixed mass
but the entire change associated with the diffusion mass transport is given by work
evaluated by computing the product of an external chemical potential and a specific
transported mass change within the system. As a result the combination of heat, work, and
any energy transport by non-thermodynamic carriers includes all the energy in transition
between a system and its surroundings. Energy by non-thermodynamic carriers is that
transported by radiant heat transfer, X-rays, gamma radiation, nuclear particles, cosmic
rays, sonic vibration, etc. Energy of this type is not usually considered as either heat or
work and must be evaluated separately in system where it is involved. Energy transport by
nuclear particles into a system ultimately appears as an increase in thermal energy within
the system and is important in thermodynamic applications to nuclear engineering.
In every application of thermodynamics, however, it is essential that we account for
all the energy in transition across the system boundary and it is only when this is done that
the laws of thermodynamics can relate this transported energy to changes in properties
within the system. In the processes we will discuss, heat and work together include all the
transported energy.

The Chemical Potential

For processes involving diffusion mass transport we can, however, define a
thermodynamic intensive driving force responsible specifically for the total energy change
accompanying the diffusion mass transfer of molecules from one region to another. This
driving force can be defined simply as a partial derivative representing the variation of the
total internal energy of a region with respect to an increment in the number of moles of one
particular species in the region when no other extensive properties are altered. This partial
derivative is an important intensive property called the chemical potential. By reason of its
definition the chemical potential is an intensive property because whenever it is multiplied
by the extensive property change in moles of a particular molecular species within a system
the result is identically the internal energy change of the system resulting only from this
change in moles, and not from the change in any other extensive property.
In elementary physics the energy per unit mass, per mole, or per particle involved in
moving the mass, mole, or particle from one region to another is generally defined as a
potential. Table I lists several types of potentials (driving forces) which are important in
thermodynamic applications. A potential therefore can always be regarded as a driving
force for a mass change. The chemical potential is a driving force of this type. Physically the
driving force represented by the chemical potential results from the same molecular actions
which give rise to a partial vapor pressure in a liquid or a partial pressure in a gas. Each of
these has the ability to expel molecules of a given type out of a multi-component phase.
The energy change within a system accompanying a change in the number of moles
of a given component of the system by molecular processes can now be defined as a type of
work which results from a difference in a chemical potential driving force between the
system and its surroundings. As is the case of other types of work, in order to evaluate
quantitatively the work of a chemical potential driving force it is first necessary to define a
system. In accordance with the principles discussed in section 14,this work is then defined
as the product of a chemical potential of a component outside the system on its external
boundary and a change in the number of moles of this component inside the system. When
the number of moles of a molecular species increases in a system, work must be done on the
system to overcome the molecular forces tending to expel molecules of this species.
Consequently, in accordance with the sign convention, the work relative to the system
receiving this increase in moles within it must be a negative number.
Although we have discussed the chemical potential as a thermodynamic driving force
for the diffusion mass transport, its utility is not confined to this particular process alone.
Because of its definition the chemical potential is a driving force for changes in moles of a
molecular species in a system not only by means of diffusion mass transfer but by any other
molecular process as well. The most important example is the role of the chemical potential
within a system as the driving force for changes in moles brought about in the system by
chemical reactions.

Energy Transport by Mass Transfer

For any region with a boundary which is penetrated by mass, a thermodynamic
analysis always requires a distinction between mass carried across the boundary by bulk
stream flow and mass carried across by diffusion processes resulting from molecular action.
In the case of bulk stream flow with no diffusion mass transport, energy is carried
into the region in two distinct ways. Part of the energy added to a region receiving mass by
stream flow is the work of a pressure which displaces a quantity of flowing fluid into the
region. The remaining part of the energy added is the internal energy content of this
quantity of fluid which enters. In a bulk stream flow process these two parts of the total
energy transport can be separated and evaluated. This is done most conveniently by
defining the system in this case as a fixed mass enclosed by moveable boundaries which are
not penetrated by mass at all. In this manner a small contiguous quantity of fluid in an
entering conduit becomes a homogeneous sub-region within the system and its energy thus
becomes a part of the total internal energy of the entire system. The boundary of this subregion
is acted upon by an external pressure which performs work on the entire system in
moving the boundary of the sub-region. When the system is defined in this way, no energy
is carried into the system in the form of the internal energy of mass crossing its boundaries.
In a region receiving mass transported by a diffusion process, part of the energy content of all molecules outside the region is used to propel some of them into the region. In
contrast to the situation in a purely bulk stream flow process, there is no way in this case to
define a system which excludes the internal energy of transported molecules from the energy
crossing the system boundary. There is no way to define a system in which the propelling
forces which induce the mass transport are a driving force for all of the energy which crosses
the system boundary in the transport process. The diffusion processes these propelling
forces result from the behavior of individual molecules and are not scalar thermodynamic
properties at all so that we cannot define an intensive thermodynamic driving force property
to represent them.

Work

Now that we have used the term "work" it is necessary to emphasize that work, like
heat, must also be regarded only as a type of energy in transition across a well defined, zero
thickness, boundary of a system. Consequently work, like heat, is never a property or any
quantity contained within a system. Whereas heat is energy driven across this boundary by
a difference in temperature, work is energy driven across by differences in other driving
forces on either side of it. Various kinds of work are identified by the kind of driving force
involved and the characteristic extensive property change which accompanied it.
Work is measured quantitatively in much the same manner as heat. Any driving
force other than temperature, located outside the system on its external boundary, is
multiplied by a transported extensive property change within the system which was
transferred across the system boundary in response to this force. The result is the numerical
value of the work associated with this system and driving force. It is important to
emphasize that the extensive property change within the system which is used in this
computation must be a transported quantity whose transfer across the system boundary
depends on a particular driving force with different values inside and outside the system.
This transported extensive property change within the system always occurs with the same
magnitude but with opposite sign in the surroundings.
Neither work nor heat results from any part of a change in an extensive property of a
system which has not been transported in this manner without alteration in magnitude
across the system boundary. A non-transported extensive property change within a system
when multiplied by an appropriate driving force property located within the system measures a form of internal energy change in the system but not work or heat.
Conventionally the quantity of work calculated by this procedure is given a positive
sign when work is done by the system on the surroundings and energy crosses the boundary
in a direction from the system to the surroundings. An energy transport in the opposite
direction, when work is done by the surroundings on the system, is given a negative sign.
It is awkward that the sign given to energy transferred as work is opposite to that given to
energy transferred as heat in the same direction, but tradition has established the convention
and it is important that it be followed consistently. Like heat, both the absolute value and the
sign of what is called work depend entirely on how the system is specified.
Several thermodynamic driving forces and their characteristic displacements are
listed in Table I. Any of these properties, other than temperature and entropy, can measure
various types of work when the driving force is located on the outer side of the system
boundary and the displacement is a transported quantity whose change is located within the
system. The product, when given the proper sign, is a type of work transfer for this system.

Entropy

As indicated in Table I, entropy is the name given to the extensive property whose
change when multiplied by temperature gives a quantity of thermal energy. In classical
thermodynamics there is no need to give any physical description of this property in terms
of molecular behavior. A change in entropy is defined simply as a quantity of thermal
energy divided by the temperature driving force which propels it so that it always produces
the thermal energy identically when multiplied by the temperature. Because temperature is
an intensive property and this product is energy, we know that the entropy must be an
extensive property. Furthermore, thermal energy is a part of the total internal energy within
a system so that the entropy change computed this way is a change in a property of the
system Thermal energy crossing a system boundary is defined as heat so that the entropy
change transported by it is simply the quantity of heat transported divided by the
temperature which transports it. This transporting temperature is the temperature of the
external or surroundings side of the system boundary. It is important to realize that this
transported entropy change may be only apart of the total entropy change within the
system. Because thermal energy can be produced within a system by other means than
adding heat to it, a thermal energy increase in the system can be greater than the heat
transported into it. In this case the entropy change within the system accompanying its
thermal energy increase will be greater than the entropy change transported into the system
with the heat flow.

Basic Principles of Classical and Statistical Thermodynamics lec (2)

7 Intensive and Extensive Properties
In discussing microstate driving forces in section 5, we noted that the force to be
applied or the force to be overcome in order to make a change in the position or motion of
any one particle in a multi-particle system depends both on the nature of the particle and on
its environment. When these remain the same then the necessary force to induce a change is
also the same, no matter how many other individual particles are present in the system.
Because a thermodynamic driving force in a system is the composite result of all the
individual particle forces, it likewise should be independent of the number of particles
present as long as they all have the same environment and individual characteristics.
Properties of a system which have this type of independence of the number of
particles present are called "intensive properties" and all the thermodynamic driving forces
are selected from among properties of this type. The test for an intensive property is to
observe how it is affected when a given system is combined with some fraction of an exact
replica of itself to create a new system differing only in size. Intensive properties are those
which are unchanged by this process, whereas those properties whose values are
increased/decreased in direct proportion to the enlargement/reduction of the system are
called "extensive properties." For example, if we exactly double the size of a system by
combining it with an exact replica of itself, all the extensive properties are then exactly
double and all intensive properties are unchanged.
As we have explained the displacements in a system induced by thermodynamic
driving forces are a summation of all the motion and position changes in all the ultimate
particles of the system. Consequently, if we alter the number of particles by changing only
the size of the system, we should then alter the overall displacement in exactly the same
proportion. This means that the overall change which we call a displacement must be a
change in an extensive thermodynamic property of the system.
If the magnitude of a displacement thus varies directly with the size of a system in
which it occurs, whereas the driving force is not affected, their product must likewise change
directly with the system size so that energy itself is always an extensive property.
8 Identification of Thermodynamic Driving Forces and Displacements
In addition to the differences between thermodynamic driving forces which arise
because the thermodynamic forces are scalar properties instead of vectors, another important
difference is that they are quite different dimensionally. This is a consequence of the fact
that the thermodynamic driving forces are defined in a quite intuitive manner.
Thermodynamic driving forces are identified empirically as the intensive property
whose difference on each side of some part of the boundary between a system and its
surroundings control both the direction and the rate of transfer of one specific extensive
property displacement across it. For example, consider the volume filled with air within a
pump and bicycle tire as a system and the inner surface of the piston of the pump as the
boundary across which a volume change is transferred. When the piston is moved the
magnitude of the volume change in the surroundings is exactly the magnitude of the volume
change of the system and the increase in volume of one is exactly the decrease in volume of
the other. We can say, therefore, that volume is an extensive property transferred across this
boundary. When only volume and no other extensive property change is transferred, then
we find by experiment that the pressure difference is the only intensive property across this
boundary that controls both the direction and rate of change of the volume. Then we define
pressure as the thermodynamic driving force. It is important that only one extensive
property be transferred across this boundary in the experiment. For example, suppose there
was a crack in the piston which allowed air to leak through it. We now can have both
volume and mass transferred across this same boundary and we observe in this case that
lowering the pressure outside the piston may not necessarily cause the volume of the system
to expand. To properly identify a driving force we must always examine the transport of
only one displacement and one characteristic type of energy.
Although the dimensional and physical nature of each thermodynamic driving force
identified in this manner are very different, the product of each with its associated
displacement always measures a distinctive type of energy and must have the characteristic
energy dimensions of force multiplied by length.
Once a particular type of energy crossing a boundary has been identified, the manner
in which it is divided into a driving force and displacement is completely arbitrary as long
as the driving force is intensive and the displacement is a change in an extensive property.
For example, in this illustration of the transfer of pressure-volume work we could have
equally well called the displacement the distance traveled by the pump piston and the
driving force a product of pressure and piston area. We would thus change the dimensions
of the driving force and displacement, but this would not affect any thermodynamic
computations where only the magnitudes and not the rates of changes in energy and
properties are to be determined. In a subject called "non-equilibrium thermodynamics
where a description of the rates of various changes is an objective, the definition of driving
force and displacement is not at all arbitrary and must be done only in certain ways.1
Some of the diversity of driving force-displacement combinations and their
dimensions, which represent various types of energy in some important thermodynamic
applications is shown in Table I. The product of the two represents a change in the energy
of a region in which both the driving force and displacement are properties. It also gives the
energy transported between a system and its surroundings when the driving force is located
on its outer boundary and the displacement is within the system.
9 The Laws of Thermodynamics
Now that we have discussed the nature of different forms of energy and properties of
matter, we must describe the basic principles of thermodynamics which are used to relate
them.
Classical thermodynamics is one of the most important examples of the axiomatic
form of the scientific method.2 In this method certain aspects of nature are explained and
predicted by deduction from a few basic axioms which are assumed to be always true. The
axioms themselves need not be proved but they should be sufficiently self-evident to be
readily acceptable and certainly without known contradictions. The application of
thermodynamics to the prediction of changes in given properties of matter in relation to
energy transfers across its boundaries is based on only two fundamental axioms, the First
and Second Laws of thermodynamics, although the total field of thermodynamics requires
two other axioms. What is called the Zeroth Law considers three bodies in thermal contact,
transferring heat between themselves, yet insulated from their external surroundings. If two
of these have no net heat flow between them, a condition defined as thermal equilibrium,
then thermal equilibrium exists also between each of these and the third body. This is
necessary axiom for the development of the concept of temperature, but if one begins with
temperature as an already established property of matter, as we will do, the Zeroth Law is
not needed. The Third Law states that the limit of the entropy of a substance is zero as its
temperature approaches zero, a concept necessary in making absolute entropy calculations
and in establishing the relationship between entropy as obtained from the statistical
behavior of a multi-particle system, and the entropy of classical thermodynamics. Because in
this work we are concerned only with predicting changes in thermodynamic properties,
including the entropy, the Third Law also will not be needed or discussed.
10 The Intuitive Perception of the First Law
The First Law of thermodynamics is simply the law of conservation of energy and
mass. The ready acceptability of this law is apparent from the fact that the concept of
conservation in some form has existed from antiquity, long before any precise demonstration
of it could be made. The ancient biblical affirmation, "What so ever a man sows, that shall
he also reap" is, in a sense, a conservation law. The Greek philosophers generally considered
matter to be indestructible, although its forms--earth, fire, air, or water-- could be
interchanged. The situation was confused in the Middle Ages by a feeling that a
combustion process actually "destroyed" the matter which burned. This was not set right
until 1774 when Lavoisier conclusively demonstrated the conservation of mass in chemical
reactions.
It is fortunate that an intuitive feeling for energy conservation is also deep-rooted
because its demonstration is experimentally more difficult than that for mass conservation
and that which is conserved is more abstract. As discussed in section 4, that which is called
energy in classical thermodynamics is a quantity which measures a combination of effort
11 The Second Law as Common Experience
The Second Law is likewise a concept which is a part of basic human experience. In its
intuitive perception the Second Law is a sense of the uniqueness of the direction of the
change which results from the action of a particular thermodynamic driving force. For
example, no one has to be told that when the earth's gravitational potential is the driving
force it will cause water to flow from a tank on top of the hill to one at the bottom, but it
alone will never cause the reverse to occur. This direction of water flow is always the same
unless we supply some work, as for example with a pump, or unless we allow a change in
the properties of some region outside the two tanks, such as the water level in some other
reservoir. We identify the earth's gravitational attraction at a given water level as a driving
force because when the water levels are the same in each tank there is no further transfer of
water and also because the rate of transfer increases with an increase in the difference in
elevation.
An analogous example occurs when heat is driven from one system to another by a
difference in their temperatures. In our earliest experience temperature is the degree of
"hotness to the touch" which in this case is different for each system. We observe that when
this temperature difference is large the rate of change of their temperatures is greater than
when it is small and when the two have the same temperature we observe no further
changes. Consequently we identify temperature as a driving force which causes something
called heat to be transferred.
No theoretical knowledge of any kind is required for us to know that if we bring two
objects into close contact and exclude any interaction between them and their surroundings,
the cold one will always get hotter and the hot one cooler but never the opposite. This
direction is always the same unless we do some work, as with a refrigerator, or allow some
energy transfer between the objects and their surroundings.
When expressed more generally to include all types of driving forces and their driven
quantities, this uniqueness of direction becomes the Second Law. This is not a concept in
any way contained within the First Law, but one involving a completely new requirement.
For example, in either the water flow or in the heat flow situations, a flow in the wrong
direction would not necessarily violate the conservation of energy or mass.

Basic Principles of Classical and Statistical Thermodynamics lec (1)

In the most general sense thermodynamics is the study of energy -- its transformations and
its relationship to the properties of matter. In its engineering applications thermodynamics
has two major objectives. One of these is to describe the properties of matter when it exists
in what is called an equilibrium state, a condition in which its properties show no tendency
to change. The other objective is to describe processes in which the properties of matter
undergo changes and to relate these changes to the energy transfers in the form of heat and
work which accompany them. These objectives are closely related and a text such as this,
which emphasizes primarily the description of equilibrium properties, must include as well a
discussion of the basic principles involved in accomplishing these two objectives.
Thermodynamics is unique among scientific disciplines in that no other branch of
science deals with subjects which are as commonplace or as familiar. Concepts such as
"heat", "work", "energy", and "properties" are all terms in everyone's basic vocabulary.
Thermodynamic laws which govern them originate from very ordinary experiences in our
daily lives. One might think that this familiarity would simplify the understanding and
application of thermodynamics. Unfortunately, quite the opposite is true. In order to
accomplish these objectives, one must almost entirely forget a life-long acquaintance with
the terms of thermodynamics and redefine them in a very scientific and analytical manner.
We will begin with a discussion of the various properties of matter with which we will be
concerned.
1 Thermodynamic and Non-Thermodynamic Properties
A property of matter is any characteristic which can distinguish a given quantity of
a matter from another. These distinguishing characteristics can be classified in several
different ways, but for the purposes of this text it is convenient to divide them into what
may be called thermodynamic and non-thermodynamic properties.
The non-thermodynamic properties describe characteristics of what are often called
the "ultimate particles" of matter. An ultimate particle from a thermodynamic view point
is the smallest subdivision of a quantity of matter which does not undergo any net internal
changes during a selected set of processes which alter properties of the entire quantity. The
ultimate particles with which we will be concerned are generally considered to be molecules
or atoms, or in some cases groups of atoms within a molecule. When the meaning is clear
we will some times delete the adjective "ultimate" and refer to them simply as "particles".
Because it has no internal changes an ultimate particle can always be regarded as a
rigid mass. Its only alterable distinguishing characteristics which could possibly be
detected, if some experimental procedure could do so, are its position and its motion. As a
result, the fundamental properties of this particle, which cannot be calculated or derived
from any others, consist only of its mass and shape plus the vectors or coordinates needed to
describe its position and motion. It is convenient to combine the mass and motion
characteristics and represent them as a momentum property. These fundamental
characteristics, mass, position, and momentum, are called "microstate" properties and as a
group they give a complete description of the actual behavior of an ultimate particle.
Everyone realizes of course, that molecules are not actually inert rigid masses. The
forces of attraction and repulsion which we ascribe to them are in reality the consequence of
variations in the quantum states of a deformable electron cloud which fills practically all the
space occupied by a molecule so that when we represent it as a rigid mass we are
constructing a model which allows us to apply classical mechanics to relate its energy
changes to changes in its microstate properties. For example, an effective model for a
complex molecule is to regard it as a group of rigid spheres of various size and mass held
together by flexible springs. The only justification for this model is that calculations of its
energy, when properly averaged, give good agreement with values of energy per molecule
obtained from experimental measurements using bulk quantities of the substance.
Constructing models is important in all aspects of thermodynamics, not only for individual
molecules, but also in describing the behavior of bulk matter.
Values which can be calculated from the microstate properties of an individual
particle or of a cluster containing only a few particles represent another group of nonthermodynamic
properties. We will refer to these derived values as "molecular" properties.
Examples are the translational, vibrational, or rotational energies of an individual molecule,
and also the calculated potential energy at various separation distances in a pair of
molecules or between other small groups of near neighbors. In some cases we wish to
calculate special functions of the potential energy within a group composed of a few
neighbors. An important feature of all of these combinations of fundamental microstate
properties is that they can produce the same value of a calculated molecular property. For
example, assigning values to the microstate properties of a molecule determines its energy
but specifying the energy of a molecule does not specify any one particular set of values for
its microstate properties.
Whereas the non-thermodynamic properties pertain to a single or to only a few
ultimate particles, the characteristics of matter which are called thermodynamic properties
are those which result from the collective behavior of a very large number of its ultimate
particles. Instead of only one or a few particles, this number is typically on the order of
Avogadro's number. In a manner analogous to the way in which molecular properties can
be calculated from the fundamental microstate properties of an individual or small group of
particles, the various thermodynamic properties likewise depend upon the vastly greater
number of all the microstate properties of the very large group. Furthermore, an even larger
number of different sets of microstate properties can produce the same overall
thermodynamic property value. In contrast to non-thermodynamic properties,
thermodynamic properties can always be measured experimentally or calculated from such
measurements.
Establishing relationships between non-thermodynamic and thermodynamic
properties of matter in equilibrium states is the task of statistical thermodynamics while the
study of relationships among the thermodynamic properties alone is generally the topic of
classical thermodynamics. In the past it has been customary for textbooks and their readers
to make a sharp distinction between the two disciplines. The historical development of
classical thermodynamics and its applications to a wide range of engineering problems took
place without any reference at all to ultimate particles or molecular properties. This
development is entirely rigorous and has the merit of establishing the validity of general
thermodynamic principles to all types of matter regardless of its molecular character.
However, the problem of predicting and correlating thermodynamic properties of an
increasing diversity of substances both in pure form and in mixtures with the accuracy
needed in modern technology requires a combination of the classical and molecular
viewpoints. It is this combination which is the objective of this text.
2 The Selection of a System
The first concept which must be understood in applying thermodynamics is the
necessity to begin with the definition of what is called a "system". In thermodynamics this
is any region completely enclosed within a well defined boundary. Everything outside the
system is then defined as the surroundings. Although it is possible to speak of the subject
matter of thermodynamics in a general sense, the establishment of analytical relationships
among heat, work, and thermodynamic properties requires that they be related to a
particular system. We must always distinguish clearly between energy changes taking place
within a system and energy transferred across the system boundary. We must likewise
distinguish between properties of material within a system and properties of its
surroundings.
In accordance with their definition, thermodynamic properties apply to systems
which must contain a very large number of ultimate particles. Other than this there are no
fundamental restrictions on the definition of a system. The boundary may be either rigid or
movable. It can be completely impermeable or it can allow energy or mass to be transported
through it. In any given situation a system may be defined in several ways; although with
some definitions the computations to be performed are quite simple, with others they are
difficult or even impossible.
For example, it is often impossible by means of thermodynamic methods alone to
make heat transfer calculations if a system is defined so that both heat transfer and
diffusional mass transfer occur simultaneously through the same area on the boundary of
the system. For processes in which mass transfer takes place only by bulk stream flow this
problem can be avoided easily by a proper definition of the system. In a flow process of this
type the system is defined so that it is enclosed by moveable boundaries with no stream flows
across them. Heat transfer then always occurs across a boundary not crossed by mass.
3 Microstates and Thermodynamic States
The state of a system is an important concept in thermodynamics and is defined as
the complete set of all its properties which can change during various specified processes.
The properties which comprise this set depend on the kinds of interactions which can take
place both within the system and between the system and its surroundings. Any two
systems, subject to the same group of processes, which have the same values of all properties
in this set are then indistinguishable and we describe them as being in identical states.
A process in thermodynamics is defined as a method of operation in which specific
quantities of heat and various types of work are transferred to or from the system to alter its
state. As we pointed out, one of the objectives of thermodynamics is to relate these state
changes in a system to the quantity of energy in the form of heat and work transferred
across its boundaries.
In discussing non-thermodynamic processes, a system may be chosen as a single
ultimate particle within a larger quantity of matter. In the absence of chemical reactions the
only processes in which it can participate are transfers of kinetic or potential energy to or
from the particle. In this case we would like to relate these energy transfers to changes in
the microstate of the system. A microstate for this one-particle system is a set of coordinates
in a multi-dimensional space indicating its position and its momenta in various vector
directions. For example, a simple rigid spherical monatomic molecule would require a total
of six such coordinates, three for its position and three for its momentum in order to
completely define its microstate.
Now consider a system containing a large number of these ultimate particles. A
microstate of this system is a set of all position and momentum values for all the particles.
For example, if there were N rigid spherical molecules we would then need 6N coordinates
to give a complete set of all the microstate properties and define a microstate for this system.
In a multiparticle system a particular microstate exists only for an instant and is then
replaced by another so that there is no experimental way to measure the set of positions and
motions which comprise one microstate among the vast number of them which occur
sequentially.
Because the microstates of a multiparticle system represent exactly what all the
particles are doing, all thermodynamic properties of the group are thus determined by them.
With this common origin all the thermodynamic properties are therefore related to each
other and we need to develop this relationship. The set of all the thermodynamic properties
of a multiparticle system its temperature, pressure, volume, internal energy, etc., is defined
as the thermodynamic state of this system.
An important aspect of this relationship between thermodynamic properties is the
question of how many different thermodynamic properties of a given equilibrium system are
independently variable. The number of these represents the smallest number of properties
which must be specified in order to completely determine the entire thermodynamic state of
the system. All other thermodynamic properties of this system are then fixed and can be
calculated from these specified values. The number of these values which must be specified
is called the variance or the degrees of freedom of the system.
4 The Concept of Energy
In elementary physics energy is often defined as "the capacity to produce work". At
a descriptive level the idea expressed is correct, but for thermodynamics which is to be
applied quantitatively this definition is not a good one because the term "work" itself
requires a more precise definition than the general idea it ordinarily conveys. A better
definition of energy from the viewpoint of thermodynamics would be "the capacity to induce
a change in that which inherently resists change". This capacity represents a combination
of an effort, expended in overcoming resistance to a particular type of change, with the
change it produces. The combination is called energy.
The effort involved is measured quantitatively by what is defined as a "driving
force" in thermodynamics. A driving force is a property which both causes and also
controls the direction of change in another property. The quantitative value of this change
is called a "displacement". The product of a driving force and its associated displacement
always represents a quantity of energy, but in thermodynamics this quantity has meaning
only in relation to a specifically defined system.
Relative to a particular system there are generally two ways of locating a driving
force and the displacement it produces. In one way both the driving force and the
displacement are properties of the system and are located entirely within it, so that the
energy calculated from their product represents a change in the internal energy of the
system. Similarly, both the driving force and its displacement could be located entirely
within the surroundings so that the calculated energy is then a change in the total energy of
the surroundings.
In another way, however, the displacement occurs within the system but the driving
force producing it is a property of the surroundings and is applied externally at the system
boundary. By definition, the boundary of a system is a region of zero thickness containing
no matter at all so that the energy calculated in this way is not a property of matter either in
the system or in its surroundings but represents a quantity of energy in transition between
the two. In any quantitative application of thermodynamics it is always important to make
a careful distinction between energy changes within a system or within its surroundings
and energy in transition between them.
5 Microstate Driving Forces
In order to explain the nature of driving forces, suppose we consider first a system
defined as a single ultimate particle of a simple fluid, either a gas or a liquid. The system in
this case is a rigid spherical mass with no possibilities for any internal changes and obeying
Newtonian mechanics. In its surroundings are similar ultimate particles of this fluid.
From a Newtonian point of view the mass of this system resists any change in its condition
of motion and a specific change occurs only with the application of an external force to
overcome the inertial resistance inherent in the mass. In the presence of mutual attraction
and repulsion between this system and neighboring particles it may be considered to resist
any displacement from a position in which this attraction and repulsion are balanced. In
this situation a force vector directed toward the center of mass must be applied for a fixed
time period to produce a change. This force is produced by the environment around the
particle chosen as the system. The mechanism for its generation is by the action of
neighboring particles in exerting attraction or repulsion or in colliding with the system.
The scalar product of the vector force generated in this manner with other vectors which
represent the resulting displacements in position and velocity of the system determine the
energy added to the system when its velocity is increased, when its position is moved away
from attracting neighbors, or when moved toward neighbors which repel it.
Since these displacements represent changes in microstate properties, we define the
force vector producing them as a "microstate driving force." According to Newtonian
mechanics this applied force is always opposed by an equal and opposite force representing
the resistance of the system to change. Although mechanically we could position these two
forces anywhere along their line of action, in terms of the system it is convenient to think of
them as opposing one another at the boundary of the system to describe energy in transition
across it and then as opposing one another within the system when we describe this
quantity of energy as the energy change of the system. An important characteristic of
microstate driving forces is that they are true force vectors in the Newtonian sense and there
is never a condition of unbalanced driving forces. This is not at all the case for what we will
define as "thermodynamic driving forces" which are the agents of change for
thermodynamic properties in multiparticle systems.
6 Thermodynamic Driving Forces
In contrast to the one-particle system which we have discussed in section 5, for
thermodynamic systems consisting of many particles we are usually as interested in
internal energy changes as we are in changes in position or motion of the entire system. In
this case we wish to define these internal energy changes in terms of thermodynamic
properties, each of which are the collective results of the enormous number of microstates for
all the ultimate particles of the system. Because the fundamental agents of change within
the system are microstate driving forces, the corresponding agents of change or driving
forces in thermodynamic systems are the composite result of all the microstate driving force
vectors in the system. However, the only case in which the collective behavior of all these
microstate driving force vectors defines a thermodynamic property is the one in which these
microstate vectors for all the individual particles are oriented in a completely random
manner in every conceivable direction. In this case their overall resultant in the entire
system is completely scalar in nature and a thermodynamic property of the system. We
define this resultant as a "thermodynamic driving force."
Likewise, the cumulative effect of all the microstate changes induced, which are also
vectors, produces in this case a completely scalar thermodynamic property change for the
multiparticle system. This overall change is the displacement induced by the
thermodynamic driving force.
Because these thermodynamic driving forces are not true vector forces in the
Newtonian sense but are scalar properties, the thermodynamic driving forces tending to
cause a change are not always balanced by equal and opposite driving forces opposing the
change. Changes in internal thermodynamic properties within a system can be controlled
as to direction, and in some instances as to their rates, by the degree of difference between
the value of a particular thermodynamic driving force property outside the system at its
boundary and a value of this same property somewhere within the system. Between
thermodynamic driving forces this difference can be of any magnitude, finite or
infinitesimal. When they are exactly equal there is then no net change induced and no
energy is transferred

the state diagram for steam

boiling points super heated state vapor state

Lectures on Heat (3)

Thermal Expansion and the Gas Law

Coefficients of Expansion
Almost all materials expand on heating—the most famous exception being water, which contracts as it is warmed from 0 degrees Celsius to 4 degrees. This is actually a good thing, because as freezing weather sets in, the coldest water, which is about to freeze, is less dense than slightly warmer water, so rises to the top of a lake and the ice begins to form there. For almost all other liquids, solidification on cooling begins at the bottom of the container. So, since water behaves in this weird way, ice skating is possible! Also, as a matter of fact, life in lakes is possible—the ice layer that forms insulates the rest of the lake water from very cold air, so fish can make it through the winter.
Linear Expansion
The coefficient of linear expansion α of a given material, for example a bar of copper, at a given temperature is defined as the fractional increase in length that takes place on heating through one degree:

Of course, α might vary with temperature (it does for water, as we just mentioned) but in fact for most materials it stays close to constant over wide temperature ranges.

Volume Expansion
For liquids and gases, the natural measure of expansion is the coefficient of volume expansion,β.

Of course, on heating a bar of copper, clearly the volume as well as the length increases—the bar expands by an equal fraction in all directions (this could be experimentally verified, or you could just imagine a cube of copper, in which case all directions look the same).
The volume of a cube of copper of side L is V = L3. Suppose we heat it through one degree. Putting together the definitions of ,αβabove,

Gas Pressure Increase with Temperature

In 1702, Amontons discovered a linear increase of P with T for air, and found P to increase about 33% from the freezing point of water to the boiling point of water.
That is to say, he discovered that if a container of air were to be sealed at 0°C, at ordinary atmospheric pressure of 15 pounds per square inch, and then heated to 100°C but kept at the same volume, the air would now exert a pressure of about 20 pounds per square inch on the sides of the container. (Of course, strictly speaking, the container will also have increased in size, that would lower the effect—but it’s a tiny correction, about ½% for copper, even less for steel and glass.)
Remarkably, Amontons discovered, if the gas were initially at a pressure of thirty pounds per square inch at 0°C, on heating to 100°C the pressure would go to about 40 pounds per square inch—so the percentage increase in pressure was the same for any initial pressure: on heating through 100°C, the pressure would always increase by about 33%.
Furthermore, the result turned out to be the same for different gases!

Finding a Natural Temperature Scale
In class, we plotted air pressure as a function of temperature for a fixed volume of air, by making several measurements as the air was slowly heated (to give it a chance to all be at the same temperature at each stage). We found a straight line. On the graph, we extended the line backwards, to see how the pressure would presumably drop on cooling the air. We found the remarkable prediction that the pressure should drop to zero at a temperature of about −273°C.
In fact, if we’d done the cooling experiment, we would have found that air doesn’t actually follow the line all the way down, but condenses to a liquid at around −200°C. However, helium gas stays a gas almost to −270°C, and follows the line closely.

We shall discuss the physics of gases, and the interpretation of this, much more fully in a couple of lectures. For now, the important point is that this suggests a much more natural temperature scale than the Celsius one: we should take −273°C as the zero of temperature! For one thing, if we do that, the pressure/temperature relationship for a gas becomes beautifully simple:
P  ∝  T
This temperature scale, in which the degrees have the same size as in Celsius, is called the Kelvin or absolute scale. Temperatures are written 300K. To get from Celsius to Kelvin, just add 273 (strictly speaking, 273.15).
An Ideal Gas
Physicists at this point introduce the concept of an “Ideal Gas”. This is like the idea of a frictionless surface: it doesn’t exist in nature, but it is a very handy approximation to some real systems, and makes problems much easier to handle mathematically. The ideal gas is one for which for all temperatures, so helium is close to ideal over a very wide range, and air is close to ideal at ordinary atmospheric temperatures and above.

Lectures on Heat (2)

Thermal Equilibrium and the Zeroth Law of Thermodynamics
Once the thermometer came to be widely used, more precise observations of temperature and (as we shall see) heat flow became possible. Joseph Black, a professor at the University of Edinburgh in the 1700’s, noticed that a collection of objects at different temperatures, if brought together, will all eventually reach the same temperature.
As he wrote, “By the use of these instruments [thermometers] we have learned, that if we take 1000, or more, different kinds of matter, such as metals, stones, salts, woods, cork, feathers, wool, water and a variety of other fluids, although they be all at first of different heats, let them be placed together in a room without a fire, and into which the sun does not shine, the heat will be communicated from the hotter of these bodies to the colder, during some hours, perhaps, or the course of a day, at the end of which time, if we apply a thermometer to all of them in succession, it will point to precisely the same degree.”
We say nowadays that bodies in “thermal contact” eventually come into “thermal equilibrium”—which means they finally attain the same temperature, after which no further heat flow takes place. This is equivalent to:
The Zeroth Law of Thermodynamics:
If two objects are in thermal equilibrium with a third, then they are in thermal equilibrium with each other.
The “third body” in a practical situation is just the thermometer.
It’s perhaps worth pointing out that this trivial sounding statement certainly wasn’t obvious before the invention of the thermometer. With only the sense of touch to go on, few people would agree that a piece of wool and a bar of metal, both at 0°C, were at the same temperature.
Measuring Heat Flow: a Unit of Heat
The next obvious question is, can we get more quantitative about this “flow of heat” that takes place between bodies as they move towards thermal equilibrium? For example, suppose I reproduce one of Fahrenheit’s experiments, by taking 100 ccs of water at 100°F, and 100ccs at 150°F, and mix them together in an insulated jug so little heat escapes. What is the final temperature of the mix?
Of course, it’s close to 125°F—not surprising, but it does tell us something! It tells us that the amount of heat required to raise the temperature of 100 cc of water from 100°F to 125°F is exactly the same as the amount needed to raise it from 125°F to 150°F. A series of such experiments (done by Fahrenheit, Black and others) established that it always took the same amount of heat to raise the temperature of 1 cc of water by one degree.
This makes it possible to define a unit of heat. Perhaps unfairly to Fahrenheit,
1 calorie is the heat required to raise the temperature of 1 gram of water by 1 degree Celsius.
(Celsius also lived in the early 1700’s. His scale has the freezing point of water as 0°C, the boiling point as 100°C. Fahrenheit’s scale is no longer used in science, but lives on in engineering in the US, and in the British Thermal Unit, which is the heat required to raise the temperature of one pound of water by 1°F.)
Specific Heats and Calorimetry
First, let’s define specific heat:
The specific heat of a substance is the heat required in calories to raise the temperature of 1 gram by 1 degree Celsius.
As Fahrenheit continues his measurements of heat flow, it quickly became evident that for different materials, the amount of heat needed to raise the temperature of one gram by one degree could be quite different. For example, it had been widely thought before the measurements were made, that one cc of Mercury, being a lot heavier than one cc of water, would take more heat to raise its temperature by one degree. This proved not to be the case—Fahrenheit himself made the measurement. In an insulating container, called a “calorimeter” he added 100ccs of water at 100°F to 100ccs of mercury at 150°F, and stirred so they quickly reached thermal equilibrium.
Question: what do you think the final temperature was? Approximately?
Answer: The final temperature was, surprisingly, about 120°F. 100 cc of water evidently “contained more heat” than 100 cc of mercury, despite the large difference in weight!
This technique, called calorimetry, was widely used to find the specific heats of many different substances, and at first no clear pattern emerged. It was puzzling that the specific heat of mercury was so low compared with water! As more experiments on different substances were done, it gradually became evident that heavier substances, paradoxically, had lower specific heats.
A Connection With Atomic Theory
Meanwhile, this quantitative approach to scientific observation had spread to chemistry. Towards the end of the 1700’s, Lavoisier weighed chemicals involved in reactions before and after the reaction. This involved weighing the gases involved, so had to be carried out in closed containers, so that, for example, the weight of oxygen used and the carbon dioxide, etc., produced would accounted for in studying combustion. The big discovery was that mass was neither created nor destroyed. This had not been realized before because no one had weighed the gases involved. It made the atomic theory suddenly more plausible, with the idea that maybe chemical reactions were just rearrangements of atoms into different combinations.
Lavoisier also clarified the concept of an element, an idea that was taken up in about 1800 by John Dalton, who argues that a given compound consisted of identical molecules, made up of elementary atoms in the same proportion, such as H2O (although that was thought initially to be HO). This explained why, when substances reacted chemically, such as the burning of hydrogen to form water, it took exactly eight grams of oxygen for each gram of hydrogen. (Well, you could also produce H2O2 under the right conditions, with exactly sixteen grams of oxygen to one of hydrogen, but the simple ratios of amounts of oxygen needed for the two reactions were simply explained by different molecular structures, and made the atomic hypothesis even more plausible.)
Much effort was expended carefully weighing the constituents in many chemical reactions, and constructing diagrams of the molecules. The important result of all this work was that it became possible to list the relative weights of the atoms involved. For example, the data on H2O and H2O2 led to the conclusion that an oxygen atom weighed sixteen times the weight of a hydrogen atom.
It must be emphasized, though, that these results gave no clue as to the actual weights of atoms! All that was known was that atoms were too small to see in the best microscopes. Nevertheless, knowing the relative weights of some atoms in 1820 led to an important discovery. Two professors in France, Dulong and Petit, found that for a whole series of elements the product of atomic weight and specific heat was the same!

The significance of this, as they pointed out, was that the “specific heat”, or heat capacity, of each atom was the same—a piece of lead and a piece of zinc having the same number of atoms would have the same heat capacity. So heavier atoms absorbed no more heat than lighter atoms for a given rise in temperature. This partially explained why mercury had such a surprisingly low heat capacity. Of course, having no idea how big the atoms might be, they could go no further. And, indeed, many of their colleagues didn’t believe in atoms anyway, so it was hard to convince them of the significance of this discovery.
Latent Heat
One of Black’s experiments was to set a pan of water on a steady fire and observe the temperature
as a function of time. He found it steadily increased, reflecting the supply of heat from the fire, until the water began to boil, whereupon the temperature stayed the same for a long time. The steam coming off was at the same (boiling) temperature as the water. So what was happening to the heat being supplied? Black correctly concluded that heat needed to be supplied to change water from its liquid state to its gaseous state, that is, to steam. In fact, a lot of heat had to be supplied: 540 calories per gram, as opposed to the mere 100 calories per gram needed to bring it from the freezing temperature to boiling. He also discovered that it took 80 calories per gram to melt ice into water, with no rise in temperature. This heat is released when the water freezes back to ice, so it is somehow “hidden” in the water. He called it latent heat, meaning hidden heat.

Lectures on Heat (1)

Heat

Feeling and seeing temperature changes
Within some reasonable temperature range, we can get a rough idea how warm something is by touching it. But this can be unreliable—if you put one hand in cold water, one in hot, then plunge both of them into lukewarm water, one hand will tell you it’s hot, the other will feel cold. For something too hot to touch, we can often get an impression of how hot it is by approaching and sensing the radiant heat. If the temperature increases enough, it begins to glow and we can see it’s hot!
The problem with these subjective perceptions of heat is that they may not be the same for everybody. If our two hands can’t agree on whether water is warm or cold, how likely is it that a group of people can set a uniform standard? We need to construct a device of some kind that responds to temperature in a simple, measurable way—we need a thermometer.
The first step on the road to a thermometer was taken by one Philo of Byzantium, an engineer, in the second century BC. He took a hollow lead sphere connected with a tight seal to one end of a pipe, the other end of the pipe being under water in another vessel.

 

To quote Philo: “…if you expose the sphere to the sun, part of the air enclosed in the tube will pass out when the sphere becomes hot. This will be evident because the air will descend from the tube into the water, agitating it and producing a succession of bubbles.
Now if the sphere is put back in the shade, that is, where the sun’s rays do not reach it, the water will rise and pass through the tube …”
“No matter how many times you repeat the operation, the same thing will happen.
In fact, if you heat the sphere with fire, or even if you pour hot water over it, the result will be the same.”
Notice that Philo did what a real investigative scientist should do—he checked that the experiment was reproducible, and he established that the air’s expansion was in response to heat being applied to the sphere, and was independent of the source of the heat.

Classic Dramatic Uses of Temperature-Dependent Effects

This expansion of air on heating became widely known in classical times, and was used in various dramatic devices. For example, Hero of Alexandria describes a small temple where a fire on the altar causes the doors to open

The altar is a large airtight box, with a pipe leading from it to another enclosed container filled with water. When the fire is set on top of the altar, the air in the box heats up and expands into a second container which is filled with water. This water is forced out through an overflow pipe into a bucket hung on a rope attached to the door hinges in such a way that as the bucket fills with water, it drops, turns the hinges, and opens the doors. The pipe into this bucket reaches almost to the bottom, so that when the altar fire goes out, the water is sucked back and the doors close again. (Presumably, once the fire is burning, the god behind the doors is ready to do business and the doors open…)
Still, none of these ingenious devices is a thermometer. There was no attempt (at least none recorded) by Philo or his followers to make a quantitative measurement of how hot or cold the sphere was. And the “meter” in thermometer means measurement
The First Thermometer
Galileo claimed to have invented the first thermometer. Well, actually, he called it a thermoscope, but he did try to measure “degrees of heat and cold” according to a colleague, and that qualifies it as a thermometer. (Technically, a thermoscope is a device making it possible to see a temperature change, a thermometer can measure the temperature change.) Galileo used an inverted narrow-necked bulb with a tubular neck, like a hen’s egg with a long glass tube attached at the tip.

He first heated the bulb with his hands then immediately put it into water. He recorded that the water rose in the bulb the height of “one palm”. Later, either Galileo or his colleague Santorio Santorio put a paper scale next to the tube to read off changes in the water level. This definitely made it a thermometer, but who thought of it first isn’t clear (they argued about it). And, in fact, this thermometer had problems.
Question: what problems? If you occasionally top up the water, why shouldn’t this thermometer be good for recording daily changes in temperature?
Answer: because it’s also a barometer! But—Galileo didn’t know about the atmospheric pressure.
Torricelli, one of Galileo’s pupils, was the first to realize, shortly after Galileo died, that the real driving force in suction was external atmospheric pressure, a satisfying mechanical explanation in contrast to the philosophical “nature abhors a vacuum”. In the 1640’s, Pascal pointed out that the variability of atmospheric pressure rendered the air thermometer untrustworthy.
Liquid-in-glass thermometers were used from the 1630’s, and they were of course insensitive to barometric pressure. Meteorological records were kept from this time, but there was no real uniformity of temperature measurement until Fahrenheit, almost a hundred years later.

Newton’s Anonymous Table of Temperatures
The first systematic account of a range of different temperatures, “Degrees of Heat”, was written by Newton, but published anonymously, in 1701. Presumably he felt that this project lacked the timeless significance of some of his other achievements.
Taking the freezing point of water as zero, Newton found the temperature of boiling water to be almost three times that of the human body, melting lead eight times as great (actually 327C, whereas 8x37=296, so this is pretty good!) but for higher temperatures, such as that of a wood fire, he underestimated considerably. He used a linseed oil liquid in glass thermometer up to the melting point of tin (232°C). (Linseed oil doesn’t boil until 343°C, but that is also its autoignition temperature!)
Newton tried to estimate the higher temperatures indirectly. He heated up a piece of iron in a fire, then let it cool in a steady breeze. He found that, at least at the lower temperatures where he could cross check with his thermometer, the temperature dropped in a geometric progression, that is, if it took five minutes to drop from 80° above air temperature to 40° above air temperature, it took another five minutes to drop to 20° above air, another five to drop to 10° above, and so on. He then assumed this same pattern of temperature drop was true at the high temperatures beyond the reach of his thermometer, and so estimated the temperature of the fire and of iron glowing red hot. This wasn’t very accurate—he (under)estimated the temperature of the fire to be about 600°C.
Fahrenheit’s Excellent Thermometer
The first really good thermometer, using mercury expanding from a bulb into a capillary tube, was made by Fahrenheit in the early 1720’s. He got the idea of using mercury from a colleague’s comment that one should correct a barometer reading to allow for the variation of the density of mercury with temperature. The point that has to be borne in mind in constructing thermometers, and defining temperature scales, is that not all liquids expand at uniform rates on heating—water, for example, at first contracts on heating from its freezing point, then begins to expand at around forty degrees Fahrenheit, so a water thermometer wouldn’t be very helpful on a cold day. It is also not easy to manufacture a uniform cross section capillary tube, but Fahrenheit managed to do it, and demonstrated his success by showing his thermometers agreed with each other over a whole range of temperatures. Fortunately, it turns out that mercury is well behaved in that the temperature scale defined by taking its expansion to be uniform coincides very closely with the true temperature scale, as we shall see later.
Amontons’ Air Thermometer: Pressure Increases Linearly with Temperature
A little earlier (1702) Amontons introduced an air pressure thermometer. He established that if air at atmospheric pressure (he states 30 inches of mercury) at the freezing point of water is enclosed then heated to the boiling point of water, but meanwhile kept at constant volume by increasing the pressure on it, the pressure goes up by about 10 inches of mercury. He also discovered that if he compressed the air in the first place, so that it was at a pressure of sixty inches of mercury at the temperature of melting ice, then if he raised its temperature to that of boiling water, at the same time adding mercury to the column to keep the volume of air constant, the pressure increased by 20 inches of mercury. In other words, he found that for a fixed amount of air kept in a container at constant volume, the pressure increased with temperature by about 33% from freezing to boiling, that percentage being independent of the initial pressure.