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.
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.