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Chapter 3 Applying Particle Models to Matter 53
Applying Particle Models
to Matter
As previously noted, we are pursuing two goals in Part 1 of this course. On the
one hand, we want to get a solid understanding of energy and how we can use this
understanding to get answers and make predictions about interesting phenomena.
In Chapter 2 we got through the basics–introducing work and potential energy–
and applied these concepts to mostly macroscopic phenomena. Now, in Chapter 3,
we turn to the development of particle models of matter. We would like to be able
to answer questions such as: Why do things melt and/or vaporize at different
temperatures? What determines heat capacities of different substances? What
aspects of these thermal properties are common to many substances and which are
unique to particular substances? What common things can we say about all kinds
of chemical bonding? Some of the most important ideas in our particle models of
matter are related to the behavior of the spring-mass motion introduced in
Chapter 2. We extend these ideas to understand the motion of atoms using a model
that has at its core the idea that atoms and molecules in liquids and solids act like
they oscillate exactly the way the spring-mass system oscillates. The relation
between force and potential energy allows us to really make sense of the forces
that act between atoms and molecules in terms of their equilibrium spacing and to
understand the differences between solids, liquids, and gases in a much more
fundamental way.
Looking back at Learning Expectations for you, the learner
Before going on, it might be helpful to think again about our expectations for you
as a learner. After you have carried out the various activities in discussion/lab,
had some intense discussions with your classmates, made several attempts at
reading this chapter, and worked the Chapter 3 FNT (for next time) homework
assignments, you should have a pretty good understanding of the ideas presented
in this chapter. You won't understand many of the ideas presented in this chapter
the first time you encounter them. This is normal–don’t give up. Your
understanding will increase as you work with the ideas, talk to others about them,
carry on a conversation with yourself about them, and apply them to new
problems and questions. These Chapters in the course packet complement other
course activities, and will be most useful if you refer to them at several stages
during your work with this material. Things that make no sense at first will
become clearer as you make more connections with other areas of knowledge,
such as ideas from chemistry you are familiar with, and as you reorganize some of
your thoughts in a little more consistent and logical way.
This is perhaps a good time to go back and reread several of the earlier parts of
this course packet. You should definitely reread again, perhaps with a slightly
different perspective now, the Preface on Meaningful Learning. Also, the
Appendix M1: Science and the Place of Models in Science should begin to
really make sense to you as we work our way through the three models in this
chapter.
3 Contents
3-1 Where we are
headed in this
chapter
3-2 Phenomena,
Data Patterns,
Questions
3-3 Intro Particle
Model of Matter
3-4 Particle Model
of Bond Energy
3-5 Particle Model
of Thermal
Energy
3-6 Looking Back
and Ahead
54 Chapter 3 Applying Particle Models to Matter
3-1 Where We Are Headed in this Chapter
A famous Nobel laureate in physics, Richard P. Feynman, in the first chapter of
an introductory physics book he wrote for Cal Tech students back in the late 50’s,
claims that the particle model of matter is the most important or powerful model
in science. Here is what he said:
“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only
one sentence passed on to the next generations of creatures, what statement
would contain the most information in the fewest words? I believe it is the
atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all
things are made of atoms—little particles that move around in perpetual
motion, attracting each other when they are a little distance apart, but repelling
upon being squeezed into one another. In that one sentence, you will see, there
is an enormous amount of information about the world, if just a little imagination
and thinking are applied.”
The Feynman Lectures on Physics, Volume I, page 1-2
Your job over the next couple of weeks is to use “a little imagination” and apply a
“little thinking” to the content of this powerful statement.
The heart of the content in Chapter 3 is the development of a full understanding of
the details contained in the Feynman quote. You already have a lot of useful ideas
about this model. Much of it you have studied in chemistry. Keep consciously
trying to integrate this new material with things you already know, like the ideal
gas model (PV=nRT). It will take mental effort, but the understanding you gain
will help your see chemical and biological concepts in a new light.
One of the areas in which our particle model of matter really shines is in
explaining the experimentally observed thermal properties of matter, e.g., the
values and trends of the specific heats of many substances in the gas, liquid, and
solid phases. One of the interesting things about science is that it is in trying to
resolve discrepancies that we push ahead and make breakthroughs. One of the
discrepancies we will meet as we look at specific heats is that values for gases as
well as solids are often lower than we would predict, especially at lower
temperatures, but tend to rise to the predicted values as the temperature rises. The
changes we need to incorporate in our model are due to the quantum mechanical
nature of matter on a microscopic scale. We introduce some quantum ideas here
and will continue to return to them throughout the course. Of course, you already
know a lot about some of the central notions of quantum mechanics from your
study of chemistry. For example, you have encountered the notion of orbital, or
quantized energy levels, for the electrons swirling about the nuclei of atoms.
When you get to the discussion in this chapter on how quantum mechanics alters
things, you should definitely connect it to what you already know about
quantization.
Through the activities in discussion/lab, those using a sophisticated computer
simulation and those involving liquid nitrogen, springs and masses, etc., and
through the activities you carry out at home, the particle model of matter
described in Chapter 3 will become part of your mental tool kit that you regularly
use as you answer scientific questions in the future.
Chapter 3 Applying Particle Models to Matter 55
3-2 Phenomena, Data Patterns, and Kinds of
Questions and Explanations
Phenomena
The particulate nature of matter provides a model that allows explanations of a
large range of phenomena that simply cannot be explained without invoking this
fundamental idea regarding what matter is. In this chapter much of the focus will
be simply developing a basic particle model, Intro Particle Model of Matter,
sufficiently far so that, with a Particle Model of Bond Energy and a Particle
Model of Thermal Energy it will be possible to develop explanations for many of
the empirically determined thermal properties of matter encountered in Chapter 1.
Specifically, how do we make sense of the range of thermal and bond energies we
encountered in Chapter 1?
Data Patterns
In addition to the sampling of heat capacity data and heats of fusion and
vaporization presented in Chapter 1, we would expect our models to provide us
with the capability of explaining the heat capacity values, both at constant
pressure and at constant volume for a large range of substances. Several of these
data patterns are presented on this and the next page.
This first graph shows the constant volume molar heat
capacity of several gases from room temperature up to
several thousand kelvin. The values of the heat
capacities have been divided by the gas constant, R.
There are several obvious trends. The monatomic
gases have the lowest molar constant-volume heat
capacity at 3/2 R and the values are independent of
temperature. Diatomic gases seem to have higher
values starting at about 5/2 R, while polyatomic gases
have significantly larger values, but also a much more
pronounced temperature dependence. These are some
of the trends our models should enable us to provide
explanations for.
9.5
8.5
7.5
6.5
5.5
4.5
Monatomic gases (He, Ar, Ne, etc.)
H2
N2
Cl2
CO2
CH4
NH3
300 500 1000 2000
T [Kelvin]
Cvm
R
1500
3.5
2.5
1.5 3
2
5
2
7
2
56 Chapter 3 Applying Particle Models to Matter
Molar values of both Cp and Cv for Monatomic, Diatomic and Triatomic Gases at Room Temperature
gas Cmp Cmv Cmv/R Cmp - Cmv (Cmp - Cmv)/R
Monatomic He 20.79 12.52 1.51 8.27 .99 Ne 20.79 12.68 1.52 8.11 .98 Ar 20.79 12.45 1.50 8.34 1.00 Li 20.79 12.45 1.50 8.34 1.00 Xe 20.79 12.52 1.51 8.27 .99
Diatomic N2 29.12 20.80 2.50 8.32 1.00 H2 28.82 20.44 2.46 8.38 1.01 O2 29.37 20.98 2.52 8.39 1.01 CO 29.04 20.74 2.49 8.30 1.00
Triatomic CO2 36.62 28.17 3.39 8.45 1.02 N2O 36.90 28.39 3.41 8.51 1.02 H2S 36.12 27.36 3.29 8.76 1.05 (This table is a version of a similar table in Physical Chemistry, Second Edition, by Joseph H. Noggle)
Several general trends are evident in the tabulated data. (Please refer to both tables of data–the table above and the table in Chapter 1.)
1 When the values of heats of melting and vaporization and specific heats of different
substances are compared, there is a wide variation in quantities measured per kilogram. However, when these same quantities are measured per mole, much of the variation is removed.
2 The molar specific heat of many solids is very similar. 3 For monatomic substances, the value of molar specific heat of liquids is similar to the values
for the solid phase. 4 The molar specific heats of polyatomic substances are greater in the liquid phase than the
solid phase. 5 The specific heats of gases measured at constant pressure are greater than at constant
volume. 6 Each type of gas, e.g., monatomic, has similar values of specific heats, and the values are
ordered from smaller to larger as we go from monatomic to diatomic to triatomic. 7 The difference between Cmp and Cmv is very similar for all gases. 8 The same kinds of regularities observed in the molar specific heats do not appear in the
melting and boiling points and in the heats of melting and vaporization. However, there are definite correlations between the melting and boiling points and the heats of melting and vaporization.
Chapter 3 Applying Particle Models to Matter 57
3-3 Intro Particle Model of Matter
(Summary on foldout #4 at back of course
packet)
In this introductory model of the particle nature of matter we focus primarily on
the force that acts between two atoms or molecules. We make extensive use of the
relation of force to the potential energy describing the interaction between two
atomic sized particles. Initially we assume the particles are electrically neutral,
but we will see how to take this into account a little later in the chapter. The next
two models will use the basic ideas established here to help us develop a much
deeper understanding of both bond energy and thermal energy.
The particle model of matter that we introduce is the familiar picture of matter as
composed of atoms and molecules. Our particle model for ordinary matter is
simple and universal. It is not restricted to a particular kind of matter, but
encompasses all ordinary matter. That is what makes this model so useful. Of
course, being very general, it can’t predict many of the details that depend on the
“particulars”, but it can predict many of the universal properties.
Construct Definitions
Particle
This label applies to microscopic constituents of matter, typically an atom or
molecule, but it could also refer, for example, to the constituents of the nucleus, if
that were the focus of interest.
Attractive and Repulsive Forces
Atomic sized particles exert forces on each other in the same way that large-scale
objects do. These forces can be attractive or repulsive, which one typically
depends on their separation.
Interaction between two particles
The basis for making sense out of how particles interact is to focus on the
interaction of only two particles at a time. There are several properties that keep
reoccurring in our description of this interaction.
Center-to-center separation
We consistently refer to the distance between particles as being the center-to-
center separation, rather than the distance between their surfaces. Usually we
will use the symbol r to indicate this separation distance.
Equilibrium position or equilibrium separation
When we are focusing on just two particles, we will find that there is often a
“special” separation, often referred to as the equilibrium separation. The
reason this separation is special is that at this equilibrium separation the
interparticle force is zero. What does this say about the slope of the PE at this
point? It has similarities to the spring-mass system in this respect. A mass
58 Chapter 3 Applying Particle Models to Matter
hanging on a spring hangs at a particular “separation” from the point at which
the spring is supported. This is a favored position. If the mass finds itself
closer to the point of support, the “spring force” pushes it away, back toward
the equilibrium position. Conversely, if it finds itself too far form the support,
the spring force pulls it back toward the equilibrium position. The exact same
thing happens with two atomic sized particles. We label the equilibrium
separation for two particles with the symbol ro.
Pair-wise Potential Energy
We will consistently refer to the potential energy between two atomic size
particles as the pair-wise potential energy, PEpair-wise. This potential energy
has a fairly generic shape to it that you need to become familiar with. In
general terms, it becomes very repulsive if the two particles begin to get too
close to each other. The potential has a minimum and becomes “horizontal”–
slope is zero–at the two particle’s equilibrium separation. As the particles
begin to separate, the potential at first “looks like” a spring-mass potential, but
then begins to flatten out and becomes perfectly flat (horizontal, so zero force
acting between the two particles here) once the separation is a few times that
of the equilibrium separation, ro. The parameter that describes how “deep” the
potential is, that is, the difference in energy between where the potential is flat
at large separations and at its lowest value where the equilibrium separation
occurs, is often called the “well depth” and designated with the lowercase
Greek letter . The well depth, , is the magnitude of this energy difference, so
is always a positive quantity.
Single Particle Potential Energy
In a solid or liquid, each particle has multiple pair-wise interactions, because it
has lots of neighbors to interact with. It will sometimes be useful to focus on just
one particle at a time, and to “add up” all the interactions it has with its neighbors
to obtain a potential energy function that describes the forces acting on just this
one particle from all of its neighbors. We call this the single-particle potential
energy to make it clear that it is not PEpair-wise.
Chapter 3 Applying Particle Models to Matter 59
Graphical Representation of the Pair-Wise Potential Energy
The PEpair-wise PE curve shown has the typical shape of almost all atomic-size
pair-wise potentials. It has a simple mathematical form, so is useful for that
reason alone. This particular shape describes rather well the interaction
between the atoms of the noble elements (He, Ne, etc) in all their phases. Note
that the equilibrium separation occurs at a slightly larger separation distance
than one particle diameter, designated by the lower-case Greek sigma, . If
these particles acted like billiard balls, there would be a little space between
them, even when they were as close as they “wanted” to get to each other.
It is customary, much to many beginners’ consternation, to define the zero of
PEpair-wise to be the value the PE has when the particles are separated by a great
difference. Also notice how steep the curve gets as the particles begin to get
closer than the equilibrium separation. Remember, a steep PE curve means a
strong force, which is repulsive in this case. We will frequently return to this
graph.
60 Chapter 3 Applying Particle Models to Matter
Meaning of the Model Relationships (Numbers below correspond to the numbered relationships on the fold-out Summary.)
1) All “normal” matter is comprised of tiny particles (atoms and molecules) that
move around in perpetual motion, attracting each other when they are a little
distance apart, but repelling upon being squeezed into one another.
This is a slightly paraphrased quote from Nobel Laureate Richard Feynman in
which he stated that if all scientific information were to be lost, these would
be the most valuable ideas to pass on to future generations.
2) The part about the particles attracting and repelling each other is most easily
visualized in terms of the slope of the pair-wise potential energy acting
between the two particles. Be sure you review the previous chapter if you
need to so that the relationship between force and PE ( |F| = |d(PE)/dr| ) is
absolutely clear to you. There is no point in going any further, if you are still
stumbling over this relationship.
3) Make sure you can use the relationship in (2) to explain to your fellow
chemistry students (or your chemistry TA) the three bulleted features of the
pair-wise potential using relationship (2) and the general shape of the pair-
wise potential.
4) When there are many particles, the phase (s, l, g) of those particles depends on
their total energy. At sufficiently high total energy, the particles are unbound
and in the gas phase. At sufficiently low energy the particles are in the liquid
or solid phase and are bound. The average particle-particle separation in the
bound state is approximately equal to the separation corresponding to the
minimum of the pair-wise potential energy. In the unbound state it is much
greater than the separation corresponding to the minimum PE.
One common mistake that many students make is to attempt to ascribe
macroscopic properties (like solid, liquid, gas) to the interaction of only a
small number of particles using PEpair-wise. The macrostate of matter, whether
it is in a solid, liquid or gas phase, for example, is due to the simultaneous
interactions of something like 1025
pair-wise interactions if we have a mole of
the substance. These ideas are not easy, so be patient. Initially, try to imagine
a solid at very low temperatures. Each particle “wants” to be at the right
distance with respect to all of its neighbors. If there is a way for the system to
“get rid” of its energy (by giving it to some colder system, for example), it
will continue to settle down and reduce its thermal energy. Eventually, all the
random motion comes to a stop (if we can keep cooling the sample) and the
particles find their “magic” places, each near the “bottom” of the PEpair-wise
with each of each neighbors.
Now, imagine we start adding energy to the sample. All the particles begin
acting like little spring-masses, oscillating back and forth around their
equilibrium positions. Eventually they move sufficiently far, so that some
“jump” out of where they are “supposed to be.” Particles at or near the surface
might even leave the sample if their vibrations get vigorous enough. Picturing
what happens when a substance melts, i.e., turns from a solid to a liquid, is
Chapter 3 Applying Particle Models to Matter 61
difficult, even for the experts. Don’t worry about picturing that transition. But
you can imagine continuing to add energy until all the particles, even 1023
particles, have sufficient energy to separate far apart from each other, causing
them to be in the gas phase. Recall what value all ~1025
pair-wise potentials
will have, if all particles are separated by many particle diameters. So, what is
the bottom line here at this point in our making sense of all this? Without
getting into a lot of detail, it should make sense to you that at some
sufficiently low temperature, everything will be a solid and at some
sufficiently high temperature, everything should be a gas. That is plenty for
right now.
5) The interactions of one particle in a liquid or solid with all of its neighbors
add together to form one three-dimensional potential energy for a particular
particle with a minimum that defines the equilibrium position of that particle
(where the net force due to all of the pair-wise interactions is zero). We refer
to this potential as the single-particle potential energy to emphasize its
distinction from the pair-wise potential energy.
6) Each particle in a solid or liquid oscillates in three dimensions about its
equilibrium position as determined by its single-particle potential.
OK, so here is where we are attempting to make our mental picture a little
clearer regarding what is happening to a single particle (which could be an
atom or a tightly bound-together molecule) when it finds itself somewhere in
the middle of similar particles in a solid or liquid. It really is acting like it is
attached to a bunch of springs with all of its neighbors (and nearby neighbors).
But here is the “really neat” thing. No matter how complicated the actual
chemical bonds are, and no matter how many there are, or in what directions
they point, they all add up to exactly what would happen if you had only three
(that’s right, only three) little springs of exactly the right strength, one going
out in each of the three x, y, and z directions of the three-dimensional space
we seem to occupy in this universe (at least on our scale and on the scale of
atoms and molecules). So the picture you want to get into your head is
something like that shown below, remembering that the spring constant of the
springs can be different in the three directions.
We will come back to this picture shortly when we make more sense of
thermal energy.
y
x
z
62 Chapter 3 Applying Particle Models to Matter
But, what about the bond energy? Well, it really does depend on the real
bonds, the real chemical bonds. However, we can develop reasonable
estimates in terms of the well depths of the pair-wise interactions for the bond
energy that work for practically all pure substances. Carrying out the analysis
to make sense of bond energy and to make sense of thermal energy is what the
next two models are about.
3-4 Particle Model of Bond Energy
(Summary on foldout #5 at back of course
packet)
New Construct Definitions
Internal energy and Mechanical energy
The internal energy, U, is the energy associated with all the kinetic and potential
energies of the particles constituting a substance. This will include the energies
associated with the formation of the various phases as well as the energies internal
to the particles themselves, such as the molecular, atomic and nuclear energies.
Mechanical energy refers to the potential and kinetic energies associated with the
motion of objects as a whole. Thus, it is often the case that the mechanical energy
of an object can be small or zero, yet the internal energy can be quite high. For
example even a baseball thrown at 90 miles/hour has much more thermal energy
at room temperature than kinetic energy due to its being thrown.
Thermal energy
Thermal energy is the sum of the potential and kinetic energies that are associated
with the disordered motions of the particles that make up an object. We will
significantly expand our understanding of this construct in Section 3-5: Particle
Model of Thermal Energy.
Bond energy and Binding energy
In solid and liquid phases there is a bond energy associated with the attractive part
of all the pair-wise potential energies acting between pairs of particles. By
convention, the binding energy is the positive energy that must be added to
separate the particles sufficiently far apart so that the bond energy has the value
zero. Since the maximum value of the bond energy occurs when the particles are
widely separated, and because of the way the pair-wise potential is defined, the
bond energy of liquids and solids must be less than zero; that is, the bond energy
is negative. Binding energy is simply the magnitude of the bond energy and is
always a positive number, even though the bond energy is negative.
Nearest neighbor (n-n) pair
In a solid or liquid, each atom or molecule will “be almost touching” about 12
other atoms or molecules. It is exactly 12 for many substances, if the atoms or
molecules are spherically shaped. You will get a chance in the discussion/lab
Chapter 3 Applying Particle Models to Matter 63
activities to check on this. It is these 12 or so atoms that form nearest neighbor
pair-wise bonds with a particular atom or molecule. These are all at nearly the
“right” distance apart so that the PE of each pair is very close to its minimum
value.
Non-nearest neighbor pairs
Do we need to worry about interactions between atoms or molecules that are not
nearest neighbors? A little bit, depending on how accurate we want our numerical
predictions to be. Look back at the pair-wise potential energy curve. Has the slope
gone totally horizontal when the particles are located two diameters from each
other. No, not totally. There are a lot of nearby neighbors that are within two
diameters of each other, so these non-nearest neighbors will still be attracting
each other a little bit and will make a contribution to the binding energy, but
typically significantly less than the nearest neighbors.
Empirically determined values
The concepts of bond energy and thermal energy are very useful in models that
help us make sense of the particulate nature of matter. However, the quantities
that are actually measured, although closely related to these ideas, are not quite
the same. That is, the H’s we encountered in Chapter 1 and the bond-energy
systems we used there based on these H’s are not precisely the same as the bond
energy defined in terms of the pair-wise interactions. However, it is rather tricky
to understand precisely how they are related. When we proceed through Chapter 4
on thermodynamics, it will be possible to sort much of this out. Until then, we
will accept that when making comparisons of the concepts in our models to
empirical data, we are making some approximations, which will always be
pointed out. These approximations typically allow us to still make numerical
comparisons to within 10 to 20 percent of the best we can do with extremely
complicated models. From a modeling perspective, this is initially a price well
worth paying in order to have a model sufficiently simple and broadly applicable
to enable us to develop a meaningful understanding of a great deal of the “how
and why” matter behaves the way it does from a particulate perspective. The
models we develop in this chapter apply, in the sense that they allow us to make
sense of phenomena and get pretty close when making numerical predictions, to a
very wide range of phenomena without getting bogged down in so many details
that we never get anywhere in our understanding. Thermodynamics is the
“science” of understanding the subtleties and the details of precisely determined
empirical data. In Chapter 4 we will get a brief introduction and a taste of the
power it provides, but at a cost of the loss of the simplicity of the models in
Chapter 3.1
Meaning of the Model Relationships (Numbers below correspond to the numbered relationships on the fold-out Summary.)
1) From a macroscopic perspective, the total internal energy of a substance,
excluding nuclear and atomic energies, is comprised of thermal energy and
bond energy. (Einternal = Ethermal + Ebond) Excluding atomic and nuclear
energies, the bond energy is often referred to as the “chemical energy,”
64 Chapter 3 Applying Particle Models to Matter
because it is the changes in this part of the internal energy that result from
changes in chemical bonds.
This relationship emphasizes that we are often only interested in changes in
energies when using the Energy-Interaction Model, so we don’t usually care
what the absolute values of the internal energy actually are.
2) If all of the particles of a substance were sitting at rest at their equilibrium
positions, the magnitude of the bond energy would be the amount of energy
that would have to be added to completely separate all of the particles, still at
rest. This positive quantity is customarily referred to as “binding energy.”
This concept can be applied to phase changes of substances without causing
chemical changes as well as to the energy required to separate a particular
molecular species into separate atoms. It also is applied in exactly the same
way to changes in nuclear processes.
There is one tricky aspect associated with directly relating bond energies to
heats involved in phase changes. It is that there can be, as we shall see in our
Particle Model of Thermal Energy, changes in the thermal energy at a phase
change as well as in the bond energy. The empirically determined H’s that
we used in the bond energy system in Chapter 1, however, do incorporate any
changes of energy in thermal energy at a phase change. Thus, the H’s are not
precisely a measure of the particle model bond energy change. For the most
part we will ignore this until we have sufficient background to make sense of
it. There are also several other rather subtle effects that we will ignore until
we are ready to make sense of them in Chapter 4.
It is important to understand that “this is how science works.” And it is
certainly the way we begin to learn science! We create models that are
sufficiently simple to make a start at making sense of the phenomena, and
then the discrepancies with empirically determined data allow us to refine the
models (as well as making them a lot more complicated) to whatever degree
we need to answer the questions we are interested in answering. In the
beginning phases of making sense of phenomena, when doing science and
when learning science, simple and more broadly applicable models are almost
always the better way to begin.
3) In terms of particle potential energies, the bond energy of a substance is the
sum of all of the pair-wise potential energies of the particles comprising the
substance calculated when all of the particles are at their equilibrium positions
corresponding to a particular physical and chemical state. In molecular
substances there will be both inter- and intra-molecular contributions to the
bond energy.
This definition of bond energy avoids the issue of the thermal energy possibly
changing, because the calculation is carried out at essentially zero kelvin
(because all particles are in their equilibrium positions as they would be at
absolute zero, if the phase actually existed at absolute zero) in both the bound
state as well as when the particles are separated. We take this to be our
technical definition of bond energy. An equivalent definition would be to say
that the energy required to separate the particles is carried out so that the
thermal energy is the same after the separation as before the separation.
Chapter 3 Applying Particle Models to Matter 65
4) By convention, all pair-wise potentials are defined to be zero when the
particles are separated sufficiently so that the force acting between the
particles is zero. Therefore, the bond energy of any condensed substance is
always negative. The maximum value of the bond energy is zero when the
particles that comprised the substance are all completely separated to large
distances.
This is sometimes hard to get our minds around. For example, think of the
oxidation of hydrogen to form water. When oxygen atoms are far away form
the hydrogen atoms, the bond energy of two hydrogen atoms and one oxygen
atom have their maximum value, which is zero. As they move close to one
another “and bond,” their bond energy becomes some negative number. It
seems like we might be saying they were bound when they were far apart,
because that is when they had their greatest bond energy. No, they were not
bonded. There were no chemical or any other kind of bonds when the atoms
are greatly separated. It is just a lot more sensible to measure bond energies
this way. It takes some getting used to, however.
5) For molecular substances that don’t disassociate, the total number of n-n pairs
times the well-depth of the pair-wise potential energy ( ) between molecules
is a rough approximation for the sum of all intermolecular pair-wise potential
energies of a substance. This approximation, however, will tend to
underestimate the binding energy. The underestimation comes in because we
have not added in the contributions of the many neighbor pairs that are in the
one to two diameter separation range.
6) The empirically determined heats of melting and heats of vaporization are
reasonable approximations to the changes in bond energy at the respective
physical phase changes.
But see comments following relationship (2).
7) The empirically determined heats of formation of various chemical species
can be used to calculate changes in bond energy when chemical reactions
occur.
Chemists use a very useful system to enable these calculations to be easily
carried out. It involves carefully defining the “starting state” of the elements
and compounds. It is something you must understand precisely, but when you
do, it is an extremely powerful method. This works, because what we are
interested in is changes in bond energy. Where the zero of energy is assigned
to be for each and every distinct element or substance, doesn’t matter, as long
as everyone agrees on the assignment and sticks with it.
Algebraic Representations
The three approximate relationships mentioned in the numbered relationships
above can be expressed algebraically. For reference purposes, we list them here.
Relationship (3) Ebond = all pairs (PEpair-wise) (calculated with all particles at
their equilibrium positions)
Relationship (5) Intermolecular Ebond – (total number of n-n pairs)
Relationship (6) | Ebond| | H m|at a phase change
66 Chapter 3 Applying Particle Models to Matter
3-5 Particle Model of Thermal Energy
(Summary on foldout #6 at back of course
packet)
Construct Definitions
Thermal Equilibrium and Equipartition of Energy Among Modes
There are several important ideas here that all go together. By thermal equilibrium
we mean that the random energy fluctuations associated with the motions of the
atoms and molecules about their equilibrium positions in a solid or liquid or their
random motions when in the gas phase, will over time, become uniformly
distributed throughout the entire sample. That is, there will be about as much
energy associated with the random energies of a small piece of the sample, but
still containing 1015
or so particles, as in any other same size small piece. This is
what we mean by thermal equilibrium on a particle basis. It is also similar to what
we would say about the temperature. If we wait for a sufficiently long time, the
temperature will become uniform throughout the sample. There would seem to be
a direct connection between temperature and the disordered random motion
associated with thermal energy. In fact there is a very definite connection.
In the Intro Particle Model of Matter we saw that each atom in a liquid or solid
acted as if it vibrated like a spring-mass in each of three dimensions. An
interesting question is how many ways does each of these particles “have
energy?” We need to think about how many ways a spring-mass has energy. It has
a KE and also a PE. We simply stated at the time that the average PE was the
same as the average KE. We will simply take this as reasonable at this point.
Because of the randomness or disordered-ness of the thermal motions of all the
little mass springs in all three of the directions in space, it is plausible that on
average, each spring would have the same average PE as would any other spring.
And also the same KE as any other spring. In fact this is exactly what happens. It
can be rigorously proven for all energies that depend on the square of a position
or speed variable. Thus, in addition to working for spring-mass systems, it works
for unbound atoms in the gas phase, which will have translational kinetic
energies, since these energies depend on the square of their translational speed. It
works for molecules that rotate, which will have rotational kinetic energies, since
these energies depend on a square of a rotational speed.
So back to our question. How many ways does each spring have “to have”
energy? The answer is two: one KE and one PE. How many ways does each
particle in a solid or liquid have to have energy? Well, there are three springs and
two ways per spring, so it must be six. Each particle in a solid or liquid has six
ways to have energy. Now combine this with what we just argued regarding
thermal equilibrium. On average, each “way to have energy” would have the same
amount of energy when averaged over a sufficiently long period of time. There is
a name, or label, for “way to have energy.” The name is “mode.” So we say that
each particle has six modes in a solid or liquid. And on average when the sample
is in thermal equilibrium, each mode has the same amount of energy (on average).
This principle is referred to as the principle of equipartition of energy.
Chapter 3 Applying Particle Models to Matter 67
Freezing out of modes
Sometimes, however, the modes don’t “get excited” due to the quantization of
energy levels. At low temperatures, the quantum splitting between energy levels,
which you are familiar with from chemistry, keeps all but the ground state level
from being populated, or having any energy. When this happens, we say that
mode is “frozen out.” It is as if it didn’t exist. Frozen out modes cannot share
thermal energy. So in the following statements, we usually put in the qualifier,
“active modes,” meaning that only active modes share the thermal energy equally
among themselves.
Heat Capacity at constant volume
Because we will want to compare values of heat capacity to our predicted values
of thermal energy from the particle model of matter, we need to be careful that we
are actually comparing the same things. We know that if a force acts through a
distance, work will be done by one physical object on another. When we make a
heat capacity measurement, we don’t want the sample doing work on the
atmosphere or the container it is in. Therefore, we specify that the sample be kept
at constant volume during the heat capacity measurement. This is designated with
a subscript “v.” The important point here is that we have a way to directly
measure the change in the thermal energy by measuring the heat capacity of a
sample at constant volume, ensuring all the heat we put into the sample goes to
changing its thermal energy and not doing some work by expanding the container
or pushing against the air in the room.
Meaning of the Model Relationships (Numbers below correspond to the numbered relationships on the fold-out Summary.)
1) In gases, the translational kinetic energy of each particle can be divided into three
independent “pieces”, each one corresponding to one of the three independent spatial
dimensions; each particle in a gas has at least these three independent modes.
Because there are no springs connecting the particles in a gas, there are only three
modes per particle, if the particle itself has no internal modes.
2) In liquids and solids, the oscillations of each particle in its single-particle potential
can be modeled as a mass held in place by three perpendicular springs. The potential
and kinetic energies that are associated with those oscillations can each be divided
into three independent modes, each one corresponding to one of the three
independent spatial dimensions; each particle in a liquid or solid has at least these six
independent modes.
3) In all phases (s, l, g) polyatomic molecules may have additional energies associated
with rotations and internal vibrations of the molecule. These might contribute
additional modes, depending on whether or not they are frozen out at the temperature
in question. At room temperature the vibrational modes of most diatomic molecules
(but not translational or rotational modes) are frozen out.
A diatomic molecule, for example, might vibrate or it might rotate. These could
contribute additional modes.
4) The thermal energy of a substance is the total energy in all the active modes of all the
particles comprising the substance.
5) In thermal equilibrium all active modes have, on average, the same amount of
energy. This principle is referred to as “equipartition of energy.”
68 Chapter 3 Applying Particle Models to Matter
6) The amount of energy, on average, in an active mode is directly proportional to the
temperature. The proportionality constant, for historical reasons, is written kB/2,
where kB is the Boltzmann constant. (kB = 1.38 10-23
J/K)
Now we get the connection with temperature and modes. Temperature is actually a
measure of the average energy in an active mode when the sample is in thermal
equilibrium.
7) The total thermal energy of an object in thermal equilibrium is equal to the product
of [the total number of active modes] and [the average energy per mode].
This last relationship gives us a precise notion of what thermal energy really is.
8) When the change in thermal energy is due solely to the addition or removal of energy
as heat, the constant volume heat capacity, CV, is given by the rate of change of
thermal energy with respect to temperature.
CV is the macroscopic variable that corresponds to how many modes there are on a
particle basis.
Algebraic Representations
The three essential relationships (6., 7, and 8) summarize the three really big ideas
here.
Relationship (6) Ethermal/mode = (1/2)kBT
This is the Big One!
Relationship (7) Ethermal (total) = (total number of active modes)
(1/2)kBT
Relationship (8) CV = dEthermal/dT,
3-6 Looking Back and Ahead
At this point we have developed the energy-interaction approach rather
completely. There are still some “kinds” of energy we have not encountered, but
when we do, we know what to do: treat it as another energy-system. We know
how to approach physical systems that involve changes in macroscopic
mechanical energies as well as changes in internal energies. We have a systematic
way of “dealing with” friction as the transfer of energy to thermal systems.
We have also refined our model of matter to a point where we can understand
most of the thermal properties it exhibits. For certain thermal properties, we can
make very definite numerical predictions with our model.
With our model of matter and understanding of energy and energy conservation,
we now can actually understand many of the fundamental concepts that underlie
much of thermal physics, thermochemistry and the properties of gases, liquids,
and solids. We have also developed a much more sophisticated understanding of
temperature. We have made a solid connection of the macroscopic concept of
Chapter 3 Applying Particle Models to Matter 69
temperature that we measure with a thermometer to our extended microscopic
model of matter.
Up to this point we have tried to avoid getting into the messy details of the
interactions of matter. What is remarkable, is how much we have accomplished
with this approach. There are, however, many questions that we cannot answer
without getting involved in the details. An example is how do we determine the
strength of bonds (or spring force constant). It turns out that the spring constants
are directly related to the frequency of vibration of the particles themselves.
Infrared spectroscopy is one way to determine these frequencies and thus the
spring constants. This is an important question that we definitely want to explore.
But before we can proceed, we need to go back and spend some time developing
the general connection between unbalanced force and change in motion. In Part 2,
we will do this, and can then come back to the question of oscillation frequency of
our oscillators.
In the meantime, we will use our model of matter and energy interaction
approach, along with some new constructs and relationships to explore other
interesting physical phenomena using a very powerful approach to understanding
interactions of a chemical and biological nature: the thermodynamic model.
70 Chapter 3 Applying Particle Models to Matter
1 A little more about the reasons for the complications here, the reasons we use
the labels we have chosen for the energy systems in Chapter 1 (“thermal energy
system” and “bond energy system”), and the difference between these
macroscopic energy systems and the bond and thermal energies defined within a
particle model in Chapter 3. Remember: material in these footnotes is
considered beyond what is necessary or even desirable for the first-time student
to worry about when initially learning the basic models.
First point: Thermal energy system and bond energy system apply to the way to
divide up macroscopically the internal energy (introduced in Chapter 4) in such a
way that when considering physical phase changes, each energy system
corresponds separately and independently to one or the other of the two
empirically observed thermal properties of matter; namely, the specific heat and
the heats of melting and vaporization. That is, we assign the observed change in
energy when the temperature changes to an energy system, that we call the
“thermal energy system,” and the magnitude of the change in that energy system
is given by the change in temperature multiplied by the heat capacity. Likewise,
we assign the observed change in energy when there is a phase change to “bond
energy system,” and the magnitude of the change in that energy system is given
by the change in mass of a particular phase multiplied by the “heat”—the change
in enthalpy—of the respective phase change.
Second Point: Changes in the energy systems defined as above are not always
exactly the same as the changes in “thermal energy” and changes in “bond
energy” defined from a particle perspective during a particular physical process
for several reasons. However, the differences are seldom greater than 20% or so,
even in the worst case, since they arise due to factors that tend to cancel each
other out (ignoring the reduction in heat capacity that typically occurs when a
liquid evaporates and the work that is done in a constant pressure measurement of
the heat of vaporization).
Third Point: It is appropriate, especially in a models approach, to initially ignore
the differences in the macroscopically defined constructs thermal and bond energy
systems and the microscopically defined thermal and bond energies. It is possible
to thoroughly understand these differences and explicitly deal with them using the
understandings of the particle models of thermal and bond energies and
thermodynamics, which allow the meaning of measurements of heats of
vaporization, for example, to be accurately understood in terms of changes in
internal energies. It is not possible to understand this in the context of Chapter 1.
Fourth Point: By taking the approach we have, students can make sense of the
macroscopic changes in energy that occur and characterize them using the
standard thermal properties of matter in a straightforward way, without getting
bogged down in details that are not necessary to understand at this level.