chapter_3

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.


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


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.


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


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.


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.


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


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


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


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


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


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,”


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.


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


3-5 Particle Model of Thermal Energy


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.


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.


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.”


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


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.


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.

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