Modern Atomic Theory
To see a World in a Grain of Sand
11.1 The Mysterious
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour 11.2 Multi-Electron
William Blake (1757-1827)
Auguries of Innocence
cientists' attempts to understand the atom have led them into the unfamiliar
world of the unimaginably small, where the rules of physics seem to be different
from the rules in the world we can see and touch. Scientists explore this world
through the use of mathematics. Perhaps this is similar to the way a writer uses
poetry to express ideas and feelings beyond the reach of everyday language. Mathematics
allows the scientist to explore beyond the boundaries of the world we can experience
directly. Just as scholars then try to analyze the poems and share ideas about them
in everyday language, scientists try to translate the mathematical description of the
atom into words that more of us can understand. Although both kinds of translation
are fated to fall short of capturing the fundamental
truths of human nature and the physical world, the
attempt is worthwhile for the occasional glimpse of
those truths that it provides.
This chapter offers a brief, qualitative introduction
to the mathematical description of electrons and
describes the highly utilitarian model of atomic
structure that chemists have constructed from it.
Because we are reaching beyond the world of our
senses, we should not be surprised that the model we
create is uncertain and, when described in normal
language, a bit vague. In spite of these limitations,
however, you will return from your journey into the
strange, new world of the extremely small with a
useful tool for explaining and predicting the behavior Chemists try to "see" the structure of matter even more
of matter. closely than can be seen in any photograph.
The presentation of information in this chapter assumes that you can already perform
the tasks listed below. You can test your readiness to proceed by answering the Review
Questions at the end of the chapter. This might also be a good time to read the Chapter
Objectives, which precede the Review Questions.
Describe the nuclear model of the atom. Describe the relationship between stability
(Section 2.4) and potential energy. (Section 7.1)
414 Chapter 11 Modern Atomic Theory
11.1 The Mysterious Electron
Where there is an open mind, there will always be a frontier.
Charles F. Kettering (1876-1958)
American engineer and inventor
Scientists have known for a long time that it is incorrect to think of electrons as tiny
particles orbiting the nucleus like planets around the sun. Nevertheless, nonscientists
have become used to picturing them in this way. In some circumstances, this "solar
system" model of the atom may be useful, but you should know that the electron is much
more unusual than that model suggests. The electron is extremely tiny, and modern
physics tells us that strange things happen in the realm of the very, very small.
The modern description of the electron is based on complex mathematics and on
the discoveries of modern physics. The mathematical complexity alone makes an
accurate verbal portrayal of the electron challenging, but our difficulty in describing the
electron goes beyond complexity. Modern physics tells us that it is impossible to know
exactly where an electron is and what it is doing. As your mathematical and scientific
knowledge increases, you will be able to understand more sophisticated descriptions of
the electron, but the problem of describing exactly where the electron is and what it is
doing never goes away. It is a problem fundamental to very tiny objects. Thus complete
confidence in our description of the nature of the electron is beyond our reach.
There are two ways that scientists deal with the problems associated with the
complexity and fundamental uncertainty of the modern description of the electron:
Analogies In order to communicate something of the nature of the electron,
scientists often use analogies, comparing the electron to objects with which we
are more familiar. For example, in this chapter we will be looking at the ways in
which electrons are like vibrating guitar strings.
Probabilities In order to accommodate the uncertainty of the electron's position
and motion, scientists talk about where the electron probably is within the atom,
instead of where it definitely is.
Through the use of analogies and a discussion of probabilities, this chapter attempts to
give you a glimpse of what scientists are learning about the electron's character.
Standing Waves and Guitar Strings
Each electron seems to have a dual nature in which both particle and wave characteristics
are apparent. It is difficult to describe these two aspects of an electron at the same time,
so sometimes we focus on its particle nature and sometimes on its wave character,
depending on which is more suitable in a given context. In the particle view, electrons
are tiny, negatively charged particles with a mass of about 9.1096 10-28 grams. In
the wave view, an electron has an effect on the space around it that can be described
as a wave of varying negative charge intensity. To gain a better understanding of this
electron-wave character, let's compare it to the wave character of guitar strings. Because
a guitar string is easier to visualize than an electron, its vibrations serve as a useful
analogy of the wave character of electrons.
11.1 The Mysterious Electron 415
When a guitar string is plucked, the string vibrates up and down in a wave pattern.
Figure 11.1 shows one way that it can vibrate; the seven images on the left represent
the position of the string at various isolated moments, and the final image on the right
shows all those positions combined. If you squint a bit while looking at a vibrating
guitar string, what you see is a blur with a shape determined by the varying intensity
of the vibration along the string. This blur, which we will call the waveform, appears
to be stationary. Although the string is constantly moving, the waveform is not, so this
wave pattern is called a standing or stationary wave. Note that as your eye moves along
the string, the intensity, or amount, of the string's movement varies. The points in the
waveform where there is no motion are called nodes.
Waveform of a
The waveform shows
the variation in the
intensity of motion at
every position along
Photo by Jack Spira
7 possible configurations
for the vibration of a
guitar string Superimposing the
waveform of the
Although many waveforms are possible, the possibilities Figure 11.2
are limited by the fact that the string is tied down and cannot Some Possible Wave-
forms for a Vibrating
move at the ends. In theory, there are an infinite number of Guitar String
possible waveforms, but they all allow the string to remain
stationary at the ends. Figure 11.2 shows various allowed Objective 2
416 Chapter 11 Modern Atomic Theory
Electrons as Standing Waves
Thus, the task is not The wave character of the guitar string is represented by the movement of the string.
so much to see what We can focus our attention on the blur of the waveform and forget the material the
no one has yet seen,
string is made of. The waveform describes the motion of the string over time, not the
but to think what
nobody has yet string itself.
thought, about that In a similar way, the wave character of the electron is represented by the waveform
which everybody of its negative charge, on which we can focus without concerning ourselves about the
electron's particle nature. This frees us from asking questions about where the electrons
Erwin Schrodinger are in the atom and how they are moving--questions that we are unable to answer. The
(1887-1961) waveforms for electrons in an atom describe the variation in intensity of negative charge
Austrian physicist and
within the atom, with respect to the location of the nucleus. This can be described
without mentioning the positions and motion of the electron particle itself.
Objective 3 The following statements represent the core of the modern description of the wave
character of the electron:
Just as the intensity of the movement of a guitar string can vary, so can the
intensity of the negative charge of the electron vary at different positions
outside the nucleus.
The variation in the intensity of the electron charge can be described in
terms of a three-dimensional standing wave like the standing wave of the
As in the case of the guitar string, only certain waveforms are possible for
the electron in an atom.
We can focus our attention on the waveform of varying charge intensity
without having to think about the actual physical nature of the electron.
Waveforms for Hydrogen Atoms
Most of the general descriptions of electrons found in the rest of this chapter are based
on the wave mathematics for the one electron in a hydrogen atom. The comparable
calculations for other elements are too difficult to lead to useful results, so as you will
see in the next section, the information calculated for the hydrogen electron is used to
describe the other elements as well. Fortunately, this approximation works quite well.
The wave equation for the one electron of a hydrogen atom predicts waveforms for
the electron that are similar to the allowed waveforms for a vibrating guitar string. For
example, the simplest allowed waveform for the guitar string looks something like
Objective 4 The simplest allowed waveform for an electron in a hydrogen atom looks like the
image in Figure 11.3. The cloud that you see surrounds the nucleus and represents
the variation in the intensity of the negative charge at different positions outside
the nucleus. The negative charge is most intense at the nucleus and diminishes with
increasing distance from the nucleus. The variation in charge intensity for this waveform
is the same in all directions, so the waveform is a sphere. The allowed waveforms for
11.1 The Mysterious Electron 417
the electron are also called orbitals. The orbital shown in Figure 11.3 is called the 1s
e negative charge is most Waveform of the 1s
Nucleus, about 0.00001 intense at the nucleus Electron
the diameter of the atom and decreases in intensity
with distance outward.
Theoretically, the charge intensity depicted in Figure 11.3 decreases toward zero as
the distance from the nucleus approaches infinity. This suggests the amusing possibility
that some of the negative charge created by an electron in a hydrogen atom is felt
an infinite distance from the atom's nucleus. The more practical approach taken by
chemists, however, is to specify a volume that contains most of the electron charge and
focus their attention on that, forgetting about the small negative charge felt outside the
specified volume. For example, we can focus on a sphere containing 90% of the charge
of the 1s electron. If we wanted to include more of the electron charge, we enlarge
the sphere so that it encloses 99% (or 99.9%) of the electron charge (Figure 11.4).
This leads us to another definition of orbital as the volume that contains a given high
percentage of the electron charge.
Most of the pictures you will see of orbitals represent the hypothetical surfaces that Objective 4
surround a high percentage of the negative charge of an electron of a given waveform.
The 1s orbital, for example, can either be represented by a fuzzy sphere depicting the
varying intensity of the negative charge (Figure 11.3) or by a smooth spherical surface
depicting the boundary within which most of the charge is to be found (Figure 11.4).
Almost all of the electron's Sphere enclosing almost 1s Orbital
charge lies within a spherical shell all of the electron's
with the diameter of this circle. negative charge Objective 4
Is the sphere in Figure 11.3 the 1s electron? This is like asking if the guitar string is
the blur that you see when the string vibrates. When we describe the standing wave
that represents the motion of a guitar string, we generally do not refer to the material
composition of the string. The situation is very similar for the electron. We are able
to describe the variation in intensity of the negative charge created by the electron
without thinking too much about what the electron is and what it is doing.
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