Chapter 11 – An Introduction to Chemistry: Modern Atomic Theory

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Chapter 11
                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.

Review Skills
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.
                                                                                           Figure 11.1
                                                                                           Waveform of a
                                                                                           Standing Wave
                                                                                           The waveform shows
                                                                                           the variation in the
                                                                                           intensity of motion at
                                                                                           every position along
                                                                                           the string.
                                                                                           Photo by Jack Spira


 7 possible configurations
 for the vibration of a
 guitar string                                              Superimposing the
                                                            produces the
                                                            waveform of the
                                                            guitar string's
                                                            standing wave.

   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
         Nobel laureate
                             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
                                    guitar string.
                                    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
                                                                                               Figure 11.3
                                                            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
                                                                                                     Objective 4
                                                         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).

                                                                                        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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