SAT Subject Test Chemistry

PART 2

REVIEW OF MAJOR TOPICS

CHAPTER 2

Atomic Structure and the Periodic Table of the Elements

THE WAVE-MECHANICAL MODEL

In the early 1920s, some difficulties with the Bohr model of the atom were becoming apparent. Although Bohr used classical mechanics (which is the branch of physics that deals with the motion of bodies under the influence of forces) to calculate the orbits of the hydrogen atom, this discipline did not serve to explain the ability of electrons to stay in only certain energy levels without the loss of energy. Nor could it explain why a change of energy occurred only when an electron “jumped” from one energy level to another and why the electron could not exist in the atom at any energy level between these levels. According to Newton’s laws, the kinetic energy of a body always changes smoothly and continuously, not in sudden jumps. The idea of only certain quantized energy levels being available in the Bohr atom was a very important one. The energy levels explained the existence of atomic spectra, described in the preceding sections.

Another difficulty with the Bohr model was that it worked well only for the hydrogen atom with its single electron. It did not work with atoms that had more electrons. A new approach to the laws governing the behavior of electrons inside the atom was needed, and such an approach was developed in the 1920s by the combined work of many scientists. Their work dealt with a more mathematical model usually referred to as quantum mechanics or wave mechanics. By this time, Albert Einstein had already proposed a relativity mechanics model to deal with the relative nature of mass as its speed approaches the speed of light. In the same manner, a quantum/wave mechanics model was now needed to fit the data of the atomic model. Max Planck suggested in his quantum theory of light that light has both particlelike properties and wavelike characteristics. In 1924, Louis de Broglie, a young French physicist, suggested that, if light can have both wavelike and particlelike characteristics as Planck had suggested, then perhaps particles can also have wavelike characteristics. In 1927, de Broglie’s ideas were verified experimentally when investigators showed that electrons could produce diffraction patterns, a property associated with waves. Diffraction patterns are produced by waves as they pass through small holes or narrow slits.

In 1927, Werner Heisenberg stated what is now called the uncertainty principle. This principle states that it is impossible to know both the precise location and precise velocity of a subatomic particle at the same time. Heisenberg, in conjunction with the Austrian physicist Erwin Schrödinger, agreed with the de Broglie concept that the electron is bound to the nucleus in a manner similar to a standing wave. They developed the complex equations that describe the wave-mechanical model of the atom. The solution of these equations gives specific wave functions called orbitals. These are not related at all to the Bohr orbits. The electron does not move in a circular orbit in this model. Rather, the orbital is a three-dimensional region around the nucleus that indicates the probable location of an electron but gives no information about its pathway. The drawings in Figures 8a and 8bare only probability distribution representations of where electrons in these orbitals might be found.

Tip 

Know the uncertainty principle

Quantum Numbers and the Pauli Exclusion Principle

Each electron orbital of an atom may be described by a set of four quantum numbers in the wave-mechanical model. These numbers give the position with respect to the nucleus, the shape of the orbital, its spatial orientation, and the spin of the electron in the orbital.

Principal quantum number (n) 1, 2, 3, 4, 5, etc.

The values of n = 1, 2, 3 . . .

 

This number refers to average distance of the orbital from the nucleus. 1 is closest to the nucleus and has the least energy. The numbers correspond to the orbits in the Bohr model. They are called energy levels.

Angular momentum (quantum number
spdf
(in order of increasing energy)

The value of  can = 0, 1, . . . ,
n − 1
 = 0 indicates a spherical-shaped s orbital
 = 1, indicates a dumbbell-shaped p orbital
 = 2, indicates a five orbital orientation d orbital

 

This number refers to the shape of the orbital. The number of possible shapes is limited by the principal quantum number. The first energy level has only one possible shape, the s orbital because n = 1 and the limit of  = (n − 1) = 0. The second has two possible shapes, the s and p. See Figures 8a and 8b for representations of these shapes.

TIP 

The principal quantum number refers to the principal energy level: 1, 2, 3, and so on. The angular momentum quantum number refers to shape.

Magnetic quantum number ()
s = 1 space-oriented orbital
p = 3 space-oriented orbitals
d = 5 space-oriented orbitals
f = 7 space-oriented orbitals
The value of  can equal −...0... + .
Spin quantum number (ms) + spin − spin
The value of m = 

 

The drawings in Figure 8a show the s-orbital shape, which is a sphere, and the p orbitals, which have dumbbell shapes with three possible orientations on the axis shown. The number of spatial orientations of orbitals is referred to as the magnetic quantum number. The possible orientations are listed. Figure 8b represents the dorbitals. Electrons are assigned one more quantum number called the spin quantum number. This describes the spin in either of two possible directions. Each orbital can be filled by only two electrons with opposite spins. The main significance of electron spin is explained by the postulate of Wolfgang Pauli. It states that in a given atom no two electrons can have the same set of four quantum numbers (n, and ms). This is referred to as the Pauli Exclusion Principle. Therefore each orbital in Figures 8a and 8b can hold only two electrons.

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Pauli Exclusion Principle: No two electrons can have the same four quantum numbers.

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s orbitals are spherical and can hold 2 electrons.

Two representations of the hydrogen 1s, 2s, and 3s orbitals. (1) The electron probability distribution; the nodes indicate regions of zero probability. (2) The surface that contains 90% of the total electron probability (the size of the orbital, by definition).

Representation of the 2 p orbitals. (1) The electron probability distribution and (2) the boundary surface representations of all three orbitals.

FIGURE 8a. Representations of s and p orbitals

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p orbitals have a dumbell shape, oriented on the x, y, and z axes, and can hold a total of 2 electrons each, making a total of 6.

FIGURE 8b. Representations of d orbitals

(Indicates orbitals in y and z planes)
Representations of the 3d orbitals in terms of their boundary surfaces.
The subscripts of the first four orbitals indicate the planes in which the four lobes are centered.

FIGURE 8b. Representations of d orbitals

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d orbitals have 5 orientations and can hold 2 electrons each, making a total of 10.

Quantum numbers are summarized in the table below.

Hund’s Rule of Maximum Multiplicity and the Aufbau Principle

It is important to remember that, when there is more than one orbital at a particular energy level, such as three p orbitals or five d orbitals, only one electron will fill each orbital until each has one electron. This principle, that an electron occupies the lowest energy orbital that can receive it, is called the Aufbau Principle. After this, pairing will occur with the addition of one more electron to each orbital. This principle, called Hund’s Rule of Maximum Multiplicity, is shown in Table 2, where each arrowhead indicates an electron ().

TIP 

Know the Aufbau Principle and Hund’s Rule of Maximum Multiplicity.

If each orbital is indicated in an energy diagram as a square (), we can show relative energies in a chart such as Figure 9. If this drawing represented a ravine with the energy levels as ledges onto which stones could come to rest only in numbers equal to the squares for orbitals, then pushing stones into the ravine would cause the stones to lose their potential energy as they dropped to the lowest potential level available to them. Much the same is true for electrons.