The Dawn of Quantum Theory. At the end of the 19th century, physics (hence the theory of chemistry) was considered nearly complete. The wave behavior of electromagnetic radiation (light) was thought to be well understood in terms of the theory of James Clerk Maxwell, and a picture of the particulate atom was slowly emerging. Only a few persistent but (seemingly) small problems remained to be cleared up. These have rather esoteric names: the problem of black body radiation; the photoelectric phenomenon; and the discrete emission and absorption spectra of atoms. Physicists thought these problems would ultimately be solved within the so-called classical (i.e., pre-Quantum) framework. Ultimately, however, this proved impossible. Only by introducing the very strange and completely unfamiliar idea of quantized energy were they able to provide explanations for these rogue experiments. In doing so, they changed forever not only the way that we think about the behavior of matter at the atomic scale, but the way we think about measurement. In this text, it will not be appropriate for us to discuss the details of the explanations of the black body problem and the photoelectric effect. Suffice it to say that the idea of quantized energy was first introduced by Max Planck in 1900 in his explanation of the black body problem; and the idea of light bundles that we call photons was proposed by Einstein in 1905 to explain the photoelectric effect.
2-1 The Nature of Light. Before exploring in detail the so-called line spectra of atoms and explaining more fully what we mean by phrases like "quantized energy," we will briefly discuss the modern view of electromagnetic radiation--light--that has grown from the work of Maxwell, Planck and Einstein. Light has no mass, and is therefore not a form of matter. It is pure energy. It travels very rapidly through space at a speed of 3.0 * 108 m/sec. This speed is so important that it is given a special symbol, c. According to the Theory of Relativity, the speed of light is the fastest possible speed, attainable only by massless things. Nothing can travel faster than this. We speak of light as if it consists of waves, somewhat like water waves. We assign to it a wavelength, l (lambda), and a frequency, n (nu), such that the product of the wavelength (a distance) and the frequency (a number of wavelengths per second that pass a given point) is the speed of light, c.
In fact, some behaviors of light are consistent with this picture. For example, light diffracts (passes through two neighboring holes in a screen and interferes with itself). This is a property usually associated with waves. However, other behaviors of light suggest that it is in some sense a particle. For example, shining light on the surface of a metal causes electrons of the metal to be ejected (this is the photoelectric effect). The way in which electrons are ejected as a function of the frequency of the light suggests that each electron is ejected by being struck by a light packet. These light packets are called photons. The energy of a photon is directly proportional to its frequency, n:
The proportionality constant, h, called Planck's constant, has the value 6.6 * 10-34 J-sec, where J stands for the energy unit, Joule (a Joule is the same as 1 kg-m2/s2). It carries Planck's name because he first introduced it in his explanation of black body radiation. This is another extremely important number in science, which we will encounter repeatedly. Apparently, then, light has both wave and particle characteristics, showing one face in some circumstances, the other face in other circumstances. Equation 2-1-2 explicitly recognizes the wave/particle duality of light by connecting energy, a particle-type quantity, with frequency, a wave quantity. The fact that light (and, as we shall see, electrons) shows both particle and wave behavior is not a problem. It simply means that our simplistic particle and wave ideas, which result from observing macroscopic (large scale) phenomena in the world around us, are inadequate for describing the behavior of very small things.
Our modern view of light is as an electromagnetic oscillation in space. This view was developed by Maxwell in the 1860s and we still hold it today. Light waves are considered to be composed of sinusoidally oscillating, mutually perpendicular electric and magnetic fields. The frequency of the oscillation of both fields is the frequency of the light. An electromagnetic oscillation is pictured in Figure 2-1. Light, or electromagnetic radiation, exhibits a huge range of frequency (hence wavelength), from high energy (1020 s-1, called gamma rays) to low energy (108 s-1, radio waves). Light that is visible to the eye occupies a very narrow slice of this range, as shown in Figure 2-2. It is impossible for us to see most electromagnetic radiation! Light having a frequency somewhat less than we can see is called infrared (IR) radiation. That having frequency somewhat higher is ultraviolet (UV).
2-2 Atomic Spectra. It was noticed as early as 1855 by Bunsen and Kirchoff that when atoms of a particular element, say hydrogen, are energized by heating or by electrical discharge, they emit (give off, or produce) light of a characteristic color. In the case of hydrogen, the light is pale magenta; in the case of neon, it is orange; and so on. Further, when this light is collimated (that is, gathered into a narrow beam by passage through a slit) and then passed through a prism, it is found that only certain wavelengths of light emerge from the prism, with wide regions of darkness at wavelengths between the specifically emitted values. The collection of emitted lines is called the emission spectrum of the element. (It is also called a line spectrum, because the emitted beams take on the shape of the slit through which they are passed.) Figure 2-3 shows the line spectrum of helium. Most light sources (for example, the sun and stars, flames, and so on) produce a continuous spectrum of wavelengths when passed through a prism, just like the rainbow produced by passage of sunlight through mist or rain. That is, all wavelengths in the visible region of the spectrum are produced, with none missing. The discrete line spectra of atoms were therefore considered unusual, and could not be readily explained.
The discrete emission spectra of certain atoms meshed nicely with an observation that had been made in 1814 by Fraunhofer. Using a high quality prism, he discovered that the spectrum of light from the sun, although essentially continuous, contains a number of dark (black) lines representing wavelengths where no light is seen. The positions of these dark lines matched exactly the positions of lines in the emission spectra of a number of elements, notably hydrogen, helium, and sodium. The eventual interpretation of the Fraunhofer lines is that they result from absorption of light at the dark wavelengths by elements present in the outer portion of the gaseous cloud of the sun. The Fraunhofer lines constitute the absorption spectrum of these elements. The absorption and emission spectra of an element are complementary. The absorption spectrum shows dark lines against a bright background. The dark lines are at wavelengths where light is absorbed, and thus not seen. The emission spectrum shows bright lines against a dark background. The positions of the lines in the two types of spectra coincide exactly. The Fraunhofer lines in the spectrum of the sun are shown in Figure 2-4.
Line spectra were studied for many years without adequate explanation. A number of scientists focussed on the spectrum of hydrogen, which appeared the simplest. The series of lines falling in the visible region, called the Balmer series, is shown in Figure 2-5. The series is so named for Jacob Balmer, who found that the reciprocal wavelengths of the lines obey equation 2-2-1.
RH is a constant with value 1.09678 * 10-2 nm-1. Note that reciprocal wavelength is closely related to frequency, hence energy, via equations 2-1-1 and 2-1-2. Balmer was unable to explain the meaning of the integers occuring in the equation. Nonetheless, their occurence, and the very simple form of equation 2-2-1, are remarkable. By recording the emission spectrum of hydrogen on light sensitive film, scientists were able to "see" not only the visible emission lines, but also emission lines in the ultraviolet and infrared regions of the electromagnetic spectrum. Thus between 1906 and 1924, four additional series of lines were discovered. All of the observed spectral series of the hydrogen atom were found to obey a generalized version of 2-2-1 called the Rydberg equation, given in 2-2-2:
The unified fit of the hydrogen spectrum provided by equation 2-2-2 suggested a regular internal structure to the atom that is changed in some way by interaction with light. Nothing was known about this internal structure in 1885, when Balmer proposed his equation. In fact, the knowledge that atoms are composed of still smaller and more fundamental units of matter was not known. The origin of the emitted wavelengths of light within the atom was therefore not understood.
The first definite knowledge of the substructure of the atom emerged from the experiments of JJ Thomson in 1897. Using a device known as a cathode ray tube, Thomson demonstrated that the "cathode ray" consists of a beam of negatively charged particles that emerge from the metal cathode of the tube under an applied voltage. He called these particles electrons. By observing the manner in which the beam deflected when subjected to magnetic and electric fields of known strengths, Thomson was able to calculate a value of about 108 for the charge-to-mass ratio of the electron. The modern value is given in equation 2-2-3:
Subsequent experiments by Robert Millikan between 1908 and 1917 provided the value 1.6 * 10-19 coulombs for the charge, leading to a mass of 9.11 * 10-28 g.
Subsequent to Thomson's work, the atom was viewed as a uniform distribution of electrons embedded in a uniform distribution of positive charge, required to be present to ensure the electrical neutrality of the atom. However, in 1911 Rutherford published the so-called nuclear model of the atom, based on the results of experiments in which he and his students bombarded thin gold foil with alpha particles, which are the nuclei of helium atoms. The results of these experiments were interpreted by Rutherford as indicating that the positive charge of the atom was concentrated in a tiny and extremely dense region at the center of the atom, which he called the nucleus. He proposed that virtually all of the mass of the atom resides in the nucleus. The electrons were then proposed to orbit the nucleus, much as the planets orbit the sun. This is essentially the picture of the atom presented in Chapter 1.
2-3 Bohr's Theory of the Hydrogen Atom. In 1913, Niels Bohr, a young Danish physicist, provided the first successful theory of the internal structure of the hydrogen atom, based on the quantum ideas of Planck and Einstein. He conceived his theory by speculating about the origin and significance of atomic line spectra. According to Bohr's interpretation of atomic spectra, the absorbed and emitted photons provide a direct measure of the gaps between discrete (quantized) energy levels in the atom. To arrive at a quantitative expression for the energies of these so-called energy levels in the hydrogen atom, Bohr made several assumptions, some quite arbitrary (i.e., the assumptions were justified only because they ultimately allowed an explanation of line spectra).
The key assumption made by Bohr is the first one, which restricts the electron to certain discrete locations within the atom, while forbidding all others. In proposing this, Bohr broke with classical electromagnetic theory, according to which an electron orbiting a positive nucleus would continuously radiate its energy away, eventually spiralling in to the nucleus. With the first assumption, Bohr superimposed Planck's quantum ideas on the electron in the atom. In restricting the momentum of the electron, he restricted in turn not only the radii of the orbits, but also their energies. Assuming that the attractive force exerted on the electron by the nucleus, given by Coulomb's Law as explained in Chapter 1, was balanced by the centrifugal force due to the electron's orbital motion, Bohr was able to show that the orbit energies are given by equation 2-3-3:
In 2-3-3, e is the electron charge, with value 1.602 * 10-19 coulombs; and e is a constant of nature called the vacuum permittivity, with value 8.85 * 10-12 J-1C2M-1. We will not be concerned with the significance of the vacuum permittivity, other than to recognize that it causes the units to work out properly. The important feature of 2-3-3 is that it shows a direct dependence of the electron energy on the quantum number n, which may have only integer values. Thus only certain energies can occur! The atom is quantized.
Equation 2-3-3 is seemingly strange in one respect--it states that the energy of the electron in the hydrogen atom is negative! What do we take this to mean? It seems reasonable to think of the energy of the electron as associated somehow with its motion. However, as we will discuss fully in a later chapter, there are two types of energy. The first is kinetic energy, energy of motion, given by the expression below.
Here m is the mass and v the velocity of a particle or object (this equation shows the relationship given earlier between the energy unit, Joule, and units of mass and velocity). The second is potential energy, energy of position. The electron in the hydrogen atom has energy of both types. It is of course not possible for kinetic energy to be negative, because the concept of negative motion makes no sense. The minimum possible value for kinetic energy is zero, corresponding to no motion. However, potential energy is often negative. The familiar example of potential energy is gravitational potential energy, associated with the position of some object in the earth's gravitational field. We normally take ground level to be the zero point of potential energy. That is, when an object, say a basketball, is sitting on the surface of the earth, its potential energy is taken to be zero. This is done for our convenience. When we raise the basketball to a higher position, we do work on it, and its potential energy increases (becomes positive). However, if the basketball were to fall into a hole in the ground, its potential energy would decrease--it would become negative--because we would have to do work on the basketball to raise it out of the hole and bring its potential energy back to zero. Thus potential energy can be either positive or negative, depending on the position of the system with respect to the agreed upon zero-point position. The electron has potential energy by virtue of the force of attraction to the positive nucleus according to Coulomb's Law:
The potential energy is electrical, rather than gravitational, but it is mathematically similar. The closer the electron gets to the nucleus, the lower its potential energy becomes, just as an object in the earth's gravitational field decreases in potential energy the closer it gets to the center of the earth. By agreement among scientists, the potential energy of the electron-proton system is taken to be zero when the force of attraction is zero. According to equation 1-1-1, this will happen when r = infinity; that is, when the electron and proton are infinitely separated. When the electron is pulled away from the nucleus to an infinite distance, the potential energy of the electron is zero. At any closer distance than this, the potential energy is negative. As it turns out, the potential energy of the electron in the hydrogen atom has larger magnitude than the kinetic energy. Since the potential energy is negative, the total energy, given by equation 2-3-3, is negative. (To give a simple example, if the potential energy were -5 kJ and the kinetic energy were 2.5 kJ, the total energy would be -2.5 kJ.). The fact that the energy of the electron in the H atom is negative is thus not problematic. It is a simple consequence of our definition of potential energy for the electron-nucleus system. We will find that electron energies in atoms are always negative, due to the potential energy contribution to the total energy.
The triumph of the Bohr theory results from applying 2-3-3 to electron transitions--processes in which the electron moves from one allowed energy level to another, with the accompanying absorption or emission of a photon. Suppose that the electron is initially in the orbit with n = ni. It absorbs a photon of energy hn and moves out to the orbit with n = nf. According to Bohr, there must be an exact match of the photon energy and the difference in energy between orbits ni and nf, according to equation 2-3-2. Thus
Substituting c/l for n and dividing both sides by hc gives the expression for 1/l in 2-3-5:
This equation is identical in form to equation 2-2-2, which describes the dependence of the wavelengths of the emission lines of hydrogen on integers of unknown significance. When Bohr calculated the value of his constant, C, from known values of the mass and charge of the electron, the vacuum permittivity, Planck's constant, and the speed of light, he obtained the value 1.09678 * 10-2 nm-1. This agrees so closely with the experimentally determined value of the Rydberg constant, RH, that Bohr's theory of hydrogen was immediately and enthusiastically accepted, despite its flagrant break with the classical view.
Figure 2-6 is a plot of the allowed energy levels of the hydrogen atom according to equation 2-3-3. Such a plot is usually called an energy level diagram. Transitions between levels are shown as arrows that point from the initial to the final level. The lines in the emission spectrum of hydrogen are due to photons that are emitted by the atom when the electron moves from a higher to a lower level. Arrows corresponding to several lines in the Balmer emission series, in which the electron moves from levels with n > 3 to the n = 2 level, are shown. The dark lines in the absorption spectrum of hydrogen are due to the absence of photons that are absorbed by the atom, causing the electron to move from a lower to a higher level. Arrows corresponding to the Balmer absorption series are also shown. Note that the difference in energy between two levels is fixed by equation 2-3-4. There is thus an exact correspondence in energy of the emission and absorption lines.
After his initial success with the hydrogen atom, Bohr attempted to apply his theory to the helium atom, which has two electrons orbiting a nucleus consisting of 2 protons and 2 neutrons. Despite years of effort and numerous modifications to his theory, Bohr was unable to extend his success beyond the hydrogen atom. Nonetheless, Bohr had taken the first step in the development of the modern theory of matter and energy on the microscopic scale--Quantum Mechanics. The second breakthrough step was made by Louis de Broglie in 1924.
2-4 Wave Particle Duality--Modern Quantum Theory. In perhaps one of the most famous and significant Ph.D theses in all of science, Louis de Broglie laid the foundation for further advances in the quantum ideas. He suggested that matter, like light, might exhibit a duality of wave and particle characteristics. In other words, he suggested that the electron, normally considered a particle, might behave sometimes as a wave! His suggestion was purely intuitive. There was absolutely no experimental evidence at the time to warrant it. Nonetheless, he was able to put it on a quantitative basis by making an analogy to a photon, for which the relationship between wavelength and momentum, p, is given in equation 2-4-1:
de Broglie suggested that the electron is governed by a similar equation. Substituting the product of mass and velocity for momentum, he obtained equation 2-4-2:
He then proceeded to show that Bohr's theory of the hydrogen atom was a consequence of this equation! His reasoning was (probably) something as follows. The allowed orbits of the hydrogen atom are those in which an integral number of electron wavelengths exactly fits the circumference of the orbit. In other words, de Broglie viewed the electron in the hydrogen atom as a standing wave, analogous to the standing waves set up in a guitar string when it is plucked. Just as the number of wavelengths must exactly fit the length of the guitar string, which is tied down at both ends, so must the electron wave fit the orbit. If this were not true, the electron wave would interfere with itself, and the electron would disappear. In quantitative terms, de Broglie's assumption is stated as in 2-4-3:
Here n is an integer and r is the orbit radius. Substituting for the electron wavelength from 2-4-2 gives 2-4-4:
This is exactly Bohr's quantized momentum assumption, from which all the results of Bohr's theory follow. It is important to realize that although de Broglie's result for the hydrogen atom is the same as Bohr's, de Broglie's approach is more satisfying, because the quantum number n arises quite naturally from the requirement that the electron be a standing wave that just fits the circumference of the orbit. This contrasts markedly with the necessity for Bohr to superimpose the quantum condition in ad hoc fashion on an otherwise quite classical picture of the hydrogen atom.
The ideas of de Broglie were not taken completely seriously at first, because the idea of the electron as a wave was difficult to accept for many scientists. Further, there was no experimental support for his ideas. However, in 1927, Davisson and Germer showed that when beams of electrons are made to impinge on a crystalline material, a diffraction pattern results. Diffraction is a property of waves, not of particles. Further, the electron wavelength extracted by Davisson and Germer from their results was just the value expected from de Broglie's relationship, equation 2-4-2. The wave nature of the electron was thus firmly established in experiment.
It is interesting and revealing to cast equation 2-4-4 in somewhat different form. Reciprocating both sides of 2-4-4 gives 2-4-5:
We recognize that 1/l is the number of wave cycles that occur per unit length (m or cm). For this reason, it is usually called the wavenumber, and is given the symbol k. Wavenumber is similar in nature to frequency, which is the number of cycles that occur per unit time. Replacing 1/l with k and multiplying both sides by h gives 2-4-6.
Clearly this equation is very similar in form to equation 2-1-2. Like that equation, it relates a particle-type quantity (momentum) to a wave-type quantity (wavenumber) through the quantum constant h. The de Broglie relation expressed as 2-4-6 thus also explicitly recognizes the wave-particle duality of matter, just as 2-1-2 recognizes that of light.
The Final Step--the Schrodinger Wave Equation. The final stone in the foundation of modern quantum mechanics was laid by Erwin Schrodinger in 1926. He realized that if the electron were indeed a wave, as de Broglie claimed, it should be governed by a mathematical relationship analogous to the equations that describe ordinary waves at the macroscopic scale. He set out to find such a relationship. The result of his effort, the Schrodinger equation, is the basis for our understanding of the quantum characteristics not only of the hydrogen atom, but of all atoms and molecules:
This intimidating-looking equation is called a second-order differential equation. The same type of equation governs three-dimensional wave motions of all types. We will not be at all concerned with the manner in which this equation was obtained, nor with how it is used or solved. That will be left for higher level texts that are concerned with specialized areas of chemistry. However, we will be interested, at least qualitatively, in the solutions to this equation, particularly those for the hydrogen atom. It is to the hydrogen atom that Schrodinger first applied his new equation, with resounding success. We now look at his results, in a simplified way.
We must first discuss the various terms and symbols in equation 2-4-7. The symbols h and m have the same meaning as in Bohr's equation: they are Planck's quantum constant and the mass of the electron. The symbol, ¶, is a partial derivative symbol, used in calculus, and will not concern us now. V is the potential energy of the electron in the electric field of the nucleus. We discussed this in talking about the Bohr Theory above. E is the total energy of the electron (potential plus kinetic); and y is the wave function of the electron. We are now in a position to summarize the major results of Schrodinger's solution for the hydrogen atom.
Table 2-1 summarizes the main shell/subshell/orbital hierarchy of the hydrogen atom.
| n (main shell) | l (subshell) | ml (orbital) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0 | 0 |
| 2 | 1 | -1, 0, 1 |
| 3 | 0 | 0 |
| 3 | 1 | -1, 0, 1 |
| 3 | 2 | -2, -1, 0 1, 2 |
| 4 | 0 | 0 |
| 4 | 1 | -1, 0, 1 |
| 4 | 2 | -2, -1, 0, 1, 2 |
| 4 | 3 | -3, -2, -1, 0, 1, 2, 3 |
A visual presentation of the location-probability of the electron in a particular orbital of the atom can be obtained by constructing plots of the mathematical expression for the orbital wave function. Two types of plots are typically made. The first shows the manner in which the location probability varies with distance of the electron from the nucleus of the hydrogen atom. These are called radial plots. Radial plots for the 1s, 2s, 2p, 3s, and 3d orbitals of the hydrogen atom are shown in Figure 2-7a. Several features of these plots should be noted.
First, there is no single particular distance at which the electron is located with respect to the nucleus in any orbital. This is in marked contrast to the Bohr theory, in which electrons are constrained to fixed circular orbits of definite radius. According to the modern view, the electron in, say, the 2s orbital can be found at any distance from the nucleus, although the probability is highest of finding it at a distance of 2.116 * 10-8 m. This corresponds to the maximum in the curve. Second, as the value of n increases, the maximum in the plot moves to larger distance, meaning that the electron tends to be further from the nucleus in shells with larger n (higher energy). This is qualitatively similar to the results of Bohr's Theory, in which the radii of allowed orbits became larger with increasing n. Third, the radial plot for the 2s orbital shows that there is zero probability of finding the electron at 1.06 * 10-8 m from the nucleus, because the radial plot goes to zero here. This region of zero probability is called a node. It is analogous to the nodes in a standing wave in a guitar string, which are locations at which the string does not move. Nodes are characteristic of wave patterns; their occurrence in the wave functions for the electron in the H atom reinforces the de Broglie picture of the electron as a wave.
The second type of plot shows the 3-dimensional shape of the orbital. These are usually referred to as angular plots. Angular plots for the 1s, 2s, 2p, and 3d orbitals are shown in Figure 2-7b1 and 2-7b2. Again, there are a number of important features of these plots. First, the angular plots for s orbitals are spherical in shape. This implies that an electron in such an orbital can be found with equal probability in any direction from the nucleus. In contrast, p orbitals are dumbbell-shaped. Each of the sphere-like portions of a p orbital is called a lobe. p orbitals are therefore said to have 2 lobes. The + and - signs in the orbital lobes refer to the sign of the wavefunction, y, in that lobe; these signs do NOT refer to electrical charge. There are three p orbitals in each main shell with n > 2, oriented along the three axes of the Cartesian coordinate system. For this reason, the three p orbitals are designated px, py, and pz. Each main shell with n > 3 has five d orbitals (remember, there is a d orbital for each possible ml value in the l = 2 subshell: -2, -1, 0, 1, 2). Four of these are very similar in appearance, each with four lobes. The four lobes alternate in the sign of the wave function. The fifth d orbital has lobes along the + and - directions of the z axis, with a donut of negative sign in the x-y plane, encircling the z axis. The subscripts on the orbitals indicate in shorthand fashion where the lobes of the orbitals are. Thus the dxy orbital is in the xy plane, with lobes halfway between the x and y axes. The dx2-y2 orbital is also in the xy plane, but has its lobes along the x and y axes. For now, you should familiarize yourself with the shapes of s and p-type orbitals.
The Heisenberg Uncertainty Principle. It may puzzle you that we have changed rather abruptly from discussing exact electron orbits in the Bohr theory to rather fuzzy probabilities in the modern quantum theory. Indeed, one of the remarkable things that we have learned from quantum theory is that we can not know the exact location of the electron in the hydrogen atom (or of any electron in any atom). We can know at best only the probability that the electron will be found in a particular region. This lack of exact knowledge is a consequence of one of the strangest and most fundamental principles of quantum mechanics, the Uncertainty Principle, proposed by Werner Heisenberg in 1927. There are numerous ways to state this principle, one of which follows: it is not possible to simultaneously know exactly both the position and the momentum of an electron (or of any other small "particle"). As we have seen, the momentum of a particle-wave is the product of its mass and velocity. We may know the position exactly, but can then have no idea what the momentum is; similarly, if we know the momentum exactly, we can have no idea what the position is. This is difficult to accept because at the macroscopic scale, we of course are able to know both quantities at the same time. For example, when you drive your car to the grocery store, you know how fast you are going (momentum) and you know exactly where you are at each moment (position). Very small particles simply do not behave in the same way as large ones. This is not something that can be explained; it simply is.
Just as Heisenberg's principle links momentum and position in uncertainty, it also links energy and time. In more mathematical terms, Heisenberg's Uncertainty relations are expressed as in equations 2-4-8.
In words, these twin relationships say that the product of the uncertainty (indicated by D) in momentum (or energy) and the uncertainty in position (time) must exceed a fundamental minimum value given by h/2p. This minimum value is so small that we do not observe the Uncertainty Principle working at the macroscopic scale. In the atomic/molecular realm, however, it is extremely important. (For those with an interest in the quantum view of nature, there is a third Heisenburg relationship involving the uncertainties in angular momentum, L, and angular position, w, written (DL)*(Dw) > h/2p. This relation is less-used than the two above and will not concern us.)
As we have stated earlier, atoms are not possible in terms of classicial physics, according to which the electrons should be attracted into the nucleus, radiating their energy away as they fall. There is no doubt whatsoever, though, that this does not happen. Electrons occupy regions of space called orbitals, with definite energies. The Uncertainty Principle allows us to understand the existence of atoms. If electrons were to spiral into the nucleus, losing all of their energy, then we would simultaneously know where they were (in the nucleus) and what their momentum is (zero). This would violate the Uncertainty Principle. The electron is a standing wave in the atom, with a definite wavelength, and consequently a definite momentum. The Uncertainty Principle therefore requires that we have no knowledge about where it is! In fact, the radial plots in Figure 2-7a show that the electron can be found with some probability at any distance (even an infinite one) from the nucleus; at any given instant we do not know where it is. The Uncertainty Principle is extremely important--it allows the existence of atoms!
With the Uncertainty Principle in mind, it is instructive to return to the radial plots for the orbitals of the hydrogen atom, shown in Figure 2-7a. Figure 2-8a shows an enlarged version of the radial plot for the 2s orbital. What does the plot tell us about the electron, and in particular about its distance from the nucleus? First, the radial plot has the visual appearance of a wave; it oscillates up and down, with clear peaks and troughs. This is certainly consistent with the de Broglie/Schrodinger view of the electron as a wave. In addition, it reveals the following specific information about where the electron is and is not:
This, then, is why we may speak only of the probability of finding the electron at some particular location in the atom: the radial probability curve forbids our knowing with certainty where it is. We can contrast this modern quantum view of where the electron is with the earlier Bohr view. Recall that Bohr postulated orbits of definite radius for the electron. A radial probability plot for the electron in the n = 2 Bohr orbit would then appear as in Figure 2-8b. According to Bohr, the probability of finding the electron is zero everywhere except at distance c. Thus the plot rises to probability = 1 at distance = c, and has the value zero everywhere else. It is interesting that the radius, c, of the Bohr orbit for n = 2 is exactly the same as the most probable distance predicted by Quantum Theory for the 2s orbital: 2.116 * 10-10 m.
2-5 Atoms with More than one Electron. Electron Spin and the Pauli Exclusion Principle. The line spectra of many atoms other than hydrogen were well-known by the time Schrodinger developed his equation. In general, these spectra were much more complex than that of hydrogen. Even the relatively simple helium atom, with only 2 electrons, demonstrated a more complex spectrum than hydrogen. It was understood that the complexity was due to the presence of more than one electron. It was also realized that, contrary to what might be expected, these electrons did not all occupy the lowest energy Bohr-type shell. If this were the case, then all atomic spectra would show the same simple pattern of line spacings (although not the same wavelengths) of the hydrogen spectrum. When Schrodinger presented his quantum mechanical view of the hydrogen atom, it was recognized that shells, subshells, and orbitals similar to those in hydrogen must exist in all atoms, and that electrons must occupy these according to certain rules. In 1925, Wolfgang Pauli proposed the most important of these rules, called the Exclusion Principle:
Prior to the discovery of the Schrodinger wave equation, improvement in the quality of instruments for measuring atomic spectra had shown that what had previously appeared to be single lines in the spectra of certain elements were in fact doublets--two closely spaced lines! In 1925 it was proposed that the doubling of lines was a consequence of a previously unsuspected type of electron motion, electron spin. The spin of the electron was proposed to be similar to the rotation of the earth on its north-south axis as it orbits the sun, as shown in Figure 2-9. (Note that this description of electron spin must be taken with a grain of salt; we have learned that we must think of the electron in the atom as a wave, not a particle. What does it mean for a wave to be "spinning on its axis"?) The quantum number associated with this motion was dubbed the electron spin quantum number, and was given the symbol ms. In order to explain the doubling of lines in atomic spectra, it was necessary to restrict this quantum number to only two values: +1/2 and -1/2. These two values are referred to as "up spin" and "down spin", respectively. This new quantum number is unusual in 2 respects. First, it can have only 2 values. Second, the allowed values are not integers. Instead, they are half integers.
The existence of the spin quantum number, with only 2 values, suggests an explanation for the Pauli Principle. Two electrons in the same orbital must have the same values of n, l, and ml. However, if they have opposite spins (one "up", the other "down"), then they will differ in at least 1 of the 4 quantum numbers that characterize them. A restatement of the Pauli Principle in these terms is as follows:
Electron Configurations. Armed with the Pauli Principle and the atomic orbitals from the Schrodinger treatment, we are in a position, for any atom, to specify the distribution of electrons among the orbitals. This distribution is called the electron configuration of the atom. We will take on faith (for the time being) several statements about the energies of shells, subshells, and orbitals:
| s subshell | 2 electrons |
| p | 6 electrons (3 orbitals, 2 per orbital) |
| d | 10 electrons (5 orbitals, 2 per orbital) |
| f | 14 electrons (7 orbitals, 2 per orbital) |
The order of subshell filling consistent with these statements is the following:
Note that there is some overlap of the high energy subshell of n = 3, 3d, with the low-energy subshell of n = 4, 4s. Similar overlaps occur more frequently as n increases.
The electron configurations are as follows. The number of electrons occupying a set of orbitals (a subshell) is indicated as a superscript:
For atoms with many electrons, it becomes tedious to write out the complete order of filling. It is customary in these cases to explicitly indicate only the electrons beyond the previous noble gas. The presence of the inner electrons is indicated by writing the noble gas symbol enclosed in brackets. Using this shorthand system, the electron configurations for Ca, Fe, and Bi become
You may be chagrined at the prospect of memorizing the order of filling given in 2-5-1. In fact, this is not necessary.
2-6 Periodic Properties. Electron Configurations from the Periodic Table. The periodic table was formulated long before the advent of quantum theory, but the ordering of elements was understood only on an experimental basis. Quantum Theory provides an explanation for the appearance of the periodic table in terms of the order of filling of orbitals. In fact, this order of filling may be read directly from the table. The long version of the periodic table is presented in color-coded form in Figure 2-10. We recognize correlations between its appearance and the order of filling given above:
We now show by example how to "read" the electron configuration of an element from the periodic table.
Now string these together in order to obtain the complete configuration. To obtain the configuration in shorthand notation, we need read only from the previous noble gas. Thus
You are certainly free to memorize the order of filling if you want to. However, with practice it is much quicker and easier to read the configuration from the periodic table.
There remains one matter with which we must deal before moving on. The electron configuration for nitrogen is given below.
Because there are three 2p orbitals, it is not clear from the configuration how the three electrons are to be distributed in them. There are several possiblities, a few of which are shown here. It is found experimentally that the last of these arrangements, with one electron in each of the 2p orbitals and all electron spins parallel, is preferred. Generally, when several electrons occupy a set of equal-energy (degenerate) orbitals, the most stable arrangement is the one with the maximum number of unpaired electrons. This statement is known as Hund's Rule. In accordance with Hund's Rule, the valence orbital occupations of carbon, oxygen, and iron are shown here.
Electron Shielding and Effective Nuclear Charge. We now return to an idea that we earlier stated as fact, without attempt at explanation. That is, that in an atom with several electrons, the energy ordering of subshells within a main shell is s < p (< d (< f)). The parentheses remind us that these subshells do not occur in shells with n < 3 or 4. We will try to develop an understanding of why this is true in terms of the radial plots for the 3s, 3p, and 3d subshells of the sodium atom shown qualitatively in Figure 2-11. The core electrons (in 1s, 2s, and 2p) are shown as a sphere that is inside the distance of maximum probability for the n = 3 shell. The sodium atom has a single electron in the n = 3 main shell, which must occupy one of the three subshells in Figure 2-11. Experimentally, it is found to occupy the 3s subshell, which we conclude must be more stable (lower energy) than the 3p and 3d subshells. Why? We can make a few observations about these plots that will hopefully tell us.
The penetration of the core electron cloud by the 3s electron is of course incomplete, so the 3s electron does not feel the full effect of the 11 positive charges in the nucleus. The core electrons shield the 3s electron to some extent from the nucleus by coming between them. The net positive charge felt by the 3s electron may be called the effective nuclear charge, Zeff. It is the difference between the number of protons in the nucleus (the atomic number, Z) and the average number of electrons between the 3s electron and the nucleus, which we symbolize S:
S, the so-called shielding factor, is related to but always less than the number of shielding electrons. It is less than the number of shielding electrons because the shielded electron penetrates the core to some extent. Calculation of the value of S for a particular atom is somewhat complex; consequently, we will not attempt it. Instead, we will discuss a quantity called the core charge, Zcore, which is an easily-calculated quantity that roughly parallels Zeff in value. The core charge of an atom is the difference between the number of protons in the nucleus and the number of core electrons. Core electrons are those in completely filled shells below the valence shell. The core charge therefore measures the net positive charge pulling on the valence electrons. The core charge is very simple to calculate. For example, the core charge for lithium is its atomic number (the number of protons) minus the number of electrons in filled shells (the atomic number of the preceding noble gas): Zcore(Li) = 3 - 2 = 1. Similarly, the core charges for carbon and fluorine are 6-2 = 4 and 9-2 = 7, respectively. Core charges for the period 3 elements are worked out below:
| Element | Z | Number of core electrons | Zcore |
|---|---|---|---|
| Na | 11 | 10 | 1 |
| Mg | 12 | 10 | 2 |
| Al | 13 | 10 | 3 |
| Si | 14 | 10 | 4 |
| P | 15 | 10 | 5 |
| S | 16 | 10 | 6 |
| Cl | 17 | 10 | 7 |
| Ar | 18 | 10 | 8 |
Note that the core charge for an element is equal to the number of valence electrons (which is the last digit of the group number). Two trends in Zcore are of importance to us. First, Zcore increases as we proceed left to right across a row of the periodic table, because the number of protons in the nucleus increases while the number of core electrons stays the same. Second, Zcore remains constant down a family of the periodic table, because the elements in a family all have the same number of valence electrons. These simple trends are the basis for the experimentally observed variations in atomic size, ionization energy, and other atomic properties discussed below.
Atomic Size. The molar volume for an element, in units of mL/mole can be calculated readily by dividing the atomic weight in g/mole by the density of a condensed phase (solid or liquid) of the element in g/mL. When molar volume is plotted as a function of atomic number, Figure 2-12 results. Examination of the plot shows two clear trends. First, molar volume tends to decrease from left to right across a period. Second, molar volume increases top to bottom down a family. If we make the reasonable assumption that there is a relationship between the molar volume and the size of an atom of the element, then the second trend is expected, but the first is counter-intuitive. Most of us would probably predict that atoms should get larger as the number of electrons increases across a period. But they apparently do not. The results of x-ray diffraction experiments have given us fairly reliable values for the radii of atoms of many elements. Some representative values are presented in Table 2-2. It is clear from the data that atomic radii do indeed decrease left to right across a row of the periodic table.
| H | 132 | ||||||
| Li180 | Be--- | B--- | C168 | N155 | O150 | F155 | Ne 160 |
| Na230 | Mg170 | Al--- | Si210 | P185 | S180 | Cl180 | Ar190 |
| K 280 | Ca--- | Se190 | Br190 | Kr 200 | |||
| Xe 220 |
This is quite readily understood in terms of the trend in Zcore discussed above. Across a row, the added electrons enter the same main shell while experiencing an ever increasing pull from the nucleus. The result is a steady shrinkage of the electron cloud. The elements of a family (column) of the periodic table have the same number of electrons in the shell of largest n. For example, the elements of group 2 all have 2 electrons in the outermost main shell. Thus the core charge is the same for all members of a family. The vertical trend in atomic size therefore reflects the increasing distance from the nucleus that accompanies an increase in the value of the principle quantum number, n.
Before ending this discussion of periodic trends in atomic size, it is useful to make a rough calculation of the size of an atom from the experimentally measured atomic volume. Mercury, a liquid metal, has a density of 13.6 g/mL and atomic weight 200.6 g/mole. The molar volume of mercury is thus
We divide by Avogadro's number to convert this to an atomic volume:
To approximate the atomic radius, we take the cube root of the atomic volume:
This very simple calculation provides a very approximate experimental value for the size of the atom: 10-8 cm, or 100 pm.
Ionization Energy, I. The energy required to remove the least tightly held electron of an atom in the gas phase is called the first ionization energy, I1, of the atom. The process of electron removal is shown for the Li atom in equation 2-6-2:
The electron removed in this case is the single 2s valence electron. Figure 2-13 shows a plot of the first ionization energy versus atomic number for the first 3 periods of the periodic table. The following trends are recognizable from the plot.
The force exerted on the outer electron by the core charge that it feels is given by Coulombs Law:
Here Zcore is the core charge felt by the electron, e is the charge of the outermost electron, and r is the average distance of this electron from the nucleus. It is clear from equation 2-6-3 that the force, and therefore the required input energy, increases as Zcore increases. We have seen above that Zcore increases steadily from left to right across a period. Thus F and I1 are expected to increase also. The overall increasing trend is readily understood. Similarly, the force decreases as r increases. Since we have seen that r increases with increasing n down a family, the decrease in I1 down the family is also understandable in terms of 2-6-3. For the time being we will not concern ourselves with the "jogs" in the left-to-right trend in I1. We will provide an explanation for them at the point that they become important.
It is of course possible to remove more than one electron from an atom. Sequential removal of the first 3 electrons from an atom, E, is represented in equations 2-6-4 to 2-6-6:
The energies required to remove the second and third electrons are called the second ionization energy, I2 and the third ionization energy, I3, respectively. No matter what particular atom E represents, it is invariably true that I3 > I2 > I1 > 0. Of course, it requires the input of energy to remove the outermost electron of the atom because the electron must be pulled away from the effective positive nuclear charge that it feels. When the first electron is removed, the remaining electrons are pulled in more tightly to the nucleus because there is now a net positive charge of 1 unit on the atom. The second electron is therefore harder to remove because it must be pulled away from the positive ion, E+. For the same reason, the third electron is even harder to remove than the second. The first 3 ionization energies for the elements through Ar are given in Table 2-3.
| Element | EA2(I-1) | EA1(I0) | I1 | I2 | I3 |
|---|---|---|---|---|---|
| H | --- | --- | 1312 | ||
| He | --- | --- | 2372 | 5250 | |
| Li | --- | 59.6 | 520 | 7298 | 11815 |
| Be | --- | -241 | 899 | 1757 | 14848 |
| B | --- | 26.7 | 801 | 2427 | 3660 |
| C | --- | 121.85 | 1086 | 2353 | 4621 |
| N | -801 | <0 | 1402 | 2856 | 4578 |
| O | -780 | 140.98 | 1314 | 3388 | 5300 |
| F | --- | 328.0 | 1681 | 3374 | 6051 |
| Ne | --- | -28.9 | 2081 | 3952 | 6122 |
| Na | --- | 52.87 | 496 | 4562 | 6912 |
| Mg | --- | -231 | 738 | 1451 | 7732 |
| Al | --- | 42.55 | 578 | 1817 | 2745 |
| Si | --- | 133.6 | 786 | 1577 | 3231 |
| P | -463 | 72.0 | 1012 | 1903 | 2912 |
| S | -590 | 200.4 | 1000 | 2251 | 3361 |
| Cl | --- | 349 | 1251 | 2297 | 3822 |
| Ar | --- | -34.7 | 1520 | 2666 | 3931 |
Study of the table provides an explanation for a statement presented in Chapter 1: elements in Groups 1, 2, and 13 of the periodic table form cations with positive charge equal to the last digit of the group number. Another way to say this is that in forming compounds, these elements lose all of the electrons in the outer main shell, but do not lose any of the electrons in lower shells. Successive values of the ionization energies for these elements show why. For Mg, I2 is larger than I1, for reasons discussed above, but by a factor of only 2. Loss of two electrons depletes the outer main shell of the magnesium atom. If further electrons are to be removed, they must be taken from the next lower (n = 2) shell, which is substantially closer to the nucleus than the n = 3 shell. Consistent with this, I3 is more than 5 times larger than I2! The cost in energy to remove the third electron is too high. Mg therefore stops at the Mg2+ cation. The sodium atom has only one electron in the outer shell. Its removal requires the energy, I1, which is only 496 kJ/mole. However, removal of a second electron from Na+ requires disruption of the n = 2 shell. Roughly 10 times more energy is required to accomplish this than is required to remove the first. Again, the cost is too high, and sodium stops at Na+.
Electron Affinity, EA. The electron affinity is closely related to ionization energy. It provides a measure of the tendency of an atom to gain, rather than lose an electron. The first electron affinity, EA1, is defined as the energy required to remove an electron from the anion, E-, forming the neutral atom E:
This equation is written analogously to equations 2-6-4 to 2-6-6; that is, it shows loss of an electron by a species, forming a related species having one more positive (or one less negative) charge. Thus the first electron affinity is sometimes called the zeroth ionization energy, I0 (because a species of zero charge is formed). The second electron affinity (also I-1) is defined analogously:
As explained above, successive removal of electrons becomes progressively more and more difficult because the electron must be pulled away from a center of increasing positive charge. This is no less true of species that are initially negatively charged. Thus I-1 and I0 fit in the expected way into the series of ionization energies:
In contrast to ionization energies, however, electron affinities are not always positive. That is, for some species, E-, the electron comes off spontaneously. No energy need be put in to cause it to happen. Instead, energy is produced as the electron comes off. As Table 2-3 shows, EA1 is positive for some elements toward the right of the periodic table, but is in fact negative for many elements. EA2 is negative for all elements.
Just as the trend in values of ionization energies serves to rationalize the observed positive charges of cations from groups 1,2, and 13, electron affinities rationalize the complementary rule: that elements from groups 15-17 tend to form anions with a number of negative charges equal to (8 - group number). Another way to say this is that these elements can add electrons to the point of filling the current main shell, despite the fact that EA is assuredly negative for any electrons past the first one added; but they will not add more electrons than this, because these would have to occupy a new main shell, further from the nucleus, where the added electrons feel a greatly diminished attraction from the nucleus.
Supplement. Experimental Evidence for the Existence of Shells--Photoelectron Spectroscopy. As we will see in more detail in Chapter 4, spectroscopy is the study of the interaction of light and matter. Our understanding of atoms and molecules is largely attributable to spectroscopy. In this section, we briefly discuss Photoelectron Spectroscopy, PES, in which light is used to eject electrons from atoms (or molecules). By subtracting the kinetic energies of the ejected electrons from the energy of light used to eject them, the ionization energies of the electrons can be calculated:
By varying the frequency of light used, electrons from various shells of the atom can be removed. This gives a measure of the energies levels of the shells. This idea is illustrated schematically in Figure 2-14.
Table 2-4 shows ionization energies obtained by photoelectron spectroscopy for the first 10 elements of the periodic table, H through Ne. Let's see what we can make of these data. First, we see that H has only one ionization energy; we would expect this, because it has only one electron. Similarly, He has only one ionization energy, even though it has two electrons, but the intensity of the photoelectron signal is twice that for hydrogen. This indicates that there are twice as many electrons susceptible to ejection by a photon of light. A single ionization energy is understandable if the two electrons are just alike; that is, if they occupy the same subshell. This is consistent with our understanding of helium, for which we write the electron configuration 1s2. We also see that the single ionization energy for He is larger than that for hydrogen. This is perfectly consistent with its greater core charge, Zcore.
For Li, there are two distinct ionization energies with intensities in the ratio 1 to 2. The first ionization energy is of relatively small magnitude, corresponding to an electron that is easily removed. The second ionization energy is quite substantial in magnitude, corresponding to two electrons that are much more difficult to remove. This is in accord with the quantum picture, in which there is a single electron in the outer n=2 shell, and two electrons in the inner filled n=1 shell. The PES for Be gives similar information. However, for boron, three types of electrons are observed, in relative numbers 1, 2, and 2. There is a single electron that is relatively easily removed. There are then two electrons that are also relatively easily removed (about the same as the single electron of hydrogen), though more difficult than the first. Finally, there is a pair of electrons that is very tightly held by the nucleus of the boron atom. These results are consistent with a picture in which the the three easily ionized electrons are in two different subshells of the n = 2 main shell. The remaining two electrons are then deeply buried in the core shell with n = 1. Proceeding from B to Ne shows up to six electrons entering the second subshell of the n = 2 main shell. Figure 2-15 shows a plot of the energy levels of the ionized electrons against the atomic number of the first 18 elements (the 10 in Table 2-4 plus the elements of period 3). (The energies of the levels are the negatives of the ionization energies, which measure the amount of energy required to separate the electron from the nucleus. Recall that the energy of the electron is considered to be zero when it is completely separated from the atom; thus the energy of the electron in the atom before its removal must be negative.) The energy of a particular subshell as a function of atomic number is indicated by connecting the appropriate points with a curve. Several things are evident from the plot.
| Element | Ionization Energy of Electron, kJ/mole | Intensity |
|---|---|---|
| H | 1310 | 1 |
| He | 2370 | 2 |
| Li | 520 | 1 |
| Li | 6260 | 2 |
| Be | 900 | 2 |
| Be | 11500 | 2 |
| B | 800 | 1 |
| B | 1360 | 2 |
| B | 19300 | 2 |
| C | 1090 | 2 |
| C | 1720 | 2 |
| C | 28600 | 2 |
| N | 1400 | 3 |
| N | 2450 | 2 |
| N | 39600 | 2 |
| O | 1310 | 4 |
| O | 3040 | 2 |
| O | 52600 | 2 |
| F | 1680 | 5 |
| F | 3880 | 2 |
| F | 67200 | 2 |
| Ne | 2080 | 6 |
| Ne | 4680 | 2 |
| Ne | 84000 | 2 |
Spreadsheet Applications
2-1. The radial wave function for the 2s orbital of hydrogen, obtained by solution of the Schrodinger equation, is given below: