This chapter, in many ways, sets the stage for a large part of what follows in the course. You have been introduced to basic ideas about atoms in Chapter 2, but now it is necessary to go into much more detail to better understand the nature of the atom and the electrons which give each element its unique properties.

Scientists have learned much of what they know today about the atom by examining how atoms interact with electromagnetic radiation - an alternating series of electric and magnetic fields which propagate through space at c, the speed of light (3.00 x 10⁸ m/s). One of the easiest ways to view electromagnetic radiation (but not the only way, as we'll see later) is as waves. There are some important differences between electromagnetic waves and more familiar waves like those on water or sound waves:

• all electromagnetic radiation, irregardless of wavelength, moves at the same speed in a vacuum - the speed of light.
• EM radiation needs no medium to exist. It can travel through a vacuum. On the other hand, water waves need water as the medium; sound waves need air or some other medium to exist. There can be no sound in a vacuum.

The number of crests (or troughs) which pass a given point per second is called the frequency and is symbolized by the Greek letter nu. There are two things which determine the frequency of a wave: the speed of the wave (higher speeds increase the frequency) and the wavelength of the wave (shorter wavelengths increase the frequency). But since all EM waves move at the same speed, the only thing which affects the frequency is the wavelength:

There are an infinite number of wavelengths possible for electromagnetic radiation. The entire collection of EM radiation is called the electromagnetic spectrum. For classification purposes, the EM spectrum is broken down into regions shown in Fig. 7.2 on page 289. There is a region at higher energy than gamma rays which isn't shown in the figure - this region is known as cosmic rays and represents the most energetic form of EM radiation (shortest wavelength). Notice what a small portion of the entire spectrum is made up of visible light. This portion is expanded in the lower part of the figure. You are responsible for knowing the relative order of energies for various regions of the spectrum (including cosmic rays), and the relative order for the colors of the visible spectrum, but you do not need to memorize the wavelength range for each region.

The German physicist Max Planck showed that EM radiation could only be emitted or absorbed by matter is discrete packets of energy called quanta (singular: quantum). He was able to show that the energy of a quantum is directly proportional to the frequency of the radiation:

(Equation 1)

where the proportionality constant is known as Planck's constant and has the value 6.6262 x 10^-34 J-sec. Planck's work led Einstein to propose that light was particulate and exists as a collection of particles called photons, which have zero rest mass but non-zero relativistic mass. The mass of a photon moving at the speed of light is given by the expression

(Equation 2)

Since h and c are constants, we see that the mass of a photon is inversely proportional to the wavelength. This equation illustrates something very important: light has both particle (hence we can talk about a photon's mass) and wave (hence we can discuss the wavelength) properties. This is known as the wave-particle duality of light. In fact, a more general concept is the wave-particle duality of matter -- matter can exhibit both particle and wave properties. The French physicist Louis de Broglie used this idea to calculate the wavelength of a particle by the expression

(Equation 3)

where m is the mass of the particle and v is its velocity. Since h is so small, the wavelengths of macroscopically sized particles are so small that they can't be measured. For particles with a very small mass, however, the wavelength can easily be measured. When an electron microscope is used to achieve very high magnifications, the electrons are exhibiting wave properties in much the same fashion as the light in an ordinary light microscope.

Atomic Spectra
An atom can either absorb or emit electromagnetic radiation (from now on, electromagnetic radiation will be referred to as "light".) When an atom absorbs light, it does so by absorbing photons. For single atoms, each photon causes the energy of an electron in the atom to change from one level to another (higher) level. The electron is said to be "promoted" to a higher level. For molecules, the absorption of light may result in a transition to a higher vibrational or rotational level. When electrons "fall" from a higher energy level to a lower-energy level, light (in the form of a single photon) is released. The energy of this photon is exactly equal to the energy difference between the two levels involved in the transition.

The absorption and emission spectra of atoms are discontinuous, or line, spectra. This means that only certain absorption lines or emission lines show up, and that these lines have specific energies. This must mean that the energies between electronic levels in atoms (and vibrational and rotational levels in molecules) must be fixed at certain values, or quantized. Since the energy difference is either absorbed as a photon or emitted as a photon, the wavelength of the photon emitted or absorbed can be calculated from the equation

(Equation 4)

Conversely, the energy difference delta E can be calculated if the frequency or wavelength of the photon is known.

The Bohr Model of the Atom
The simplest atom, the hydrogen atom, is sometimes described in terms of the Bohr model of the atom. According to this model,

1. The electron moves in circular orbits around the nucleus (Sommerfield later modified this to elliptical orbits)
2. The electron is only allowed to be in certain orbits. In other words, its angular momentum is quantized, viz.

(Equation 5)

where L is the angular momentum and n is an integer from 1 to infinity. From the ideas of the Bohr model, we can derive expressions for the electron's energy and radius in various shells, or energy levels, defined by the value of n:

The expression for the electron's energy is:

(Equation 6)

where m and e are the mass and charge of the electron, respectively, and h is Planck's constant. This expression looks complicated, but notice that everything on the right-hand side is a constant except n. Thus, we can write

(Equation 7)

The proportionality constant is R_H (a form of the Rydberg constant, with the value 2.178 x 10^-18 J.) The inverse proportionality means that the energies of the shells become less and less negative as n increases. Furthermore, it is not a linear decrease, because of the n² term.

The energy of the electron in the first five shells is therefore

Notice that the energy differences between adjacent shells continually decreases, meaning that the lines in the spectrum get closer and closer together (see Figs. 7.6(b) and 7.8(c).) It is important to remember the sign convention for electron energies:

1. A negative energy for the electron means that the electron is bound in a shell. The farther a shell gets from the nucleus, the more the energy of the electron approaches zero.
2. An energy of zero for the electron means that the electron is infinitely far from the nucleus (shell with n=infinity) and not moving.
3. A positive energy for an electron means that the electron is unbound and moving with a kinetic energy of 1/2 mv² .

The expression from the Bohr model for the radius of the electron's orbit is given by

(Equation 8)

Again, when all constants are collected together on the right-hand side, we have the expression

(Equation 9)

where a₀ is known as the first Bohr radius and has the value 0.529 angstroms (recall that an angstrom is 10^-10 meter). The first Bohr radius is, of course, the radius of hydrogen's electron in the K shell. The L-shell orbit has a radius of four times as much, or about 2.12 angstroms. The M-shell orbit is nine times a₀ , or 4.76 angstroms. Note that the radius of a shell gets geometrically larger as the shell is farther from the nucleus.

Keep in mind that the energies of shells get geometrically closer together as we move to shells farther from the nucleus, but the radii of the shells get geometrically farther apart.

Since we have the expressions for the energies of each shell, the energy differences between shells (and, consequently, the energy of the photon absorbed or emitted for the transition between shells) are given by the Rydberg Equation:

(Equation 10)

where R_H is the constant mentioned above (2.178 x 10^-18 J), and n_f and n_i represent the final and initial shells, respectively. Notice that if n_f is less than n_i , it means that the electron fell from a higher energy shell to a lower energy shell, and indeed the sign of delta E ends up being negative (energy released). If the wavelength of the photon released is wanted, simply ignore the negative sign on delta E and use Equation 4 above. (We already know that energy was released, and keeping the negative sign would give an impossible negative wavelength.)

The Quantum Mechanical Model of the Atom
The basis for this model is the idea that if electrons behave as waves, then their behavior can be described by applying wave equations to the electrons. The Austrian scientist Erwin Schroedinger developed the Schroedinger Wave Equation, which relates a wave function for the electron to its energy and it approximate position in space. When the wave equation is solved, there are several acceptable solutions, and certain numbers, with a very well-defined range of values, provide these valid solutions. These numbers are called quantum numbers. The four quantum numbers, and a description of what they represent in the atom, are:

1. n, the principal quantum number. This quantum number represents the main shell, analogous to the value n in the Bohr Model. It determines the approximate distance of the electron from the nucleus. For the hydrogen atom and any one-electron ion, n completely determines the energy of the electron. n is allowed to have any positive value from 1 to infinity.
2. l, the angular momentum quantum number (sometimes called the azimuthal quantum number) represents the subshell, or the shape of the electron cloud. The allowed values for l start with zero and go up to a maximum of (n-1). The type of subshell is determined by the value of the quantum number l (that's the letter "el", not the number one):

   Value of l	Name of subshell
   0	s
   1	p
   2	d
   3	f
   4	g
   etc.…	alphabetic after f

3. m_l , the magnetic quantum number, represents the type of orbital, or the orientation of the electron cloud with respect to a particular direction. An orbital is the region of space where an electron is likely to be found. The allowed values for m_l are -l,-(l-1),-(l-2),-(l-n)…0,…1,2,3,…l. For example, the allowed values of m_l for the d subshell (for which l=2, see table above) are -2,-1,0,1,2. This means that there are 5 orbitals in the d subshell. Every orbital can hold a maximum of two electrons.
4. s, the spin quantum number. This very loosely can be viewed as indicating the spin of an object with only two possible values: clockwise or counterclockwise. The problem with this approach, of course, is that the quantum numbers are derived from an approach which views the electrons as waves, not particles which are spinning. There are only two allowed values for the spin quantum number: +1/2 and -1/2.

Energies of Orbitals
As mentioned above, the energy of an electron in any shell or subshell of a one-electron atom (H) or ion (He⁺ , Li²⁺ , etc.) is completely determined by the quantum number n. This means that all orbitals in the n=2 (L) shell (i.e., 2s and all three 2p) have the same energy (are degenerate) in these species. Likewise, all of the orbitals of the n=3 (M) shell (i.e., 3s, the three 3p, and the five 3d) are degenerate.

In many-electron atoms, however, the subshells within a given shell have different energies as a result of electron-electron repulsion. For any given shell n, the order of subshell energies is always ns < np < nd < nf. For example, the order for the M shell is 3s < 3p < 3d. It is important to realize, however, that an orbital in a shell with higher n can be lower in energy than an orbital in a shell with lower n. For example, the 4s orbital has a lower energy than the 3d orbital because of two factors:

1. the energy of the n=4 shell is not too different from the energy of the n=3 shell. Recall that the energies of shells become closer and closer together as n increases.
2. the s subshell can penetrate closer to the nucleus than the d subshell, thus feeling a higher effective nuclear charge, resulting in lower energy (more stability). The order of decreasing penetrating ability is

The order of increasing orbital energies for multielectron atoms is:

Electron Configurations
The configuration of an atom is a specification of how many electrons occupy each type of orbital. To obtain the configuration, assume that electrons fill the orbitals in the order shown above, until the appropriate number of electrons have been used. There are three important exceptions to the order above that you are expected to know (there are many more, but it is pointless to memorize every one; the three mentioned here are commonly-encountered atoms):

• ₂₄Cr: 1s²2s²2p⁶3s²3p⁶4s¹3d⁵ (instead of 1s²2s²2p⁶3s²3p⁶4s²3d⁴)

• ₂₉Cu: 1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰ (instead of 1s²2s²2p⁶3s²3p⁶4s²3d⁹)

(Notice that the configurations above can be abbreviated as [Ar]4s¹3d⁵ and [Ar]4s¹3d¹⁰, respectively.)

• ₄₇Ag: [Kr]5s¹4d¹⁰ (instead of [Kr]5s²4d⁹)

Valence Electrons
By definition, valence electrons are those electrons in an atom which have the largest value for the quantum number n. Thus ₂₀Ca, with a configuration of 1s²2s²2p⁶3s²3p⁶4s², has two valence electrons (the two 4s electrons), and ₃₅Br, with a configuration of 1s²2s²2p⁶3s²3p⁶4s²3d¹⁰4p⁵, has seven valence electrons (the two 4s and the five 4p electrons).

For representative elements (s and p block elements), the number of valence electrons is always the (Roman numeral) group number. For example, sulfur is in group VIA and has six valence electrons.

The Periodic Table
The periodic table is arranged according to increasing atomic number, and this arrangement gives rise to a tabular format with the following features:

1. Horizontal arrangements (called periods) contain elements with the same value for the largest principal quantum number, n. For example, the first period, consisting of H and He, represents the filling of the K shell. The fourth period, consisting of the elements ₁₉K through ₃₆Kr, represents elements with valence electrons in the n=4 shell.
2. Vertical arrangements (called groups or families) consist of elements with the same number of electrons in the same types of subshells, and therefore these elements behave the same chemically. Some general configurations for classes of elements include the following:

Periodic Trends
The periodic table can be used to predict general trends in many chemical and physical properties. The trends you are expected to know are these:

1. Atomic radius: The atomic radius increases from top to bottom down a group and decreases from left to right across a period.
2. Ionization Potential: The first ionization potential decreases from top to bottom down a group and increases from left to right across a period, with a few exceptions.
3. Electron Affinity: The first electron affinity generally becomes more exothermic (electron gain is more favored) from left to right across a given period, with a few exceptions. There is generally no well-defined trend for electron affinities down a group.

Last modified October 29, 1997