TOPICS
1.4 Successes in early quantum theory
We saw in Section 1.2 that electrons in an atom occupy a region of space around the nucleus. The importance of electrons in determining the properties of atoms, ions and molecules, including the bonding between or within them, means that we must have an understanding of the CHEMICAL AND THEORETICAL BACKGROUND
Box 1.1 Isotopes and allotropes
Do not confuse isotope and allotrope! Sulfur exhibits both iso-topes and allotropes. Isoiso-topes of sulfur (with percentage naturally occurring abundances) are 3216S (95.02%), 3316S (0.75%),3416S (4.21%),3616S (0.02%).
Allotropes of an element are different structural modifications
of that element. Allotropes of sulfur include cyclic structures, e.g.
S6(see below) and S8(Figure 1.1c), and Sx-chains of various lengths (polycatenasulfur).
Further examples of isotopes and allotropes appear throughout the book.
Fig. 1.1 Mass spectrometric traces for (a) atomic Ru and (b) molecular S8; the mass:charge ratio is m/z and in these traces z¼ 1. (c) The molecular structure of S8.
Chapter 1 . Successes in early quantum theory 3
electronic structures of each species. No adequate discussion of electronic structure is possible without reference to quantum theory and wave mechanics. In this and the next few sections, we review some of the crucial concepts. The treatment is mainly qualitative, and for greater detail and more rigorous derivations of mathematical relationships, the references at the end of Chapter 1 should be consulted.
The development of quantum theory took place in two stages. In the older theories (1900–1925), the electron was treated as a particle, and the achievements of greatest signif-icance to inorganic chemistry were the interpretation of atomic spectra and assignment of electronic configurations.
In more recent models, the electron is treated as a wave (hence the name wave mechanics) and the main successes in chemistry are the elucidation of the basis of stereochemistry and methods for calculating the properties of molecules (exact only for species involving light atoms).
Since all the results obtained by using the older quantum theory may also be obtained from wave mechanics, it may seem unnecessary to refer to the former; indeed, sophisticated treatments of theoretical chemistry seldom do. However, most chemists often find it easier and more convenient to con-sider the electron as a particle rather than a wave.
Some important successes of classical quantum theory
Historical discussions of the developments of quantum theory are dealt with adequately elsewhere, and so we focus only on some key points of classical quantum theory (in which the electron is considered to be a particle).
At low temperatures, the radiation emitted by a hot body is mainly of low energy and occurs in the infrared, but as the temperature increases, the radiation becomes successively dull red, bright red and white. Attempts to account for this observation failed until, in 1901, Planck suggested that energy could be absorbed or emitted only in quanta of mag-nitude E related to the frequency of the radiation, , by equation 1.1. The proportionality constant is h, the Planck constant (h¼ 6:626 1034J s).
E¼ h Units: E in J; in s1or Hz ð1:1Þ c¼ Units: in m; in s1or Hz ð1:2Þ The hertz, Hz, is the SI unit of frequency.
Since the frequency of radiation is related to the wave-length, , by equation 1.2, in which c is the speed of light in a vacuum (c¼ 2:998 108m s1), we can rewrite equation 1.1 in the form of equation 1.3 and relate the energy of radiation to its wavelength.
E¼hc
ð1:3Þ
On the basis of this relationship, Planck derived a relative intensity/wavelength/temperature relationship which was in good agreement with experimental data. This derivation is not straightforward and we shall not reproduce it here.
One of the most important applications of early quantum theory was the interpretation of the atomic spectrum of hydrogen on the basis of the Rutherford–Bohr model of the atom. When an electric discharge is passed through a sample of dihydrogen, the H2 molecules dissociate into atoms, and the electron in a particular excited H atom may be promoted to one of many high energy levels.
These states are transient and the electron falls back to a lower energy state, emitting energy as it does so. The consequence is the observation of spectral lines in the emission spectrum of hydrogen; the spectrum (a small part of which is shown in Figure 1.2) consists of groups of discrete lines corresponding to electronic transitions, each of discrete energy. As long ago as 1885, Balmer pointed out that the wavelengths of the spectral lines observed in the visible region of the atomic spectrum of hydrogen obeyed equation 1.4, in which R is the Rydberg constant for hydrogen, is the wavenumber in cm1, and n is an integer 3, 4, 5 . . . This series of spectral lines is known as the Balmer series.
Wavenumber¼ reciprocal of wavelength; convenient (non-SI) units are ‘reciprocal centimetres’, cm1
¼1
¼ R
1 221
n2
ð1:4Þ R¼ Rydberg constant for hydrogen
¼ 1:097 107m1¼ 1:097 105cm1
Other series of spectral lines occur in the ultraviolet (Lyman series) and infrared (Paschen, Brackett and Pfund series). All
Fig. 1.2 Part of the emission spectrum of atomic hydrogen. Groups of lines have particular names, e.g. Balmer and Lyman series.
lines in all the series obey the general expression given in equation 1.5 where n’ > n. For the Lyman series, n¼ 1, for the Balmer series, n¼ 2, and for the Paschen, Brackett and Pfund series, n¼ 3, 4 and 5 respectively. Figure 1.3 shows some of the allowed transitions of the Lyman and Balmer series in the emission spectrum of atomic H. Note the use of the word allowed; the transitions must obey selection rules, to which we return in Section 20.6.
¼1
¼ R
1 n2 1
n’2
ð1:5Þ
Bohr’s theory of the atomic spectrum of hydrogen
In 1913, Niels Bohr combined elements of quantum theory and classical physics in a treatment of the hydrogen atom.
He stated two postulates for an electron in an atom:
. Stationary statesexist in which the energy of the electron is constant; such states are characterized by circular orbits about the nucleus in which the electron has an angular momentum mvr given by equation 1.6. The integer, n, is the principal quantum number.
mvr¼ n
h 2
ð1:6Þ where m¼ mass of electron; v ¼ velocity of electron; r ¼ radius of the orbit; h¼ the Planck constant; h=2 may be written as h.
. Energy is absorbed or emitted only when an electron moves from one stationary state to another and the energy change is given by equation 1.7 where n1and n2 are the principal quantum numbers referring to the energy levels En1 and En2respectively.
E¼ En2 En1¼ h ð1:7Þ
If we apply the Bohr model to the H atom, the radius of each allowed circular orbit can be determined from equation 1.8.
The origin of this expression lies in the centrifugal force acting on the electron as it moves in its circular orbit; for the orbit to be maintained, the centrifugal force must equal the force of attraction between the negatively charged electron and the positively charged nucleus.
rn¼"0h2n2
mee2 ð1:8Þ
where "0¼ permittivity of a vacuum
¼ 8:854 1012F m1
h¼ Planck constant ¼ 6:626 1034J s n¼ 1; 2; 3 . . . describing a given orbit me¼ electron rest mass ¼ 9:109 1031kg
e¼ charge on an electron (elementary charge)
¼ 1:602 1019C
From equation 1.8, substitution of n¼ 1 gives a radius for the first orbit of the H atom of 5:293 1011m, or Fig. 1.3 Some of the transitions that make up the Lyman and Balmer series in the emission spectrum of atomic hydrogen.
Chapter 1 . Successes in early quantum theory 5
52.93 pm. This value is called the Bohr radius of the H atom and is given the symbol a0.
An increase in the principal quantum number from n¼ 1 to n¼ 1 has a special significance; it corresponds to the ionization of the atom (equation 1.9) and the ionization energy, IE, can be determined by combining equations 1.5 and 1.7, as shown in equation 1.10. Values of IEs are quoted per mole of atoms:
One mole of a substance contains the Avogadro number, L, of particles:
L¼ 6:022 1023mol1
HðgÞ "HþðgÞ þ e ð1:9Þ
IE¼ E1 E1¼hc
¼ hcR
1 12 1
12
ð1:10Þ
¼ 2:179 1018J
¼ 2:179 1018 6:022 1023J mol1
¼ 1:312 106J mol1
¼ 1312 kJ mol1
Although the SI unit of energy is the joule, ionization energies are often expressed in electron volts (eV) (1 eV ¼ 96:4853 96:5 kJ mol1).
Impressive as the success of the Bohr model was when applied to the H atom, extensive modifications were required to cope with species containing more than one electron; we shall not pursue this further here.