Archived Lecture Notes #1 - Atomic and Electronic Structure

3.091 - Introduction to Solid State Chemistry Lecture Notes No. 1 ATOMIC AND ELECTRONIC STRUCTURE ************************************************** Sources for Further Reading: 1. 2. 3. Davies, D.A., Waves, Atoms and Solids, Longman, 1978. DeKock, R.L., Chemical Structure and Bonding, Benjamin, 1980. Brown, T.L., Chemistry, The Central Science, 5th Ed., Prentice Hall, 1991. ************************************************** 1. ATOMS The familiar model of an atom is that of a small nucleus composed of protons and neutrons surrounded by rapidly moving electrons. Typically, the atomic diameter is on the order of l0–10 m while that of the nucleus is on the order of 10–15 m. Protons and neutrons have about the same mass (1.00728 and 1.00867 amu respectively) and each is about 1800 times as heavy as an electron. A neutron is electrically neutral, but a proton has a positive charge (+1.6 x 10–19 coulomb*) which is exactly the opposite of the negative charge of an electron. In a neutral atom, the number of electrons around the nucleus equals the number of protons in the nucleus. The number of protons in the nucleus (the “atomic number”, Z) characterizes a chemical element. Atoms of a given element all have the same number of protons, yet may have different masses. The atomic mass number of an atom, A, is given by A = Z + N, where N is the number of neutrons in the nucleus. Since an element is characterized solely by Z, it follows that atoms of a given chemical element may have a varying number of neutrons. Subspecies of chemical elements with the same Z but differing N and A are called isotopes. The atomic weight of an element is the weighted average of the atomic masses of the various naturally occurring isotopes of the element, and the atomic weight scale is based on a value of exactly 12, after the carbon isotope that has an atomic mass number of 12. 2. NUCLEI The nucleus of an atom weighs less than the sum of the weights of its isolated component particles. The difference between the actual mass and that of the ____________________ *Generally adopted International Unit System. LN–1 components is called the mass defect. The mass defect, Δm, is related to the binding energy within the nucleus, ΔE (in Joules), through Einstein’s equation: ΔE = Δm c2 where c is the velocity of light (in m.s–1) and Δm is the mass defect (in kg). The nuclear forces which bind protons and neutrons together are strong, and the binding energy per nuclear particle (nucleon) amounts to about 1.4 x 10–12 Joules. The greatest nuclear binding energy is found in nuclei of medium atomic number (such as Fe) where N is approximately equal to Z. For nuclei of larger atomic number, such as uranium, N is about equal to 1.5 Z, and the binding energy per nucleon is less. As a consequence of this decreased nuclear stability, some isotopes (of uranium, for example) are unstable. That is, if the uranium isotope, 92U235 (Z = 92, A = 235), is bombarded with neutrons, the following reaction can take place: 92U 235 + n → 39Y94 + 53I140 + 2n Here the reaction products are smaller nuclei of increased stability. (In the above convention, subscripts indicate the atomic number and the superscripts the mass number.) Notice that one incident neutron generates fission products including two neutrons - the basis for chain reactions in nuclear reactors and nuclear explosions. Heavy nuclei (even light nuclei) which have an unfavorable ratio in the number of protons and neutrons can spontaneously decay by the emission of α particles (helium ions) or β particles (electrons). These nuclei are referred to as radioactive. The rate at which the decay of such unstable nuclei takes place varies greatly and is indicated by the half-life of the material. In one half-life period, half of the unstable nuclei will have emitted radiation and thus will have changed their character (atomic number). In two half-life periods, only 1/4 of the nuclei will have survived. In three half-lives, only 1/8 of the original nuclei remain, etc. For example, the half-life of gamma-emitting “radio” cobalt, 27Co60 (used for X-ray therapy), is 5.3 years, whereas that of radioactive 6C14 is 5700 years. (Much more radiation is emitted per second by a given number of Co60 atoms than by the same number of C14 atoms.) 3. EXTRA-NUCLEAR ELECTRONS IN ATOMS The first atomic theory in quantitative agreement with some experimentally determined facts was proposed in 1913 by Niels Bohr. He postulated (for atomic hydrogen) that: • The (extra-nuclear) electron can assume only distinct (quantized) energy levels or states. • In such energy levels, the electrons in motion will not radiate (loose) energy; on changing energy levels, radiation equivalent to the energy difference between the 2 LN–1 levels is involved. Erad = E2 – E1 = hν • The stable states of the atom involve motion of the electrons in circular orbits. • The angular momentum of electrons in orbit (mvr) is an integral number of h/2π units: mvr � n h 2� •Newtonian mechanics applies to orbiting electrons. A pictoral view of the Bohr atomic model visualizes electrons orbiting at a velocity (v) in well-defined spherical orbitals of radius (r) around the nucleus. The angular momentum of the orbiting electrons can, as stated above, only assume certain values of n x h/2π where n (called the principal quantum number) can assume any positive integer value, i.e.1, 2, 3, 4, etc. to infinity (fig. 1). (emission of radiation (return of electron from excited to less excited or ground state) excited electron-state of short lifetime (v) E3 E2 E1 nucleus (e+) r3 r2 ro proton (p) rotating electron in groundstate (e­) electron orbital radius (r) n=1 n=2 n=3 (absorption of radiation with excitation of electron to excited state - of short lifetime) Figure 1 The Hydrogen Atom According to Bohr. The Hydrogen Atom Assuming coulombic interaction and the applicability of Newtonian mechanics, the following values may be directly obtained for the hydrogen atom: 3 LN–1 • The radius of an electron orbit (spherical) in hydrogen is given by: n 2h 2� o (n = 1, 2, 3, 4, etc.) �me 2 rn = n2 x constant = n2(0.529 x 10–10) m rn � It can be seen that the smallest electron orbit in hydrogen - the stable orbit - is given for n = 1 at ro = 0.529 Å. It is also seen that r increases with n2 (r = n2ro). • The energy of an orbiting electron characterized by the principal quantum number (n) is given as: 4 � � 12 x constant [Joule] E n � � 12 x me 2 2 n 8h � o n This relationship indicates that electron orbits with increasing principal quantum number (n) assume decreasing negative values (1/n2) with the limiting value (n → ∞) being zero. For convenience, it is frequently customary to express electronic energies in terms of wave numbers (ν) with the units of [m–1]. This conversion may readily be made since E = hν. Thus: 4 [m �1] � � 1 � � 12 x me 3 � n 8h c� 2o Upon substituting numerical values for all the constants, we obtain: � � � 10973500 � � 12 x 10973500 [m �1] n2 n (where n is again the principal quantum number which may assume the values 1, 2, 3, 4, etc. The value of 10973500 is referred to as the Rydberg constant (R; also RH and R∞) to honor the man who obtained its value from spectroscopic studies prior to the establishment of the Bohr model.) The energy changes associated with electronic transitions and the accompanying emission of radiation (or absorption of radiation) may simply be calculated as 4 LN–1 follows: Emission � �E � (E n 1 � E n2) � h� Here n2 is the principal quantum number of the outer orbit and n1 is that of the inner orbit. In terms of ν, the wave number, we obtain for emission of radiation: �� � � �� 10973500�� � � ��� 10973500 � �� � n 21 n 22 � � �� �� or � � �E � 10973500 � 12 � 12� x hc �n2 n1� � � �E � R� 12 � 12� x hc �n2 n1� � � �E � 2.1798 x 10 �18 � 12 � 12� J �n2 n1� �E � E n2 � E n 1 Absorption � �E � E n 2 � E n1 � � � � R� 12 � 12� �n1 n2� Electronic transitions (fig. 2) are conveniently represented in so-called term schemes. Multi-Electron Systems The limitations of the Bohr atom became apparent soon after its establishment. For example, the uranium atom would require 92 electronic orbits and it would have to assume a size which is incompatible with the experimental facts. The first modifications of the Bohr model were made by Sommerfeld, who introduced additional quantum conditions: Taking into account that individual sharp spectral lines split into several lines (of slightly differing λ) if light emission takes place in a magnetic field (Zeeman effect), the following picture emerged from the Bohr atom: electronic spherical orbits (shells) 5 LN–1 Balmer series of transitions terminating at n=2 Paschen series of transitions terminating at n=3 Lyman series of transitions terminating at n=1 n=5 E4 - r4=16ao=8.464Å n=4 E3 r3=9ao=4.761Å n=3 E2 n=2 r2=4ao=2.116Å E1 n=1 + ao=0.529 Å En= -R.1/n2 → n=∞ n=5 n=4 Paschen series (I.R.) n=3 Balmer series (visible) n=2 E∞= 0 E5= -R.1/25 E4= -R.1/16 E3= -R.1/9 E2= -R.1/4 Lyman series (ultraviolet) n=1 ΔE ΔE ΔE ΔE E1= -R 5→1 4→1 3→1 2→1 Figure 2 Electron Transitions and Related Optical Spectra in Atomic Hydrogen 6 LN–1 may have suborbitals (orbits) which are elliptical of varying eccentricity - controlled by an additional quantum number, the orbital quantum number (l). The eccentric orbitals may assume only particular spatial orientations controlled by the magnetic number (m), in an external magnetic field. The observed magnetic behavior of the elements required an additional quantization (Uhlenbeck) - the introduction of the spin quantum number (s). According to the Bohr–Sommerfeld quantum theory, electrons move about the nucleus of an atom in well-defined orbits, each of which is characterized by four numbers, called quantum numbers. An electron moving in an orbit close to the nucleus has a larger (negative) energy than one in an outer orbit. Energy is therefore necessary to move an electron from an inner to an outer orbit (with smaller negative energy). Conversely, energy is released (as radiation) during the transition of an electron from an outer to an inner orbit. The (electromagnetic) radiation released consists of discrete entities or quanta, which are called photons. The energy of emitted photons is given by the difference in energy between the two orbits involved and is related to the frequency, ν, or the wavelength, λ, of the electromagnetic radiation by: E photon � E 2 � E 1 � h� � h(c��) (c = λν) where h is called the Planck’s constant, E2 and E1 are the energies of the outer and inner orbits, respectively, and c is the velocity of light. Through (spectroscopic) observations of the frequencies or wavelengths of radiation emitted by an excited atom (atom with an excited electron), the differences of electron energy levels can be determined. Figure 1 shows such energy levels obtained from measurements of the wavelengths of radiation emitted by atomic hydrogen excited electrically in a gas discharge tube. To specify the spatial location and energies of electrons in an atom requires the use of four quantum numbers. The “principal quantum number”, (n), determines the size of the shell in which a particular electron orbits around the nucleus; it also largely determines its energy. The value of n is restricted to integral values: n = 1, 2, 3 and so forth. A value of n = 1 signifies that the electron exists in the lowest energy state and its orbit is in the innermost allowed shell, as close to the nucleus as possible. Successively higher energy states are represented by n = 2, 3, etc. It should thus be clear that the principal quantum number of the outermost orbiting electron determines, in principle, the size of a given atom. For convenience, letters are frequently used to characterize the electronic shells of n = 1, 2, 3, 4, etc. In letter notation: n = 1 = K shell n = 2 = L shell n = 3 = M shell etc. 7 LN–1 The second quantum number, (l), is referred to as the “orbital” or “azimuthal quantum number”. It specifies the angular momentum of orbiting electrons and, to a minor extent, their energy. “l” can only assume integral values ranging from l = 0 to l = (n–1) (where n is the principal quantum number). Accordingly, for n = 1 (the K shell), l can only assume a value of 0. For n = 2, either l = 0 or l = 1 is possible, etc. For convenience again, letters are used to specify the electronic state (orbit) in a given shell (characterized by n) corresponding to the second quantum number: l l l l = = = = 0 1 2 3 = = = = s (orbit) p (orbit) d (orbit) f (orbit) An electron with l = 0 is in an s–quantum state (in short, it is an s-electron), one with l = 1 is in a p-state, etc. Thus, an electron with a principal quantum number n = 3 and an orbital quantum number l = 1 is called a 3p electron (the electron in the M shell is in a p orbital). The third quantum number, (m), called the “magnetic quantum number”, controls the number of allowed spatial orientations (“degeneracy”) of each orbit characterized by l in a given shell [characterized by (n)]. (Degenerate states are of identical energy. The “degeneracy” disappears in the presence of a magnetic field where different spatial orientation of orbits assume different energy values.) The total number of allowed orbital orientations for any orbital characterized by l is (2l + 1), corresponding to m values of l, (l – 1), (l – 2), ..., 0, –1, –2, ..., –l. An electronic state with l = 0 necessarily has only m = 0 and thus has no directional orientation in space; it is a spherical orbit. On the other hand, a p-state (l = 1) allows m values of –1, 0, +1. The resulting three possible orientations are perpendicular to one another (as shown in fig. 4). The first three quantum numbers (n, l and m) define atomic electron orbitals. They are related respectively to the size, shape, and spatial orientation of the orbital. The fourth quantum number, s, (called the “electron spin quantum number”), can be interpreted as determining for an orbiting electron the direction of electronic spin around its own axis. “s” can assume values of +1/2 (parallel spin) and –1/2 (anti-parallel spin). For notational purposes, a positive s is conveniently represented as ↑ and a negative s as ↓. We may now summarize our findings about the electronic states in atoms. Any electron in an atom is defined by four quantum numbers, viz.: • The principal quantum number (n), which may have any positive integral value except zero. (In practice, because of the instability of heavy nuclei, atoms containing electrons with n values greater than 7 are unknown.) 8 LN–1 • The orbital or azimuthal quantum number (l), which is an integer and may have any positive value less than n, including zero. 0�l