AtomicStructureReading

AtomicStructureReading - Some basic concepts l Fundamental...

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Unformatted text preview: Some basic concepts l Fundamental particles I Atomic number, mass number and isotopes I An overview of quantum theory I Orbitals of the hydrogen atom and quantum numbers I The multi-electron atom, the aufbau principle and electronic configurations l The periodic table Ionization energies and electron affinities I Lewis structures 1.1 Introduction Inorganic chemistry: it is not an isolated branch of chemistry If organic chemistry is considered to be the ‘chemistry of carbon’. then inorganic chemistry is the chemistry of all elements except carbon. In its broadest sense. this is true. but of course there are overlaps between branches of chemistry. A topical example is the chemistry of the fill/cr- enes (see Section 13.4) including C60 (see Figure 13.5) and C70; this was the subject of the award of the 1996 Nobel Prize in Chemistry to Professors Sir Harry Kroto. Richard Smalley and Robert Curl. An understanding of such molecules and related species called nanom/ms involves studies by organic. inorganic and physical chemists as well as by physicists and materials scientists. Inorganic chemistry is not simply the study of elements and compounds; it is also the study of physical principles. For example. in order to understand why some compounds are soluble in a given solvent and others are not. we apply laws of thermodynamics. If our aim is to propose details of a reaction mechanism. then a knowledge of reaction kinetics is needed. Overlap between physical and inorganic chemistry is also significant in the study of molecular structure. In the solid state, X-ray diffraction methods are routinely used to obtain pictures of the spatial arrangements of atoms in a Valence bond theory Fundamentals of molecular orbital theory The octet rule Electronegativity Dipole moments MO theory: heteronuclear diatomic molecules Isoelectronic molecules Molecular shape and the VSEPR model Geometrical isomerism molecule or molecular ion. To interpret the behaviour of molecules in solution, we use physical techniques such as nuclear magnetic resonance (NMR) spectroscopy; the equivalence or not of particular nuclei on a spectroscopic timescale may indicate whether a molecule is static or under- going a dynamic process (see .S'm'ri'on ,7./'/), In this text, we describe the results of such experiments but we will not. in general. discuss underlying theories; several texts which cover experimental details of such techniques are listed at the end of Chapter I. The aims of Chapter 7] In this chapter. we outline some concepts fundamental to an understanding of inorganic chemistry. We have assumed that readers are to some extent familiar with most of these concepts and our aim is to give a point of reference for review purposes. 1.2 Fundamental particles of an atom An mom is the smallest unit quantity of an element that is capable ofexistence, either alone or in chemical combination with other atoms of the same or another element. The . fundamental particles of which atoms are composed are the proton. elem-on and neutron. 2 Chapter 1 0 Some basic concepts Table 1.1 Properties of the proton. electron and neutron. Proton Electron Neutron Charge/C +1.602 x 10*[9 —1 .602 x 10"9 0 Charge number (relative charge) 1 —1 0 7 Rest mass/kg 1.673 >< 10—27 9.109 x 10‘31 1.675 X 10“7 Relative mass 1837 1 1839 A neutron and a proton have approximately the same mass and, relative to these, an electron has negligible mass (Table 1.1). The charge on a proton is positive and of equal mag- nitude, but opposite sign, to that on a negatively charged electron; a neutron has no charge. In an atom of any element, there are equal numbers of protons and electrons and so an atom is neutral. The nucleus of an atom consists of protons and (with the exception of protium; see Syriion 9.3) neutrons, and is positively charged; the nucleus of pro- tium consists of a single proton The electrons occupy a region of space around the nucleus. Nearly all the mass of an atom is concentrated in the nucleus, but the volume of the nucleus is only a tiny fraction of that of the atom; the radius of the nucleus is about 10715m while the atom itself is about 105 times larger than this. It follows that the density of the nucleus is enormous, more than 1012 times than of the metal Pb. Although chemists tend to consider the electron, proton and neutron as the fundamental (or elementary) particles of an atom, particle physicists would disagree, since their research shows the presence of yet smaller particles. if , isotopes-g, ‘ Nuclides, atomic number and mass number A nuclide is a particular type of atom and possesses a charac— teristic atomic number, Z , which is equal to the number of pro- tons in the nucleus; because the atom is electrically neutral, Z also equals the number of electrons. The mass number, A. of a nuclide is the number of protons and neutrons in the nucleus. A shorthand method of showing the atomic number and mass number ofa nuclide along with its symbol, E, is: Mass number——> A e‘g‘ zoNe E 5- Element symbol 10 Atomic number ——> Z Atomic number = Z : number of protons in the nucleus = number of electrons Mass number = A = number of protons + number of neutrons Number of neutrons : A — Z Relative atomic mass Since the electrons are of minute mass, the mass of an atom essentially depends upon the number of protons and neu- trons in the nucleus. As Table 1.1 shows, the mass of a single atom is a very small, non-integral number, and for convenience we adopt a system of relative atomic masses. We define the atomic mass unit as l / 12th of the mass of a lgC atom so that it has the value 1.660 X 10 ’27 kg. Relative atomic masses (A) are thus all stated relative to lEC:12.0000. The masses of the proton and neutron can be considered to be ~1u where u is the atomic mass unit (1 u x 1.660 x 10‘27 kg). Isotopes Nuclides of the same element possess the same number of protons and electrons but may have different mass numbers; the number of protons and electrons defines the element but the number of neutrons may vary. Nuclides of a particular element that differ in the number of neutrons and, therefore, their mass number, are called isotopes (see Appemlix 5). Isotopes of some elements occur naturally while others may be produced artificially. Elements that occur naturally with only one nuclide are monotopic and include phosphorus, ‘gP, and fluorine, “3F. Elements that exist as mixtures of isotopes include C (léC and QC) and o (‘30, 'go and 130). Since the atomic number is constant for a given element. isotopes are often distinguished only by stating the atomic masses, eg. 12C and 13C. Worked example 1.1 Relative atomic mass Calculate the value of AIr for naturally occurring chlorine if the distribution of isotopes is 75.77% $0 and 24.23% flCl. Accurate masses for 35C] and 37Cl are 34.97 and 36.97. The relative atomic mass of chlorine is the weighted mean of the mass numbers of the two isotopes: Relative atomic mass, 75.77 24.23 Ar ~ ( 100 X 34.97) +( '00 x 36.97) - 35.45 CHEMICAL AND THEORETICAL BACKGROUND Chapter 1 0 Successes in early quantum theory 3 Box 1.1 Isotopes and allotropes Do not confuse isotope and allotropel Sulfur exhibits both iso- topes and allotropes. Isotopes of sulfur (with percentage naturally occurring abundances) are ?%S (95.02%), (0.75%). 123 (4.21%), fgs 10.02%). Allotropes of an element are different structural modifications Self—study exercises 1. 1f.~1r for CI is 35.45. what is the ratio 01'35C1:37C| present in a sample of C1 atoms containing naturally occurring C1? IAm‘. 3.17:1] 2. Calculate the Value of Ar for naturally occurring Cu if the distribution of isotopes is 69.2".) “Cu and 30.8"» “Cu; accurate masses are 62.93 and 64.93. Mm. 63.5] 3. Why in question 2 is it adequate to write “Cu rather than 63C ., 29 u. 4. Calculate Ar for naturally occurring Mg if the isotope distribu- tion is 78.99% “Mg. 10.00% 25Mg and 11.01% 2“Mg; accurate masses are 23.99. 24.99 and 25.98. IAm‘. 24.31] Isotopes can be separated by mass .8’])((‘ll'0l71€ll“1‘ and Figure 1.1a shows the isotopic distribution in naturally occurring Ru. Compare this plot (in which the most abundant 100 ” 75" 50‘ 25‘l Relative abundance Relative abundance 96 98 100 102 104 Mass number of that element. Allotropes of sulfur include cyclic structures, e.g. S6 (see below) and 83 (Figure He), and Sx-chains of various lengths (polycatenasulfur). Further examples of isotopes and allotropes appear throughout the book. 1% Part of the helical chain of SW isotope is set to 100) with the values listed in Appendix 5. Figure l.lb shows a mass spectrometric trace for molecular St. the structure of which is shown in Figure l.lc; five peaks are observed due to combinations of the isotopes of sulfur. (See problem 1.3 at the end of this chapter.) Isotopes of an element have the same atomic number. Z. but different atomic masses. 1.4 Successes in early quantum theory We saw in Section 1.2 that electrons in an atom occupy a region 01‘ 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 256 258 260 Mass number (1)) (C) (a) m Fig. 1.1 Mass spectrometric traces for (a) atomic Ru and (b) molecular 53; the mass:charge ratio is m]; and in these traces : : l. (c) The molecular structure of S8. 4 Chapter 1 ' Some basic concepts electronic structures ofeach species. No adequate discussion of electronic structure is possible without reference to quantum theory and ware III(’('/ltlllft‘.\‘. 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 ofChapter I should be consulted. The development of quantum theory took place in two stages. In the older theories (I900 I925). the electron was treated as a particle. and the achievements ofgreatest signif— icance to inorganic chemistry were the interpretation of atomic spectra and assignment of electronic conligurations. In more recent models. the electron is treated as a wave (hence the name wary I)I(’(‘/I(Illf(‘.\') and the main successes in chemistry are the elucidation of the basis ofstereochemistry and methods for calculating the properties of molecules (exact ()II/_t‘ 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 (‘/ll.\'.\'f('tl/ 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 .315 related to the frequency of the radiation. 1/. by equation 1.]. The proportionality constant is It. the Planck constant (It : (3.626 x It) “l .I s). Ali : ln/ Units: 15 in .I: l/ in s l or Hz (1.1) (' 4- /\1/ Units: /\ in m: 1/ in s l or Hz (1.2) The hertz. Hz. is the SI Ltnit of freqtiency. I‘IJIIIICI seltcs Since the frequency of radiation is related to the wave- length. /\. by equation 1.2. in which (‘ is the speed of light in a vacuum (t' : 2.998 X 10“. ms 1). we can rewrite equation 1.1 in the form of equation 1.3 and relate the energy of radiation to its wavelength. Alf : ’1‘! (1.3) ,\ 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 RutherfordiBohr model of the atom. When an electric discharge is passed through a sample of dihydrogen. the H3 molecules dissociate into atoms. and the electron in a particular excited H atom may be [)I'OIHUIt’l/ 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 (If.\'('l'(’lt’ 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. 17 is the wavenumbcr in cm I. and n is an integer 3. 4. This series of spectral lines is known as the Bu/ntvr .vt'rim. Wavenumber : reciprocal of wavelength; convenient (non- SI) units are ‘reciprocal centimetres‘. cm I 1 l /:—:R —7——1 ' A (2— n—) l l R : Rydberg constant for hydrogen : 1.097 X itfm ‘ : 1.097 X 105cm Other series of spectral lines occur in the ultraviolet (Lyman series) and infrared (Paschen. Brackett and Pfund series). All VISIBLE ULTRAVIOLET ~> Increasing energy Fig. 1.2 Part of the emission spectrum of atomic hydrogen. Groups of lines have particular names. e.g. Balmer and Lyman series. Fig. 1.3 : lines in 1 equation the Balm Pfund se some of series in of the VI rules, to Bohr's hydrog In 1913, and clas: He state: 0 Statit is con abou‘ mom the p. mvr = when radiu be w1 \ 11 O N 11 ,..i l,_\i‘.1.msct:c:t / l; *8” n=1 '— Chapter 1 - Successes in early quantum theory 5 N 11 N n : oo : the continuum Fig. 1.3 Some of the transitions that make up the Lyman and Balmer series in the emission spectrum of atomic hydrogen. lines in all the series obey the general expression given in equation 1.5 where n’ > n. For the Lyman series, n : 1. for the Balmer series, 11 : 2. and for the Paschen. Brackett and Pfund series. 17 : 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. , 1 1 1 V:X:R<H_Z_ITZ> (1.5) 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: o Stationary stares exist 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 mor given by equation 1.6. The integer. n. is the principal quantum number. l1 nun 711<TF> (1.6) where "1 : mass of electron; 11 : velocity of electron; r : radius of the orbit; lz : the Planck constant; li/27r may be written as l7. 0 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 111 and n2 are the principal quantum numbers referring to the energy levels E,“ and En: respectively. AE : E”: i E,“ : lzu (1.7) 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. 7 EUIZZI/F rn : (1.8) 7 Trim 6” permittivity of a vacuum 1| where 50 : 8.854 x1042 Fm'l l7 : Planck constant : 6.626 X 10’34 Js n : 1,27 3 . . . describing a given orbit my 2 electron rest mass : 9.109 X 10’31 kg 6 : charge on an electron (elementary charge) : 1.602 X 104°C From equation 1.8. substitution of n :1 gives a radius for the first orbit of the H atom of 5.293 X IO’Hm. or 6 Chapter 1 0 Some basic concepts 53.03 pm, This \alue is called the Bohr rat/I'm ol‘ the H atom and is giyen the symbol a”, .\n increase in the principal quantum number l‘rom II c l to H \ has a special signilicance: it corresponds to the ioni/ation ol‘ the atom (equation 1.9) and the ionization energy. 11-). can be determined by combining equations l5 and I.7. as shown in equation |.l(l. Values ol‘ Ilis are quoted per NIH/U u/ ulmm: ()ne mole ol‘ a substance contains the Ay'ogadro number. 1.. ol‘ particles: 1. ~ 6.032 *» ltlzl mol llth >l'l (gt ~ e (|.*)l (1.1m \ 5 t‘ / l ‘l: E k A l tal— l /l_ ml V 3.179 l() I\i \' 1m s in " (m2: -- Wilma 1.311 |()(‘.l mol 7 1312 kl mol Although the SI unit ol‘ energy is the joulc. ioni/ation energies are ol‘ten expressed in electron \olts (e\") (le\" : 90.4%} t 96.5 k.l mol I). linprcssiye as the success ol‘ the Bohr model was \\ hen applied to the H atom. extensiye modilications “etc required to cope \\ith species containing more than one electron; \\e shall not pursue this t'urther here. 1.5 An introduction to wave mechanics The wave—nature of electrons The quantum theory ol‘ radiation introduced by M ax Planck and Albert liinstcin implies a particle theory ol‘ light. in addition to the nave theory ol‘ light required by the phenomena ol‘ interference and dill‘raction. in l924. Louis de Broglie argued that it" light were composed ol‘ particles and yet shoyyed \\aye—like properties. the same should be true ot‘ electrons and other particles. This phenomenon is rel‘erred to as n'urt' pul'lie/t' t/im/t'lr. The de Broglie relation— ship (equation l.ll) combines the concepts ol‘ classical mechanics with the idea ol‘ \yaye-like properties by show- ing that a particle \\ith momentum Mr (I): : mass and 1' ' \eloeity ol‘ the particle) possesses an associated \\‘ay'e ol‘yyayelength /\. II . ,\ 7. I)”. \\ here It is the Planck constant (1.] 1) An important physical obscry'ation \\hich is a consequence ol‘ the de Broglie relationship is that electrons accelerated to a \‘elocity ot‘ (1 - It)“ in s l (by a potential ol‘ l()()\7) haye an associated \\ayelength ot‘ alltlpm and such electrons are dill‘racted as they pass through a crystal. This phenom- enon is the basis ol‘ electron dill‘raetion techniques used to determine structures ol‘ehemical compounds (sec l. The uncertainty principle 11‘ an electron has \\ay‘e—like properties. there is an important and dillicult consequence: it becomes impossible to knoyy exactly both the momentum and position ol‘ the electron a! l/It‘ yume [Initial in Iilllt‘. This is a statement ot‘ l leisenberg‘s mut'rmimy principle. In order to get around this problem. rather than trying to deline its e\act position and momen- tum. \\e use the pru/w/a/irr ti/ Hut/mg I/lt’ electron in a giy'cn yolume ol‘ space, The probability ol‘ linding an electron at a giy en point in space is determined l‘rom the t‘unction 1': \\‘here 1' is a mathematical l‘unction \\hich describes the behayiour oli an electron—\\ay'e: t,‘ is the n'ui'e/i/nt'rimr The probability ol‘ linding an electron at a giyen point in space is determined from the l'unction (" \\‘here 1.‘ is the u'urc/um‘liu/l. The Schrédinger wave equation Information about the \\ayet‘unction is obtained l‘rom the Schrodinger \\aye equation. \\hich can be set up and solved either exactly or approximately: the Schrodinger equation can be soly'ed t’.\'(l('l[l'(1/l[l [or a species containing a nucleus and only ()lIt’ electron (e.g. 1ll. glle ). i.e. a In't/ro‘eeu-/i/\t' system. A /1_i‘t/I'ugt'Ii-li/rc alum or [on contains a nttclctts and only one electron. The Schrodinger \\a\'e equation may be represented in sey eral l‘orms and in Box 1.3 we examine its application to the motion ol~ a particle in a one—dimensional box; equation l.l2 giyes the l‘orm ol‘ the Schrodinger \\aye equation that is appropriate l'or motion in the \* direction: . I s d‘t‘ br‘m . r . (liil‘it-itt (1.12) dy’ /I' \\‘here Ill : mass. I5 T total energy and l' 7 potential energy of the particle. ()l‘ course. in reality. electrons moye in threc—dimensional space and an appropriate l‘orm ol‘ the Schrodinger \\'a\e equation is gi\ en in equation 1.13, t' 4')“1' 47‘1‘ (Sf/H ‘ - .A (If Mr .0 (1.131 (11“ 6):“ /i“ Solving this equation \\ill not concern us. although it is usel‘ul to note that it is adyantageous to \\ork in spherical polar coordinates (l’igure 1.4). When \\e look at the results obtained l‘rom the Schrodinger \\a\e equation. \\e talk in terms ol‘ the I't/t/[u/ (Illt/ align/Hr parts 0/ I/l(’ ll'tlt‘t’flt/lt‘l/Ull. Fig. 1.4 D: angular C0( are: bx. Ham . Fig. 1.4 Definition of the polar coordinates (r. l). (i) angular coordinates. H and o are measured in 111 CHEMICAL AND THEORETICAL BACKGROUND The ditfraction of electrons by molecules illustrates the fact that the electrons behave as both particles and waves. Electrons that have been accelerated through a potential clill‘erence of SOkV possess a wavelength of 5.5 pm and a monochromated (i.e. a single wavelength) electron beam is suitable for ditl‘raction by molecules in the gas phase. The electron diffraction apparatus (maintained under high vacuum) is arranged so that the electron beam interacts with a gas stream emerging from a nozzle. The electric fields of the atomic nuclei in the sample are resrionsible for most of the electron scattering that is observed. Electron diiTraction studies ot‘ gas phase samples are concerned with molecules that are continually in motion, which are. therefore. in random orientations and well separated from one another. The diffraction data therefore mainly provide information about intrumolecular bond parameters (contrast with the results of X-ray difi‘raetion. see Bar 5.5). The initial data relate the scattering angle of the electron beam to intensity. After corrections have been made for atomic scattering. Ili()l(’(’I(/(II' Scattering data are obtained. and from these data it is possible (Via Fourier transformation) to obtain interatomic distances between all possible (bonded and non—bonded) pairs of atoms in the gaseous molecule. Converting these distances into a three- dimensional molecular structure is not trivial. particularly for large molecules. As a simple example. consider electron difi‘raction data for BC]; in the gas phase. The results give bonded distances BrrCl : 174 pm (all bonds of equal length) and non—bonded distances Cl~~Cl:301pm (three equal distances): ti Point defined as (I i 0 rad Point defined as 0 : 0 rad Box 1.2 Determination of structure: electron diffraction Chapter 1 0 An introduction to wave mechanics By trigonometry. it is possible to show that each Cl» B—~Cl bond angle. 0. is equal to 120 and that BC]; is therefore a planar molecule. Electron dim‘action is not confined to the study of gases. Low energy electrons (10~200eV) are diffracted from the surl‘ace of a solid and the dill‘raction pattern so obtained provides information about the arrangement of atoms on the surface of the solid sample. This technique is called lair e/Iergr electron diffraction (LEED). Further reading E.A.V. Ebsworth. D.W.H. Rankin and S. Cradock (1991) Structural Methods in Inorganic ('licnit.\'trr. 2nd edn. CRC Press. Boca Raton. FL A chapter on dill‘raction methods includes electron difi‘raetion by gases and liquids. C. Hammond (2001) The Basics of (rmtal/ngrap/n‘ and Diffraction. 2nd edn. Oxford University Press, Oxford Chapter ll covers electron difl‘raction and its applications. // is measured along llt|\ .n't‘ l'his point has polar coordinates (t: ll (.7) 0 Is measured along this are l‘or a point shown here in pink: r is the radial coordinate and (7' and o are dians (rad). Cartesian axes (.\‘. _t' and :) are also shown. 7 8 Chapter 1 0 Some basic concepts CHEMICAL AND THEORETICAL BACKGROUND The following discussion illustrates the so—called particle in a one-(Iii:icnxionul /)().\‘ and illustrates quantization arising from the Schrodinger wave equation. The St'lu't‘it/ingvr wave equation for the motion ofa particle in one dimension is given by: dZL‘ + 872m d.\'3 It2 (/5— 1,7120 where m is the mass. E is the total energy and V is the potential energy of the particle. The derivation of this equation is considered in the set of exercises at the end of Box 1.3. For a given system for which V and m are known. we can use the Schrodinger equation to obtain values of E (the u/Ion'tu/energies ol't/tvpurtic/v) and t" (the it'urLf/itm'litm). The wavefunction itselfhas no physical meaning. but U2 is a probability (see main text) and for this to be the case. u' must have certain properties: 0 L' must be finite for all values of .\‘: o L‘ can only have one value for any value ot‘x; dt,‘ , . . L' and d— must Vary Collllnuously 215 .\' Varies. .\‘ Now. consider a particle that is undergoing simple—harmonic wave-like motion in one dimension. i.e. we can fix the direction of wave propagation to be along the .\‘ axis (the choice ofx is arbitrary). Let the motion be further constrained such that the particle cannot go outside the fixed. vertical walls ol‘a box of width u. There is no force acting on the particle wit/tin the box and so the potential energy. V. is zero: it‘we take V 2 0. we are placing limits on x such that 0 S .\' g a. i.e. the particle cannot move outside the box. The only restriction that we place on the total energy Eis that it must be positive and cannot be infinite. There is one further restriction that we shall simply state: the boundary condition for the particle in the box is that t' must be zero when X 2 (l and .\' 2 u. Now let us rewrite the Schrodinger equation for the specific case of the particle in the one-dimensional box where V 2 0: 1 a d't' Xn‘mE : 2 — L‘ d.\'2 It2 which may be written in the simpler form: ‘3 ‘3 (l-L‘ 1 1 Xn‘mE . 2 —/\"l.' where k" 2 ~—,— d.\‘- Ir The solution to this (a well-known general equation) is: t,‘ 2 .4 sin Air + Beos k.\' where A and B are integration constants. When .\' 2 0. sin k.\' 2 0 and cos kx‘ 2 1: hence. L' 2 B when x 2 0. How- ever. the boundary condition above stated that l.‘ 2 0 when .\' 2 t). and this is only true if B 2 0. Also from the boundary condition. we see that z.‘ 2 0 when x 2 o. and hence we can rewrite the above equation in the form: 1.: 2 .4 sin kit 2 0 Box 1.3 Particle in a box Since the probability. up. that the particle will be at points between .\' 2 0 and .\‘ 2 a cannot be zero (i.e. the particle must be somewhere inside the box). A cannot be zero and the last equation is only valid if: kt: 2 Im where n 2 l. 2. 3 . . .: it cannot be zero as this would make the r . '7 . . probability. ti". zero meaning that the particle would no longer be in the box. Combining the last two equations gives: . imx t)‘ 2 A sm u and. from earlier: w [(3/12 "3/12 E — -, z ‘\ 87r'm Sma- where n 2 I. 2. 3 . . .: n is the quantum number determining the energy of a particle of mass or confined within a one- dimensional box ol‘ width (1. So. the limitations placed on the value of 15' have led to quantized energy levels. the spacing of which is determined by m and (I. The resultant motion ofthe particle is described by a series of standing sine waves. three of which are illustrated below. The wavefunction U‘z has a wavelength of (I. while . , . (1 3a wavefunctions 'L‘l and (H3 possess wavelengths ol 3 and 7 respectively. Each of the waves in the diagram has an amplitude of zero at the origin (i.e. at the point 11 2 0): points at which in 2 0 are called not/cs. For a given particle of mass m. the separations of the energy levels vary according to I72. i.e. the spacings are not equal. [5.‘ T 1 n W 3 v ’ Xniu“ f If '7 7 Z , 19' II‘ If 7 . 'l n l — Kim!2 H .\‘ axts u Self-st Consid wave-L tion all w=A where 1. If aw dxze and this A ((9. (1)) a‘ Li'Cartesian( Having s o The \ equat: regior c We c2 cular o Thec the St A wave; detailed atomic ‘ and an defined 1.6.41 The qt An aton integral t principal gen atorr with val values a solved. Two I angular + The rad numbers m,. and Aim, (0' it ‘tfisiwaiteiirbpaga mp , _ mainstream: in _ _ and this is represented in equation 1.14 where R(r) and A((9, <25) are radial and angular wavefunctions respectively.T TpCartesianixiyi E wradial(r)ql):1ngular(6> : Roll/“6% (1.14) Having solved the wave equation, what are the results? 0 The wavefunction it: is a solution of the Schrodinger equation and describes the behaviour of an electron in a region of space called the atomic orbital. 0 We can find energy values that are associated with parti— cular wavefunctions. o The quantization of energy levels arises naturally from the Schrbdinger equation (see Box 1.3). A wavefunction ’(i) is a mathematical function that contains detailed information about the behaviour of an electron. An atomic wavefunction 111 consists of a radial component, R(r), and an angular component, A(6, at). The region of space defined by a wavefunction is called an atomic orbital. " [Atomic orbitals, The quantum numbers n, [and m, An atomic orbital is usually described in terms of three integral quantum numbers. We have already encountered the principal quantum number, n, in the Bohr model of the hydro- gen atom. The principal quantum number is a positive integer with values lying between the limits l g n g 00; allowed values arise when the radial part of the wavefunction is solved. Two more quantum numbers, 1 and m,, appear when the angular part of the wavefunction is solved. The quantum lThe radial component in equation 1.14 depends on the quantum numbers n and 1, whereas the angular component depends on 1 and m,, and the components should really be written as R,,Y,(r) and AIJIl/(g‘t L - fimnieiharmonic ‘ ‘ ‘ * - a for ,thepwave is: 7. *_ Chapter 1 0 Atomic orbitals 9 2‘. If the’par ete- inthe‘bcx is of massm‘and moves with y T Vein-guy ‘y’v'what 3“ kinetic ‘31“?ng K5? Using the :de‘ _ “Bro ‘__ie,:equation (1“ U it write an expression for K}; in L terms_efm,tg_and )L - V ‘ . , I , . ,, 3. The equation‘iyoli derived partifl) applies only to a _ _ - :, particle moving ‘ingaspace in whichi‘thepotential energy, V, is Constant; and thewparticle eanfbe regarded as ‘ ‘ passesstng: only kinetic energy; :KE. 'Ifw the‘potential _ energy oftthe‘particle: does ‘ varygthe total‘energy, ‘ ' - :_.E = XE + V- Using.‘-thi;s:_.inforination ans your answers : L “to parts (1) and (23), derive thelSclirédinger equation“ (stated on p. for a particle; in a oneadimensional box. _ number I is called the orbital quantum number and has allowed values ofO, 1, 2 . . . (n — 1). The value of] determines the shape of the atomic orbital, and the orbital angular momentum of the electron. The value of the magnetic quantum number, m,, gives information about the directionality of an atomic orbital and has integral values between +1 and —l. Each atomic orbital may be uniquely labelled by a set of three quantum numbers: n, l and m]. Worked example 1.2 Quantum numbers: atomic orbitals Given that the principal quantum number, n, is 2, write down the allowed values of I and m,, and determine the number of atomic orbitals possible for n 2 3. For a given value of n, the allowed values of l are 0,1,2...(n — 1), and those ofml are 21. . .0...+l. For n 2 2, allowed values ofl 2 O or 1. For I 2 0, the allowed value of m] 2 0. For]: 1, allowed values of m; 2 —l, 0, +1 Each set of three quantum numbers defines a particular atomic orbital, and, therefore, for n 2 2, there are four atomic orbitals with the sets of quantum numbers: n22, [20, m120 n22, l2l, 111,2)1 n22, [21, m120 n22, I21, m,=+1 Self—study exercises 1. If m, has values of ~1,0.+ I. write down the corresponding value of /. Mus. I 2 l l 2. [fl has values 0, 1, 2 and 3, deduce the corresponding value of”. lAns. n 2 4] 10 Chapter 1 0 Some basic concepts 3. l‘or II I. what are the allowed \alnes Ill! and III,'.’ I ll:\./ 01m, 4)] 4. (oinpletc the following sets of quantum numbers: (at n 4. / U. ml 1th) I: 3.! l. ml |l/i\.ta)tl:(lil LU. l| The distinction among the l}‘/)(‘.\ ol‘ atomic orbital arises from their .\'/l(l/)<’.\ and .invalue/rim. The tour t_vpes ol‘ atomic orbital most commonly encountered are the .v. p. (I and /' orbitals. and the corresponding values ol‘lare t). l. 2 and 3 respectivelv. Fach atomic orbital is labelled with values ol‘ n and /. and hence we speak ol‘ l.\'. 3v. 2/). 3s. 3/). 31/. 4s. 4/1. Jal. 4/etc. orbitals. For an .s’ orbital. / : 0. For a p orbital. : 1. For a t/ orbital. / r: 3. For air/orbital. /: 3. Worked example 1.3 Quantum numbers: types of orbital l sing the rules that gowrn the values of the quantum numbers II and I. write down the possible t)pes ol~ atonIic orbital for n l. 2 and 3. The allowed values ol‘l are integers between 0 and (H i I). For H : l. / '7 t). The onl_v atomic orbital l‘or n : l is the l.\' orbital. For I] : Z. / : t) or l. The allowed atomic orbitals for n : 2 are the 3s and 2/; orbitals. For n : 3. / i (J. l or 3. The allowed atomic orbitals l‘or n : 3 are the 3s. 3p and 3d orbitals. Self-study exercises . \\ rite down the possible t_vpes of atomic orbital for n 4. | Ins. 4v. 4/). 4d. 4/ I -. \\l|icl1 atomic orbital has values of n 4 Hull / 2‘.’ l lm. Jill 2,4 . (liu- the three quantum numbers that describe a Zv atomic orbital. I l/Iv. I] ll 0.11:, ll] 4. \\ Iiicli quantum number distinguishes the 3\ and 5\ atomic orbitals? |,tnv, III I)t(gt'nerule orbitals possess the same energy. Now consider the consequence on these orbital types ol‘ the quantum) nunil‘m' 12,3. I'm: .111 .\ Ul'ht'lttl. / 7 (l illld II]/ can onl) equal ll. This means that for any value ol‘ II. there is onl) one .v orbital; it is said to be singlv degenerate. For a p orbital. / '7 l. and there are three possible m, values: VI. 0. I. This means that there are three p orbitals tor a gi\ en value of u when I] I: the set ot/r orbitals is said to be triva or three-Told degenerate. For a (/ orbital. I: Z. and there are live possible values ol‘ 111,: 3. ~ I. l). 7 l. 7:. tneaning that for a given value 01‘ n (n 3 3). there are live (/ orbitals; the set is said to be live—Told degenerate. As an exercise. _vou should show that there are seven /' orbitals in a degenerate set for a given value ol‘ H (II J- 4). For a given value ol‘n tn 2 I) there is one s atomic orbital. For a given value ol‘n (n 3 2) there are three p atomic orbitals. For a given value oh] (/1 Z 3) there are live (1 atomic orbitals. For a given value ofn (n 3 4) there are sevenf atomic orbitals. The radial part of the wavefunction, R(r) The mathematical l‘oi'ms ol‘ some ol‘ the wave l‘unctions for the H atom are listed in Table l2. Figure [.5 shows plots ol‘ the radial parts ol‘ the wavel‘unction. Rtr). against dis- tance. r. from the nucleus for the l.\' and 2s atomic orbitals ol‘ the hydrogen atom. and Figure 1.6 shows plots of Rtr) against 1' for the 3p. 3/). 4/) and 3d atomic orbitals: the nucleus is at r a. (t. From Table 1.3. we see that the radial parts of the wave— l‘unctions decav exponentially as 1' increases. but the decay is slower for n : 2 than for n : I. This means that the likeli- hood ol‘ the electron being further from the nucleus increases as 11 increases. This pattern continues for higher values ol‘n. The exponential decav can be seen clearly in Figure l.5a. Several points should be noted from the plots ol‘ the radial parts ol‘ w avel‘unctions in Figures 1.5 and to: o .s atomic orbitals have a finite value ol‘ Rtr) at the nucleus: o for all orbitals other than .\'. er) :7 t) at the nucleus: o for the l.\ orbital. Rtr) is always positive; tor the first orbital ol‘ other t_vpes (re. 3/). 3d. 4f). Rll') is positive evervwhere except at the origin: o for the second orbital ol‘a given t_vpe (i.e. 2s. 3/). 41/. 5/). Rtr) mav be positive or negative but the wavcl‘unction has onl_v one sign change; the point at which Rtr) : t) (not including the origin) is called a radial node: o for the third orbital ol‘ a given type (i.e. 3s. 4/). 5d. (if). Rtr) has two sign changes. i.e. it possesses two radial nodes. as orbitals have (n ‘ l) radial nodes. up orbitals have (n i 2) radial nodes. m/ orbitals have (II — 3) radial nodes. n/orbitals have {/1 4) radial nodes. Table 1 forms ( Aton ls 7s 2m- 2p: 2p.- 3‘ For tr but for The Let us in thr descri ing I’ll (see 8 electrt points which spend cal re descri 7 er)' Fir: depict Fig. 1 atom «vale: 1259 Chapter 1 0 Atomic orbitals 11 Table 1.2 Solutions of the Schrodinger equation for the hydrogen atom which define the ls. 2.x and 3/) atomic orbitals. lior these forms ofth solutions. the distance r from the nucleus is measured in atomic units. Atomic orbital n l m, ‘ lt‘ ll 0 So i 1 ,, . 2X 3 0 ll g- (I g r) e “ 2 (/3 t ,. t 2/7, I l [l 7 7,: re ' :V6 ‘p. 2 0 71,: 'e’ " 3 E :V/(i “p 3 l rel lgi'e": ‘ 3\//() Radial part of the wavefunction. Rtr‘)? Angular part of wavefunction. AW. (1’)) '3 ,— _\/it V3 tsin (I cos 1 :l i . , \(Mstn (ism o) fi if _'.\/ ll t.‘. .3iz‘\7/ttm for the Ix dlUllllL orbital. thc l0]mll.d lot Rh) is actually. _ i c ._ No but for the h_\drogen atom. / f l and in] W l atomic unit. Other functions are \lltllltll‘l§' simplified. The radial distribution function, 4wr2R(r)2 Let us now consider how we might represent atotnic orbitals in three-dimcnsional space. We said earlier that a useful description ofan electron in an atom is the pro/mhi/iir (ii/imi— iiig Illt’ U/t't'll‘U/l in a gi\ en \olume of space. The function in: (see i l is proportional to the pru/m/ii/iir (leiis'iit‘ of the electron at a point in space. By considering \‘Ltlttes of tr: at points around the nucleus. we can define a stir/aw imii/it/urr \t'hich encloses the Volume ofspace in which the electron will spend. say. 05% of its time. This ell‘ectchly gi\es its a physi— cal representation of the atomic orbital. since (A: may be described in terms of the radial and angular components Rm: and AW. (1):. First consider the radial components. A Useful w a)’ of depicting the probabilit} densin is to plot a radial (/i.\lt‘i/)Illfnlt >7 5 ’7 .3 “ n2 ‘ 7 ‘\ \\ () ‘l—-——~—‘—-—-——‘—1 (l 12 Distance r from nucletts 'atomic units (a) junction (equation HS) and this allows us to emisagc the region in space in which the electron is fottnd. Radial distribution function ; 4fil‘2Rll'): (1.15) The radial distribution functions for the l.\. is 211M 3x atomic orbitals of hydrogen are shown in Figure 1.7. and Figure lb‘ shows those of the 3s. 3/) and 3d orbitals. liar/i function is zero at the nucleus. following from the 1‘: term and the fact that at the nucleus ‘: (J. Since the function depends on R(rl:. it is always positive in contrast to Rtr). plots for which are shown in Figures 1.5 and 1.6. E'tch plot of 4m':R(r): shows at least one maximum Value for the function. corresponding to a distance from the nucleus at which the electron has the highest probability of being found. Points at which -l>Tl':R(I'): :0 (ignoring r:tl) correspond to radial nodes where Rtr) ; (l. l l 7 \ '1! l .y. (atomic units) ‘ - ’1 RU) id I /’ t. x /_,/e Distance r from nucleus atomic units (13) Fig 1.5 Plots of the radial parts of the warefunction. R(r). against distance. r. from the nucleus for (a) the l\ and (b) the 3» atomic orbitals of the h_\drogen atom: the nucleus is at r : t). The Vertical scales for the two plots are different but the horizontal SCdlL‘S Lll‘L‘ lllC same. 12 Chapter 1 0 Some basic concepts CHEMICAL AND THEORETICAL BACKGROUND Box 1.4 Notation for 1,1;2 and its normalization Although we use 1,": in the text. it should strictly be written as 1a" where t" is the complex conjugate of (3'. In the X— direction. the probability of finding the electron between the limits .\' and (.\‘ + d_\‘) is proportional to z;‘(.\')z;"(.\')dx. in three-dimensional space this is expressed as z,'L"dT in which we are considering the probability of finding the electron in a volume element d7: For just the radial part of the wavefunction. the function is R(r)R’(r). in all of our mathematical manipulations. we must ensure that the result shows that the electron is mmmr/wrc L“ 0.2 — ().l5 4] g: 0 l 1 0,05 0.05 r Fig. 1.6 Plots of. radial parts ol‘ the \\a\el'unction Rtr) ag The angular part of the wavefunction, AU). 0) Now let us consider the angular parts ot‘ the \yayel‘unctions. git/lira). l‘or dill‘erent types of atomic orbitals. These are im/epem/wzl ol‘ the principal quantum number as Table [.2 illustrates l‘or II : l and Z. Moreoyer. l‘or .\ orbitals. .ltII. 0) mike-)3 O (l '1! Fig. 1.7 Radial distribution l‘unctions. 471‘: eri (ie. it has not vanished!) and this is done by marina/[jug the wavel‘unction to unity. This means that the probability of finding the electron somewhere in space is taken to be I. Mathematically. the normalization is represented as follows: (32 dr : l or more correctly dr 2 l and this efl‘ectively states that the integral (f) is over all space (dT) and that the total integral of in: (or 1:1,") must be unity. 2/. Distance r from nucleus atomic units kainst r for the 2/7. 3/). 4p and 3d atomic orbitals; the nucleus is at r it is independent ol‘ the angles 0 and r.) and is ol‘ a constant Value. Thus. an .s‘ orbital is spherically symmetric about the nucleus. We noted aboye that a set ol‘p orbitals is triph degenerate: by conyention they are given the labels /)\_ p“ and pr From Table 12. we see that the angular part ol‘the It. \yayel‘unction is independent ol‘ 0; the orbital can he it) 15 20 Distance 1' from the nucleus atomic units ‘. for the l.\. 2x and 3} atomic orbitals ol‘ the hydrogen atom Fig. 1.8 Radial distribution functions, 47rr2R(r)2, for the 3s, 3p represented as two spheres (touching at the originf, the centres of which lie on the z axis. For the px and p}, orbitals, A(9, (1')) depends on both the angles 6’ and o; these orbitals are similar to p: but are oriented along the x and y axes. Although we must not lose sight of the fact that wave- functions are mathematical in origin, most chemists find such functions hard to visualize and prefer pictorial representations of orbitals. The boundary surfaces of the s and three p atomic orbitals are shown in Figure 1.9. The different colours of the lobes are significant. The boundary surface of an s orbital has a constant phase, i.e. the amplitude of the wavefunction associated with the boundary surface of the s orbital has a constant sign, For ap orbital, there is one phase change with respect to the boundary surface and this occurs at a nodal plane as is shown for the p: orbital in Figure 1.9. The amplitude of a wavet‘unction may be pOsitive or negative; this is shown using + and ~ signs, or by shading the lobes in different colours as in Figure 1.9. Just as the function 47rr2R(r)2 represents the probability of finding an electron at a distance r from the nucleus, we use a function dependent upon A((), of to represent the prob— ability in terms of 6 and o. For an s orbital, squaring A(E), o) causes no change in the spherical symmetry, and the surface boundary for the s atomic orbital shown in Figure 1.10 looks similar to that in Figure 1.9. For the p orbitals hOWever, going from A(9,o’) to A(6,¢)2 has the effect of elongating the lobes as illustrated in Figure 1.10. Squaring A09, o5) necessarily means that the signs (+ or —) disappear, l in order to emphasize that a is a continuous function we have extended boundary surfaces in representations of orbitals to the nucleus, but for p orbitals, this is strictly not true if we are considering @95% of the electronic charge. Chapter 1 0 Atomic orbitals 13 20 Distance r from the nucleus/ atomic units and 3d atomic orbitals of the hydrogen atom. but in practice chemists often indicate the amplitude by a sign or by shading (as in Figure 110) because of the impor- tance of the signs of the wavefunctions with respect to their overlap during bond formation (see Section [.13). Finally, Figure 1.11 shows the boundary surfaces for five hydrogen-like d orbitals. We shall not consider the mathema— tical forms of these wavefunctions, but merely represent the orbitals in the conventional manner. Each cl orbital possesses two nodal planes and as an exercise you should recognize Where these planes lie for each orbital. We consider dorbitals in more detail in Chapters 19 and 20, and f orbitals in Chapter 24. Orbital energies in a hydrogen—like species Besides providing information about the wavefunctions, solutions of the Schrodinger equation give orbital energies. E (energy levels), and equation 1.16 shows the dependence of E on the principal quantum number for hydrogen-like species. k E:—: n- k : a constant: 1.312 x 103 kJmol’I (1.16) For each value of It there is only one energy solution and for hydrogen—like species, all atomic orbitals with the same principal quantum number (eg. 3s. 3p and 3d) are degenerate. Size of orbitals For a given atom, a series ot'orbitals with different values of n but the same values of l and m, (eg. ls, 2s, 3s, 4s, . . .) differ in 14 Chapter1 0 Some basic concepts Px Z Z \ j y —, y %y X J X py 1’: Fig. 1.9 Boundary surfaces for the angular parts of the Is and 2/) atomic orbitals of the hydrogen atom. The nodal plane shown in grey for the 2]): atomic orbital lies in the xy plane. 2 Z Z \ « y y y x X x R. 15- p: Fig. 1.10 Representations of an s and a set of three degenerate p atomic orbitals. The lobes of the px those of the p_,. and p: but are directed along the axis that passes through the plane of the paper. Z Z - y y X x dry dxz Z d xz—yz orbital are elongated like 2 dz; Fig. 1.11 Representations of a set of five degenerate (1 atomic orbitals. their relative size (spatial extent). The larger the value of n, the larger the orbital, although this relationship is not linear. An increase in size also corresponds to an orbital being more diffuse. The spin quantum number and the magnetic spin quantum number Before we place electrons into atomic orbitals, we must define two more quantum numbers. In a classical model, an electron is considered to spin about an axis passing through it and to have spin angular momentum in addition to orbital angular momentum (see Box 1.5). The spin quan- tum number, 5, determines the magnitude of the spin angular Chapteri t Atomic orbitals 15 momentum of an electron and has a value of Since angular momentum is a vector quantity, it must have direction, and this is determined by the magnetic spin quantum number, my, which has a value of +% or ~ Whereas an atomic orbital is defined by a unique set of three quantum numbers, an electron in an atomic orbital is defined by a unique set of four quantum numbers: n. 1, ml and ms. As there are only two values of ms, an orbital can accommodate only two electrons. An orbital is fully occupied when it contains two electrons which are spin-paired; one electron has a value of ms 2 +12 and the other, ms : — 16 Chapter 1 0 Some basic concepts Worked example 1.4 Quantum numbers: an electron in an atomic orbital Write down two possible sets of quantum numbers that describe an electron in a ZS atomic orbital. What is the”. physical significance of these unique sets? The 2s atomic orbital is defined by the set of quantum numbers n : 2, I: 0. m, : 0. An electron in a 2s atomic orbital may have one oftwo sets of four quantum numbers: n : 2, Z:- 0. m, 2 0, ms = +% or n:2, 1:0, m,=0, mszi% If the orbital were fully occupied with two electrons, one electron would have m, : +%, and the other electron would have m.s : 7%, i.e. the two electrons would be spin- paired. Self-study exercises 1. Write down two possible sets of quantum numbers to describe an electron in 3 3s atomic orbital. [Amen—3,] 0,m, 0,m_‘. :%;nfi3,l—0,m1=0, _ i 1 ms a 2] 2. If an electron has the quantum numbers n = 2, 1 = 1, m, = —1 and m, : +% which type of atomic orbital is it occupying? [Ans 2p] 3. An electron has the quantum numbers n : 4, l : 1, m, = 0 and m, : +%. Is the electron in 3 4s, 417 0r 4d atomic orbital? [Ans. 4p] 4. Write down a set of quantum numbers that describes an electron in a SS atomic orbital. How does this set of quantum numbers differ if you are describing the second electron in the same orbital? [Ans.n:5,l=0,m,:0,ms : +%0r _%l E— The ground state of the hydrogen atom So far we have focused on the atomic orbitals of hydrogen and have talked about the probability of finding an electron in different atomic orbitals. The most energetically favourable (stable) state of the H atom is its ground state in which the single electron occupies the Is (lowest energy) atomic orbital. The electron can be promoted to higher energy orbitals (see Section 1.4) to give excited states The notation for the ground state electronic configuration of an H atom is Is] , signifying that one electron occupies the 1s atomic orbital. The helium atom: two electrons ) The preceding sections have been devoted to hydrogen-like species containing one electron, the energy of which depends only on n (equation 1.16); the atomic spectra of such species contain only a few lines associated with changes in the value of it (Figure 1.3). It is only for such species that the Schrodinger equation has been solved exactly. The next simplest atom is He (Z : 2), and for its two elec- trons, three electrostatic interactions must be considered: 0 attraction between electron (l) and the nucleus; 0 attraction between electron (2) and the nucleus; 0 repulsion between electrons (l) and (2). The net interaction will determine the energy of the system. In the ground state of the He atom, two electrons with m. : +% and ~% occupy the Is atomic orbital, i.e. the electronic configuration is Isl. For all atoms except hydro- gen—like species, orbitals of the same principal quantum number but differing / are not degenerate. If one of the 152 electrons is promoted to an orbital with n = 2, the energy of the system depends upon whether the electron goes into a 2s or 2]) atomic orbital, because each situation gives rise to different electrostatic interactions involving the two electrons and the nucleus. However, there is no energy distinction among the three different 2)) atomic orbitals. If promotion is to an orbital with n = 3, diflcrent amounts of energy are needed depending upon whether 3s, 317 or 3d orbitals are involved. although there is no energy difference among the three 3p atomic orbitals, or among the five 3d atomic orbitals. The emission spectrum of He arises as the electrons fall back to lower energy states or to the ground state and it follows that the spectrum contains more lines than that of atomic H. In terms of obtaining wavefunctions and energies for the atomic orbitals of He. it has not been possible to solve the Schrodinger equation exactly and only approximate solutions are available. For atoms containing more than two electrons, it is even more diflieult to obtain accurate solutions to the wave equation. In a mulli—electron atom, orbitals with the same value of H but different values of Z are not degenerate. Ground state electronic configurations: experimental data Now consider the ground state electronic configurations of isolated atoms of all the elements (Table 1.3). These are experimental data, and are nearly always obtained by analysing atomic spectra. Most atomic spectra are too complex for discussion here and we take their interpretation on trust. We ha\e already seen that the ground state electronic configurations of H and He are lsI and is: respectiver The ls atomic orbital is fully occupied in He and its con- figuration is often written as [He]. In the next two elements. Li and Be. the electrons go into the 2x orbital. and then from B to Ne. the 2/) orbitals are occupied to give the electronic configurations [He]2.v:2/)’” (In : l 76). When 111:6. the energy level (or ,v/ic/l) with n : 2 is fully occupied. and the configuration for Ne can be written as [Ne]. The filling of . the 3.x and 3/) atomic orbitals takes place in an analogous sequence from Na to /\r. the last element in the series having the electronic configuration [Ne]3.\'32/)(‘ or [Ar]. With K and Ca. successive electrons go into the 4s orbital. and (a has the electronic configuration [Ar]4.\‘:. At this point. the pattern changes. To a first approximation. the it) electrons for the next 10 elements (Sc to Zn) enter the 3t/ orbitals. giving Zn the electronic configuration 4.\'j3(/|”. There are some irregularities (see ) to which we return later. From Ga to Kr. the 4/) orbitals are filled. and the electronic configuration for Kr is [Ar]4.v131/'”4/7(‘ or [Kr]. From Rb to Xe. the general sequence of filling orbitals is the same as that from K to Kr although there are once again irregularities in the distribution of electrons between .\ and (I atomic orbitals (see )_ From Cs to Rn. electrons enter/orbitals for the first time; Cs. Ba and La have configurations analogous to those of Rb. Sr and Y. but after that the configurations change as we begin the sequence of the /(l/II/I(tll()f(/ elements (see )3" Cerium has the configuration [Xej4/"(is35d' and the filling of the seven 4f'orbitals follows until an electronic configuration of [XeH/“(istt/l is reached for Lu. Table 1.3 shows that the St/ orbital is not usually occupied for a lanthanoid element. After Lu. successive electrons occupy the remaining 5(/ orbitals (Hf to Hg) and then the 6/) orbitals to Rn which has the configuration [XeH/Hos:5(/”'6/)(‘ or [Rn]. Table l.3 shows some irregularities along the series of (l-block elements. For the remaining elements in Table 1.3 beginning at fran— cium (Fr). filling of the orbitals follows a similar sequence as that from ('s but the sequence is incomplete and some of the heaviest elements are too unstable for detailed investigations to be possible. The metals from Th to Lr are the ut't/tmt't/ elements. and in discussing their chemistry. Ac is generally considered with the actinoids (see ). A detailed inspection of Table 1.3 makes it obvious that there is no one sequence that represents accurately the occupation of diiTerent sets of orbitals with increasing atomic number. The following sequence is tip/Warmlatch‘ } true for the relative energies (lowest energy first) of orbitals in lit’l/ll't/f atoms: is «:T -3 < 2/) < 3.\' < 3/) < 4s 6 31/ < 4/) < 5.\' < 41/ < 5/) <_ (w < 5d % 4/ < 6/) < 7.\' < ()(f "NV 5) "The llfl’AC recommends the names lanthanoid and aetinoid in . preference to lanlhanide and actinide; the ending ‘—ide‘ usually implies a negati\el§ charged ion. Chaptert - Many-electron atoms 17 The energies of ditferent orbitals are close together for hiin values 0ft] and their relative energies can change significantly on forming an ion (see ). Penetration and shielding Although it is not possible to calculate the dependence of the energies of orbitals on atomic number with the degree of accuracy that is required to obtain agreement with all the electronic configurations listed in Table 1.3. some useful information can be gained by considering the difi‘erent strewn/1g c/fi't'ts that electrons in dill‘erent atomic orbitals have on one another. Figure LIZ shows the radial distribu— tion functions for the is. Is and 2/) atomic orbitals of the H atom. (It is a common approximation to assume hydro— gen-like wavefunctions for multi-electron atoms.) Although values of 471'3Rtr): for the is orbital are much greater than those of the 2s and 2/2 orbitals at distances relati\el_\ close to the nucleus. the values for the 3s and 3/) orbitals are still significant. We say that the 3x and 2/) atomic orbitals penetrate the l.\' atomic orbital: calculations show that the 2.\ atomic orbital is more penetrating than the 2/) orbital. Now let us consider the arrangement of the electrons in Li (Z: 3). In the ground state. the is atomic orbital is fully occupied and the third electron could occupy either a 3r or 2/) orbital. Which arrangement will possess the lower energy? An electron in a 2x or 2/) atomic orbital experiences the effective c/it/rge. 2c”. of a nucleus partly .\'/Ift’/(/t‘(/ by the is electrons. Since the 2/1 orbital penetrates the is orbital less than a 2s orbital does. a 2/) electron is shielded more than a 2s electron. Thus. occupation of the is rather than the 2/; atomic orbital gives a lower energy system. Although we should consider the energies of the electrons in atomic orbitals. it is common practice to think in terms of the orbital energies themselves: Lil’s) < [3(2/2). Similar argu— ments lead to the sequence [5(33‘) < L’(3/i) < 1;‘(3t/) and Ef4.\') < [Stet/i) < [:‘(4z/) < 1;‘(4/‘). As we move to atoms of elements of higher atomic number. the energy ditTerences between orbitals with the same value of I: become smaller. the validity of assuming hyd rogen—like w avefunetions becomes more doubtful. and predictions of ground states become less reliable. The treatment above also ignores electron relectron interactions within the same principal quantum shell. A set of empirical rules (Slatet’s rules) for estimating the ellccti\'c nuclear charges experienced by electrons in diil‘crent atomic orbitals is described in Box |.o. 1.8 The periodic table In l869 and 1870 respectively. Dmitri Mendeler and Lothar Meyer stated that the /1/'n/tertitav of [/16 t’/t’lilt’lll.\' (wt be l'(’/1/'(’.\'t'/I/t‘(/ t/A‘ portal/iv V/illlt‘llitlllfi of I/](’//' (Ito/titt- were/Its. and set out their ideas in the form of a period/r tub/e. As new elements have been discovered. the original form of ...
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This note was uploaded on 03/28/2008 for the course CHEM 333 taught by Professor Asdfsadf during the Spring '08 term at UNC.

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AtomicStructureReading - Some basic concepts l Fundamental...

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