This preview shows pages 1–17. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 multielectron 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, Xray 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. elemon 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 , isotopesg, ‘ 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, nonintegral 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 Sxchains 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‘ \yayelike 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'Iili/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‘ititt (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 Xray 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
wavelike 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 onedimensional 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 (lL‘ 1 1 Xn‘mE . 2 —/\"l.' where k" 2 ~—,—
d.\‘ Ir The solution to this (a wellknown 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 Selfst Consid
waveL
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 Veinguy ‘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. Selfstudy exercises . \\ rite down the possible t_vpes of atomic orbital for n 4.
 Ins. 4v. 4/). 4d. 4/ I . \\licl1 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 threeTold 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 threedimcnsional 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 threedimensional 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
hydrogenlike 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 hydrogenlike
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 spinpaired; 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.
Selfstudy 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 hydrogenlike 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 (lblock 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  Manyelectron 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—
genlike wavefunctions for multielectron 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 ...
View
Full
Document
This note was uploaded on 03/28/2008 for the course CHEM 333 taught by Professor Asdfsadf during the Spring '08 term at UNC.
 Spring '08
 asdfsadf
 Atom

Click to edit the document details