atomic structure and periodic trends. n lectures: week1: w9 am; week2: w9 am (icl) & 11 am (dp),...
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Atomic Structure and Periodic Atomic Structure and Periodic TrendsTrends
Atomic Structure and Periodic Atomic Structure and Periodic TrendsTrends Lectures:
week1: W9 am; week2: W9 am (ICL) & 11 am (DP), F 9 am (DP); week 3: 9 am (ICL) Books:• Inorganic Chemistry by Shriver and Atkins• Physical Chemistry by P.W.Atkins , J. De Paula • Essential Trends in Inorganic Chemistry D Mingos • Introduction to Quantum Theory and Atomic Structure by P. A. Cox Other resources:
– Web-pages: http://timmel.chem.ox.ac.uk/lectures/– Ritchie/Titmuss, Quantum Theory of Atoms and Molecules, Hilary Term
Why study atomic electronic Why study atomic electronic structure?structure?
All of chemistry (+biochemistry etc.) ultimately boils down to molecular electronic structure.
Reason: electronic structure governs bonding and thus molecular structure and reactivity
Before it is possible to understand molecular electronic structure we need to introduce a number of concepts which are more easily demonstrated in atoms.
atomic structure molecular structure chemistry
The Periodic TableThe Periodic Table
Mendeleyev was the first chemist to understand that all elements are related members of a single ordered system. From his table he predicted the properties of elements then unknown, of which three (gallium, scandium, and germanium) were discovered in his lifetime.
Mendeleyev, Dmitri Ivanovich (1834-1907)
This course......This course......
will introduce new concepts gradually starting with the “simplest”:
H-Atom Energy levels, Wavefunctions, Born Interpretation, Orbitals
Many electron atoms Effects of other electrons, Penetration, Quantum Defect
The Aufbau Principle Electronic Configuration of atoms and their ions
Trends in the PT Ionisation Energy, Electron Affinity, Size of atoms and ions
The H-AtomThe H-Atom
H1
1.008
The Hydrogen AtomThe Hydrogen Atom
The simplest possible system and the basis for all others:
+ -r
vElectron orbits the proton under the influence of the Coulomb Force:
F
Re
vis
ion
H-Atom: consider the energyH-Atom: consider the energy
r4π
evE
0
22
21
Total
+ -r
v
221
KE vE
r4π
eE
0
2
PE
ETotal =
c.m.
Energy Levels?Energy Levels?
Consider 3 approaches: Classical Bohr Model (old quantum theory) Full Quantum: Schrödinger Equation
Classically: There is no theoretical restriction at all on v or r
(there are infinitely many combinations with the same energy).
Hence the energy can take any value - it is
. As a result transitions should be possible everywhere
across the electromagnetic spectrum. So lets see the spectrum...
r4π
evE
0
22
21
Total
The H-atom Emission SpectrumThe H-atom Emission Spectrum
Infra-red
Ultra-violet XUVvisible
scre
en
lens lens
prism
+
-H2
Principles of Quantum MechanicsPrinciples of Quantum Mechanics Quantization Quantization
Energy levels
The Rydberg FormulaThe Rydberg Formula
22
21 n
1
n
1c yRv
In 1890 Rydberg showed that the frequencies of all transitions could be fit by a single equation:
Re
vis
ion
Bohr Theory (old quantum)Bohr Theory (old quantum) Bohr explained the observed frequencies by
restricting the allowed orbits the electron could occupy to particular circular orbits(by quantizing the angular momentum).
His theory gives energy levels:
ch8ε
e where
n
hcE
320
4
2
y
y
RR
1
234
n
The problem with Bohr TheoryThe problem with Bohr Theory
Bohr theory works extremely well for the H-atom.
However; it provides no explanation for the
quantization of energy -it just happens to fit the observed spectrum
But, more seriously: it just doesn’t work at all for any other atoms
Quantum mechanical PrinciplesQuantum mechanical Principlesand the Solution of the and the Solution of the Schrödinger EquationSchrödinger Equation
Principles of Quantum MechanicsPrinciples of Quantum Mechanics Quantization Quantization
Energy levels2
2
mE v
Quantum mechanicsClassical mechanics
is continuous
Principles of Quantum MechanicsPrinciples of Quantum MechanicsIt’s all about probabilityIt’s all about probability
In classical mechanicsPosition of object specified
In quantum mechanicsOnly
of object at a particular location
Principles of Quantum MechanicsPrinciples of Quantum Mechanics
How do we describe the How do we describe the electrons in atoms?electrons in atoms?
You know: Electrons can be described as (characterised by mass, momentum, position…)
p = h/ De Broglie
e-
h = Planck’s constant = 6.626 10-34 Js
However: Electrons can also be described as
(characterised by wavelength, frequency, amplitude)
and show properties such as interference, diffraction
The Davisson Germer ExperimentThe Davisson Germer Experiment
Proving the wave properties of electrons (matter!)Intensity variation in diffracted beam shows constructive and destructive interference of wave
Principles of Quantum Mechanics Principles of Quantum Mechanics
The WavefunctionThe Wavefunction
(position, time
In quantum mechanics, an electron, just like any other particle, is described by a
Contains all information there is to know about the particle
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The Results of Quantum MechanicsThe Results of Quantum Mechanics
Schrödinger equation:
EV(r)
2μ2
2
where ,zyx 2
2
2
2
2
22
is the wavefunction,V(r) the potential energy andE the total energy
Mor
e on
th
an in
Hil
ary
Ter
m
Spherical Polar CoordinatesSpherical Polar Coordinates
Instead of Cartesian (x,y,z) the maths works out easier if we use a different coordinate system:
r
y
x
z x = r sin cosy = r sin sin z = r cos
(takes advantage of the spherical symmetryof the system)
So Schrödinger’s Equation So Schrödinger’s Equation becomes.....becomes.....
EV(r)
2μ2
2as before
But now2
22
22 11
r
rrr
where
sinsin
1
sin
12
2
22
with r4ππ
eV(r)
0
2
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e on
th
an in
Hil
ary
Ter
m
We separate the wavefunction into 2 parts: a radial part R(r) and an angular part Y(,),
such that =
The solution introduces 3 quantum numbers:
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....which can be solved exactly for the H-....which can be solved exactly for the H-atom atom with the solutions called with the solutions called orbitalsorbitals, more , more specifically, specifically, atomic orbitals.atomic orbitals.
....which can be solved exactly for the H-....which can be solved exactly for the H-atom atom with the solutions called with the solutions called orbitalsorbitals, more , more specifically, specifically, atomic orbitals.atomic orbitals.
We separate the wavefunction into 2 parts: a radial part R(r) and an angular part Y(,),
such that =R(r)Y(,)
The solution introduces 3 quantum numbers:
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The quantum numbers;The quantum numbers;
quantum numbers arise in the solution; R(r) gives rise to:
the principal quantum number, n Y(,) yields:
the orbital angular momentum quantum number, l and the magnetic quantum number, ml
i.e., =Rn,l(r)Yl,m(,)
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The values of The values of nn,, l l, &, & m mll
n = 1, 2, 3, 4, .......
l = 0, 1, 2, 3, ......(n-1)
ml = -l, -l+1, -l+2,..0,..., l-1, l
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You are familiar with these.....You are familiar with these.....
n is the integer number associated with an orbital
Different l values have different names arising from early spectroscopy
e.g., l =0 is labelled
l =1 is labelled
l =2 is labelled
l =3 is labelled etc...
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Looked at another way.....Looked at another way.....
n =1 l=0 ml=0 1s orbital (1 of)n =2 l=0 ml=0 2s orbital (1 of)
l=1 ml=-1, 0, 1 2p orbitals (3 of)n =3 l=0 ml=0 3s orbital (1 of)
l=1 ml=-1, 0, 1 3p orbitals (3 of)l=2 ml=-2,-1,0,1,2 3d orbitals (5 of)
etc. etc. etc.Hence we begin to see the structure behind
the periodic table emerge.
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Exercise 1;Exercise 1;
Work out, and name, all the possible orbitals with principal quantum number n=5.
How many orbitals have n=5?
The Radial WavefunctionsThe Radial Wavefunctions
The radial wavefunctions for H-atom are the set of Laguerre functions in terms of n, l
a0 (=0.05292nm)is the-the most probable
orbital radius of an H-atom 1s electron.
R(r)
1s
2s
2p
3s 3p3d
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R(r)
R(r)
Revisit: The Born InterpretationRevisit: The Born Interpretation
The square of the wavefunction,
at a point is proportional to the
of finding the particle at that point.
In 1D: If the amplitude of the wavefunction of a particle at
some point x is xthen the probability of finding the
particle between x and x +dx is proportional to
(x)(x)dx (x)
is the complex conjugate of
2
2
dx
R(r)2R(r)1s
2s
2p
3s
3d
1s
2s
2p
3s
3p
3d
3p
Radial Wavefunctions and the Radial Wavefunctions and the Born InterpretationBorn Interpretation
At long distances from the nucleus, all wavefunctions decay to .
Some wavefunctions are zero at the nucleus (namely, all but the l=0 (s) orbitals). For these orbitals, the electron has a zero probability of being found .
Some orbitals have nodes, ie, the wave function passes through zero; There are such radial nodes for each orbital.
In 3D: If the amplitude of the wavefunction of a particle
at some point (x,y,z) is x,y,zthen the probability of
finding the particle between x and x+dx, y+dy and z+dz,
ie, in a volume dV = dx dy dz is given by
(x,y,zdV
dxdy
dz
x
z
y
It follows therefore that
2
2(x,y,z)
is a probability density.
In spherical coordinates2(r
More important to know probability of finding More important to know probability of finding
electron at a given distance from nucleuselectron at a given distance from nucleus!!
dxdydz
x
z
y
dV=dxdydz
Cartesian coordinates not very useful to describe orbitals!
…..in a shell of
dA = r2sindddA=dxdydV = r2sindddr
Surface element
Volume element
The Surface area of a The Surface area of a spheresphere is hence: is hence:
22
0 0
22
0 0
2
2
sin
sin
( cos cos0)(2 0)
4
r d d
r d d
r
r
For spherically symmetric orbitals:
the radial distribution function is defined as
2
P(r)= r2 (r)
and P(r)dr is the probability of finding the electron in a shell of radius r and thickness dr
Construction of the radial Construction of the radial distribution functiondistribution function
For spherically symmetrical orbitals
P(r)
Radial distribution function P(r)Radial distribution function P(r)Im
por
tan
t
Born interpretationprob = d
Hence plot P(r) = r2R(r)2
P(r)=4r
for spherical symmetry)
Pn(r) has
nodes
So what do we So what do we learn?learn?
R(r)
R(r)
P(r)
r
r
r
The 2p orbital is on average closer to the nucleus, but note the 2s orbital’s high probability of being
3d vs 4s3d vs 4sP(r)
The Angular WavefunctionThe Angular Wavefunction
The Ylm(,) angular part of the solution form a set of functions called the
n.b. These are generallyimaginary functions -we usereal linear combinations topicture them.
The Shapes of Wavefunctions The Shapes of Wavefunctions (Orbitals)(Orbitals)
Shapes arise from combination of radial R(r) and angular parts Ylm(,) of the wavefunction.
Usually represented as boundary surfaces which include of the probability density. The electron does not undergo planetary style circular orbits.
These are the familiar spherical s-orbitals and dumbell-shaped p-orbitals, etc...
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Electron densities representationsElectron densities representations
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+-
++-
of corresponding wavefunctions (white areas)
Signs of corresponding wave functions
R(r)
R2
r
*
*Please note that R and R2 are arbitrarily scaled.
The s-orbitals:R2
R(r)
Boundary model of an s orbital within which there is 90% probability of finding the electron
Boundary ModelBoundary Model
The p-orbitals: boundary surfacesThe p-orbitals: boundary surfaces- actually imaginary functions but linear combinations give the familiar dumbells
px pzpy
Imp
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different signs of the wavefunction
One Nodal plane
The d-orbitals : boundary surfacesThe d-orbitals : boundary surfacesIm
por
tan
t
Again, different shades denote different signs of the wavefunction
dxz dxydyzdxz
dz2dx-y
2 2
z z z
z
x x x
x
y
yyy
The d-orbitals : boundary surfacesThe d-orbitals : boundary surfaces
Again, different shades denote different signs of the wavefunction
dxz dxydyzdzx
TwoNodal planes
dxz dxydyzdxz
Two Nodal planes which split orbital into 4 lobes, orbitals lie in a plane perpendicular to the two nodal panes and point between the axes
z
x
y
z
x
y
z
x
y
The d-orbitals : boundary surfacesThe d-orbitals : boundary surfaces
22 2
z
x
y
Two Nodal planes which split orbital into 4 lobes, orbitals lie in xy-plane, pointing along the x and y axes, nodal planes at 45o to xz- and yz-planes
Cylindrical symmetry, two angular nodes which take the form of cones at 54.7o and 125.3o to the z-axis.
54.7o
125.3o
The energies of orbitalsThe energies of orbitals
In the case of the H-atom (& only the H-atom) the energy is determined exclusively by the principal quantum number, n:
ch8ε
μe32
0
4
yRwhere Ry is the Rydberg constant
Ry for H-atom = 109 677 cm-1 (= Ionization energy, 13.6eV)
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H-atom Energy H-atom Energy LevelsLevels
i) All levels with the same n
i.e., E(3s)=E(3p)=E(3d)
ii) All energies are negative(because the electron is bound)iii) n= by definition has energy zero, hence E- E1= ionization energy (13.6 eV)= Ry (109 677 cm-1)
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The Ionization EnergyThe Ionization Energy
Corresponds to the total removal of an electron (i.e., transition to n=)
Since in the ground state H-atom n=1 is the highest occupied atomic orbital, the ionization energy is given by:
hc1
11hcEnergyIonization yy RR
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See later for Koopman’s theorem
Other AtomsOther Atoms
A) A) Other single electron Other single electron atoms/ions (H-like atoms)atoms/ions (H-like atoms)
The ions He+, Li2+, Be3+, B4+,... etc. are said to be with the H-atom (same electron configuration).
Again ns, np, nd etc. levels are degenerate. However the attractive Coulomb force is much
bigger due to the Z+charge of the nuclei. The energy levels are given by a similar
expression to that for the H-atom with the inclusion of a :
2
2 hcE
n
RZ yn
Note that exponential function decays faster as Z
Orbitals contract with increasing Z
Orbitals contract with increasing Z
2s
H-like atoms/ions : SummaryH-like atoms/ions : Summary The orbitals contract with increasing Z The effect of the Z2 term is to increase the energy level
spacing. E.g., The energy level spacings in the He+ spectrum are approx.
4 times those in the H-atom whilst in Be3+ they are 16 times larger (Z=4).
This is only approximate because of the slight mass dependence of the Rydberg constant.
Ry(H) = 109 677cm-1 Ry(mass) = 109 737 cm-1
Compare ionisation energies for IE(H) = 13.6eV, IE(He+) = 54.4eV, IE(Li2+)=122.4eV
B) Multi-electron Atoms B) Multi-electron Atoms
Schrödinger equation cannot be solved analytically anymore (apart from He)
Need to develop an approximate picture for multi-electron atoms
2+
- -
Helium atom
The orbital approximation in The orbital approximation in quantum mechanicsquantum mechanics
r1, r2, …) = r1)(r2)….
Total wavefunction of many electron atom
Each electron is occupying its individual orbital with nuclear charge modified to take account of all other electrons’ presence (repulsion!)
The orbital approximation in wordsThe orbital approximation in wordsMake multi electron atom look like a one electron atomAssumes every electron to be on its own experiencing an effective nuclear charge, Zeff
Then the orbitals for the electrons take the form of those in hydrogen but their energies and sizes are modified by simply using an effective nuclear charge
One electron atom
2+
- - - -Zeff
Two-electron atom Orbital Approximation
The Concept of Electron SpinThe Concept of Electron Spin
The solution of the Schrödinger equation above accounts very well for the structure of the H-atom spectrum and, as we will see, for other atoms too.
However under very high resolution “fine structure” is observed in the transitions which is not explained using this approach.
Dirac extended wave mechanics to include Special relativity and an extra coordinate - time.
Solution of the Dirac equation yields a new intrinsic angular momentum -
The Electron SpinThe Electron Spin
An electron has The projection of this spin is also quantized
(by analogy with orbital angular momentum; l and ml) such that the spin projection quantum number, ms=½, (spin up or ) and ms=-½, (spin down or )
This is the final theoretical plank behind the structure of the periodic table.
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Pauli Pauli ExclusionExclusion Principle Principle
(as all electrons necessarily have the same s=1/2) Hence, no individual orbital may be occupied by more
than 2 electrons Electrons occupying the same orbital must be “paired up”.
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no two electrons in the same atom can have the same 4 quantum numbers n, l, ml, ms
The Pauli Principle and SymmetryThe Pauli Principle and Symmetry
The Pauli Exclusion Principle applies to any
pair of identical (spin quantum
number s = half integer) – protons, electrons,
neutrons have s = ½ and are fermions.
BUT, any number of (s = integer)
can occupy the same orbital 12C (s = 0) and
the photon (s = 1)are bosons
Now, as we proceed from H to Now, as we proceed from H to other atoms, we need to considerother atoms, we need to consider
1. The Pauli Exclusion Principle
2. The coulombic repulsions between the electrons
2+
- -
Two-electron atom
B)B) Atoms with one outer electron Atoms with one outer electron
Easiest: alkali metals Li, Na, K, Rb but also Be+, Mg+, Ag (4d105s1) as only one outer electron.
But still, the situation is no longer as simple as the H-atom due to the inner electrons - the “core”;
The core electrons will
tend to the outer
electron from the full
nuclear charge.
Shielding and PenetrationShielding and Penetration If an electron is always outside the core it experiences
only a net charge of nucleus and core. If, however, the electron spends much of its time close to
the nucleus (within the core) it will experience a larger nuclear attraction and have a lower energy (more tightly bound).
Hence the energy of the outer electron depends on how much it the core region.
This in turn depends on the type (s, p, d, f etc.) of orbital it is in.
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Recall the radial distribution Recall the radial distribution functions.....functions.....
An e- in the 3s orbital spends more time close to the nucleus than an electron in 3p and is thus more tightly bound (lower energy).
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RamificationsRamifications
The energy of a given quantum state is now no longer simply a function of its principal quantum number but also of its penetration into the core region which depends on the orbital shape (and thus l).
i.e., E=En,l
In general the energies of sub-shells of the same principal quantum number n lie in the order
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ZZeffeff – the effective nuclear charge – the effective nuclear charge To account for the effects of penetration and shielding we
use
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2
2
2
2 hc)(hcE
n
RZ
n
RZ yyeffnl
where is the shielding parameter and Z is the charge of the nucleus.
Zeff is a function of n and l as electrons in different shells and subshells approach nucleus to different extents
Trends in Trends in ZZeff eff
The Grotrian diagram for NaThe Grotrian diagram for Na
Note:
•Different l levels have different energy.
•The H-atom levels are marked on the RHS
•Note more rapid stabilisation of 4s with respect to 3d due toH-Atom
energylevels
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Radial Distribution FunctionRadial Distribution Function3d vs 4s3d vs 4s
The Aufbau Principle and the The Aufbau Principle and the Structure of the Periodic TableStructure of the Periodic Table
Electron ConfigurationsElectron Configurations
To obtain a ground state configuration for an atom we apply the Pauli exclusion and the Aufbau principle which states that electrons are added to orbitals in increasing order of energy.
n lPossible # of e
n = 22p
n = 3
3s
3p
3d
l = 1 (p)
l = 0 (s)
l = 0 (s)
l = 1 (p)
l = 2 (d)
# of orbitals2l+1
n = 1 l = 0 (s) 1s
2s
Principles of how to build up electron Principles of how to build up electron configurationsconfigurations
The Aufbau Principle - “The building-up”principle
When establishing the ground state configuration of an atom start at the energetic bottom and work your way up
2p
3s
3p
3d
1s
2sNB: The energy ordering of the orbitals changeswith the number of electrons.
4s
Principles of how to build up electron Principles of how to build up electron configurationsconfigurations
(1) The Pauli Exclusion Principle - No two electrons in one atom may have the same set of four quantum numbers (that is they must differ in one or more of n, l, ml , ms)
Differ in Differ in
2p
1s
2s
2p
1s
2s
Differ in
2p
1s
2s
Differ in
2p
1s
2s
Remember, when two electrons share Remember, when two electrons share one orbital, their magnetic spin one orbital, their magnetic spin
quantum numbers must bequantum numbers must be up down
Paired electron spins
Principles of how to build up Principles of how to build up electron configurationselectron configurations
(2) HUND’s Rule - When electrons occupy orbitals of the same energy, the lowest energy state corresponds to the configuration with the greatest number of unpaired electrons
2p
1s
2s
2p
1s
2s
E(Paired) E(Unpaired)
Energetically favored
Maximizing the number of parallel spins - Maximizing the number of parallel spins - The exchange interactionThe exchange interaction
Quantum mechanical in origin Arguments based on the fact that total
wavefunction has to be with respect to exchange of the electrons (Pauli)
Nothing to do with the fact that electrons are charged!
Result is that each electron pair with parallel spins leads to a lowering of the electronic energy of the atom
Now, let’s start rememberingNow, let’s start remembering
(1) The Pauli Exclusion Principle (only two anitparallel spins in one orbital)
(2) Hund’s Rule (parallel spin configuration of lower energy for degenerate orbitals)
(3) Aufbau Principle (from energetic bottom to energetic top)
2HeConfiguration 1s2
Full ShellInertNoble GasUnlikely to form bondsor ions
1HConfiguration 1s1
Can lose one electron to from stable ion H+
Can form single bond
H2O, H2
3Li – Lithium – remember: can’t have s3 (Pauli)•Configuration 1s22s1
•Easily ionised to Li+ (1s2) •Alkali Metal•Under standard conditions: lightest metal and least dense element
4Be- Beryllium•Configuration 1s22s2
•Be2+ (1s2) stable ion•Alkaline Earth Metal•Toxic: replaces Magnesium from Magnesium activated enzymes due to stronger coordination ability
5B - Boron
Config 1s22s2 2p1
Forms stable covalently bonded molecular networksMainly tervalentLewis acidity of many of its compounds and multicentre bondingChemistry highly diverse and complex
Number of Electrons: 5
6C - CarbonConfig: 1s22s2 2p2
Diamond, graphite, amorphous, fullerenes (e.g., C60)Forms (usually) four bonds (tetravalent), see CH4
Non-metallicCarbon's unique characteristic of bonding to itself is responsible for complex molecules composed of long chains of carbon atoms, the skeleton of life
7N - Nitrogen
Config 1s22s2 2p3
In compounds typically forms three bonds (trivalent), see NH3, N2
Non-metal
N2: •Gaseous, odourless, tasteless•78% of air is N2
•Very inert at room temperature (and below) due to strong triple bond
8OConfig 1s22s2 2p4
member of chalcogen groupNormally considered divalent but other oxidations states vary widelyhugely electronegative (see later)
Colourless, odourless, highly reactive gas,Created biologically from CO2 by green photosynthesizing plantsParamagnetic (attracted by a magnetic field)
O2:
Exercise 3;Exercise 3;
Name the electron configurations of the F and Ne atoms. Which is more stable
and why?
9F•Config
•Member of the halogen group•Most reactive of all elements•Forms F readily•highly electronegative!•Does not exist in nature in the elemental state at all because of high reactivity
10Ne - NeonConfig
Full Outer shellNoble GasUnlikely to form bonds or ions
And now the cycle repeats itself….
Remember:
Li: 1s22s1 Na: 1s22s2 2p63s1Ne: 1s22s2 2p6
1
2
34
5
67
s1 s2 p1 p2 p3 p4 p5 p6
s-block p-block
The Third PeriodThe Third Period
1s22s22p63s1Na: cf. Li (second period) 1s22s1
1s22s22p63s2Mg: cf. Be (second period) 1s22s2
1s22s22p63s2 3p1Al: cf. B (second period) 1s22s22p1
1s22s22p63s2 3p2Si: cf. C (second period) 1s22s22p2
1s22s22p63s2 3p3P: cf. N (second period) 1s22s22p3
1s22s22p63s2 3p4S: cf. O (second period) 1s22s22p4
1s22s22p63s2 3p5Cl: cf. F (second period) 1s22s22p5
1s22s22p63s2 3p6Ar: cf. Ne (second period) 1s22s22p6
And what now?And what now?
3s 3p 3d
4s
Here, again, is the picture in the H-atom
The 3d electron is more firmly bound than the 4s because of its lower principle quantum number
Energy
Remember: Grotrian diagram for NaRemember: Grotrian diagram for Na
•Note more rapid stabilisation of 4s with respect to 3d due to difference in penetration!
H-Atomenergylevels
Remember: Remember: Radial Distribution FunctionRadial Distribution Function
3d vs 4s3d vs 4s
4s through the inner shells to some extent and from N on all the way to Ca, it is more stable (stronger bound) than 3d
The energy of individual (singly The energy of individual (singly occupied orbitals)occupied orbitals)
Hence, from N Hence, from N
3s
3p
3d4s
Energy
4s is lower in energy than 3d
3s
3p
3d4s
Ar
1s22s22p6
K Ca
3s
3p
3d4s
Ar
1s22s22p6
Careful though – the actual energy ordering is now:Careful though – the actual energy ordering is now:
3s
3p
3d4s
Ar
1s22s22p6
3s
3p
3d4s
Ar
1s22s22p6
1s22s22p63s23p64s23d1 1s22s22p63s23p64s23d2
And now we start filling the d-block!
Sc
Ti
1s22s22p63s23p64s23d1
1s22s22p63s23p64s23d2
Order in which they were filled
Energy Ordering
And all the way to Zn
Order in which they were filled
1s22s22p63s23p6
Energy Ordering
1s22s22p63s23p63d14s2
Ti 1s22s22p63s23p63d24s2
Important for formation of ions!Important for formation of ions!
Ti3+ 1s22s22p63s23p63d24s2
Ti3+ 1s22s22p63s23p63d1
ScTi
1s22s22p63s23p63d14s2
1s22s22p63s23p63d24s2
V 1s22s22p63s23p63d34s2
Cr 1s22s22p63s23p6
Mn 1s22s22p63s23p63d54s2
FeCo
1s22s22p63s23p63d64s2
1s22s22p63s23p63d74s2
Ni 1s22s22p63s23p63d84s2
Cu 1s22s22p63s23p6
Zn 1s22s22p63s23p63d104s2
Ar
Now, we should really understand the Now, we should really understand the Structure of the periodic tableStructure of the periodic table
1
2
34
5
67
s1 s2 p1 p2 p3 p4 p5 p6
s-block p-block
d1 d2 d3 d4 d5 d6 d7 d8 d9 d10
d-block
4f
5f
Electron Configuration of BaElectron Configuration of Ba1s22s22p63s23p64s23d104p65s24d105p66s2
1
2
34
5
67
4f
5f
The LanthanidesThe Lanthanides
TheActinidesTheActinides
3d
4d
5d
6d
4f
5f
Inner Transition Metals
(f1-14)
Transition Metals
(d1-10)
Noble Gases
Halogens(p5)
2p
3p
4p
5p
6p
7p
1s
2s
3s
4s
5s
6s
7s
AlkalineEarth Metals (s2)
Alkali Metals (s1)
3d
4d
5d
6d
2p
3p
4p
5p
6p
7p
1s
2s
3s
4s
5s
6s
7s
4f
5f
The Periodic Table of the Elements
Periodic TrendsPeriodic Trends
Periodic TrendsPeriodic Trends1) Effective Nuclear Charge - Zeff
2s – 3s: Zeff greater for 3s probably due to actual higher overall charge
Periodic Trends – (Periodic Trends – (ZZeffeff//n)n)22
Increase across periods unchanged(but note that relative slopes are different)for (Zeff/n)2)
BUT (Zeff/n)2
H(1)Li(2s)Na(3s)
Zeff
11.261.85
10.400.38
M(g) M+(g) + e(g)
Ionization is the process that removes an electron from the neutral gas phase atom, eg,
Li(g) Li+(g) + e(g) I1 = 520 kJ/mol
M+(g) M2+(g) + e(g)
Li+( g) Li2+(g) + e(g) I2 = 7300 kJ/mol
1st IE
2nd IE
These reactions require energy (endothermic).
Periodic Trends
2) The Ionization Energies (I) within the Periodic Table
I1 = E(M+, g) – E(M, g)
I2 = E(M2+, g) – E(M+, g)
a) Electron Impact
M
Light (Eh) M e ejected from Mif Eh high enough
I = Eh - Eelkin
+V
Determining Ionisation Energies
e accelerated through potential
b) Photo Electron Spectroscopy
M(g) + e(g) M2+(g) + 2 e(g)
Trends in Ionisation Energies
General increase across periods
But kinks?Let’s recall….
Rcall: The orbital energies
2
2
2
2 hc)(hcE
n
RZ
n
RZ yyeffnl
Koopman’s Theorm
I orbital energy
Together with our trend in (Zeff/n)2 across the period, that explains the general trend beautifully
But the kinks?
11stst Ionization Energies in the 1 Ionization Energies in the 1stst Period PeriodH He Li Be N O F NeB C
Recall: Maximizing the number of parallel Recall: Maximizing the number of parallel spins - The exchange interactionspins - The exchange interaction
Quantum mechanical in origin Arguments based on the fact that total
wavefunction has to be antiparallel with respect to exchange of the electrons (Pauli)
Nothing to do with the fact that electrons are charged!
Result is that each electron pair with parallel spins leads to a lowering of the electronic energy of the atom
Space for extra Notes
B
Al GaIn Tl
F
ClBr
Ionisation Energy /eV
2 3 4 5 6n
Moving on through the Periodic Table
Group 13
Group 17I
As
Note large increase in IE from K to Cu (10 extra units of nuclear charge badly shielded by d electrons and again from Rb to Ag). Between Cs and Au the 4f shell is filled giving a total increase of 24 units of nuclear charge!
14121086420
IE(eV)
H
LiNa
K Rb Cs
Cu AgAu
Moving on through the Periodic Table
1 3 5n
More on Ionization EnergiesMore on Ionization EnergiesIonization Potentials tend to for the successively heavier elements within a period as the number of protons in the nucleus increases and electrons are successively added to the same shell (Zeff increases at constant n).However, some irregularities (penetration of p vs s and due to exchange energy contributions) occur.Ionization Potentials tend to for the successively heavier elements in a group in the periodic table (as Zeff increases but n also increases and does so faster) .Note transition metal and lanthanide contraction affect these trends. The trends in second IE are similar but shifted by one atomic number:
The second IEs are than the first IE for that element.Also, note particularly large increases as we start to take electrons from inner shells, e.g., the first, second and third ionisation energies of beryllium are: 899 kJ mol-1, 1756 kJ mol-1 and 14846 kJ mol-1.
More Periodic TrendsMore Periodic Trends3) Electron Affinities (EAs) within the Periodic Table
X -(g) X(g) + e
Cl- (g) Cl(g) + e E = 348 kJ/mol
The amount of energy needed to remove an electron from a negative ion = amount of energy released when a neutral atom in its ground state gains an electron.
Together with IEs, EAs tell us about chemical bonding: if M has a low ionization energy an X a high EA, then it is likely that
+ X - M + X M+
MX will be ionic
A positive electron affinity tells us that X -(g) has a lower (more favorable) energy than the neutral atom, X(g).
3) The Electron Affinities
1s22s2 2p51s22s2 2p6
F - F
1s22s2 2p31s22s2 2p2
C C-
Note: EA always less than IE due to extra electron repulsion on adding an electron!
4) Atomic Radii
2s1
2s22p6
3s1
3s23p6
Decrease along period
Increase down group
Again, the Lanthanide Contraction
Nb(Z=41) and Ta(Z=73) have identical atomic radii
4) Atomic Radiia) Atomic radii generally decrease moving from left to right within the periods
(nuclear charge keeps on increasing but electrons are added to the same shell),
eg, going from Li (1s2 2s1) 157pm to F (1s2 2s22p6) 64pm; for both n = 2
b) Atomic radii generally increase down the group with increasing atomic
number as electrons are occupying more and more distant electron shells, eg,
going from Li (1s2 2s1) 157pm to Cs (1s2 2s22p63s23p64s23d104p65s24d105p66s2)
272pm
c) There is a large increase as electrons go into next shell (like
between He and Li or Ne to Na)
d) All anions are larger than their parent atoms and all cations are smaller,
compare Be2+(27 pm) and Be (112pm), I(206pm) and I(133) – please note that
ionic radius depends on coordination number of ion
e) Ionic radii generally decrease with increasing positive charge
on the same ion (Tl+, 164pm > Tl3+, 88pm)
5) Electronegativity
•The electronegativity of an atom is a measure of its power when in chemical combination to attract electrons to itself•With few exceptions, electronegativity increases across the periodic table and decreases down a group,•F is far more electronegative than I•F is far more electronegative than Li
Appendices
1) Revisit: The Born Interpretation1) Revisit: The Born Interpretation
2 1d The wave function is normalised so that:
where the integration is over all space accessible to the electron. This expression simply shows that the probability of finding the electron somewhere must be 1 (100%).
2) R
adia
l Wav
efu
nct
ion
s
r
z
y
x
dr
rdrsind
d
r
rsinThe radius of the latitude is
Remember that the arc length, s, is given by s = r with in radians)
The Volume Element follows hence as dV = rsindrddrr2sindddr
3) Volume Element in spherical coordinates3) Volume Element in spherical coordinates
The Surface Element follows hence as dA = rsindrdr2sindd
4) Pauli Principle4) Pauli Principle
fermion2,1) = fermion (1,2)
boson2,1) = boson (1,2)
When the labels of any two identical fermions are exchanged, the total wavefunction changes sign.
When the labels of any two identical bosons are exchanged the total wavefunction retains the same sign.
Two particles (fermions)Two particles (fermions)
1,2) =(1)2
Total wave function of particles 1 and 2
Space wave functions of particles 1 and 2 residing in the same orbital (characterized by the same n, l, ml)
Total Spin wave function of particles 1
and 2
Now exchanging labelsNow exchanging labels
As it is just a product and a x b = b x a!
1,2) =(1)2
And in that is easy: (1)2(2)1
2,1) =(2)1exch
ange
But in the spin wave functions But in the spin wave functions ??
+
But in the spin wave functions But in the spin wave functions ??
-
ex
ex
ex
antisymmetric
symmetric
symmetric
In analogy:In analogy:
+ex
symmetric
Hence, the only allowed overall Hence, the only allowed overall wavefunction is:wavefunction is:
With an antiparallel arrangement of spins
1,2) =(1)2
-
5) Slater’s Rules5) Slater’s Rules
Approximate method for estimating the effective nuclear charge
Zeff = Z - S
Where Z is the actual nuclear charge and S is a shielding constant.
Computing SComputing S
1. Divide orbitals into groups
(1s) (2s2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f)
Note s and p with same n grouped together
2. There is no contribution to S from electron to the right of the one being considered.
3. A contribution is added to S for each electron in the same group as the one being considered – except in the (1s) group where the contribution is 0.30.
3. If the electron being considered is in an ns or np orbital, the electrons in the next lowest shell (n-1) each contribute 0.85 to S. Those electrons in lower shells ((n-2) and lower) contribute 1.00 to S.
4. If the electron being considered is an nd or nf orbital, all electrons below it in energy contribute 1.00 to S.
ExampleExampleFor P:
Zeff = 15 – 4 x 0.35 – 8 x 0.85 – 2 x 1.00 = 4.8
Trends right, actual values bads and p orbitals treated the same – huge differences for the orbitals in terms of penetration!!!!!