Download Atomic Structure and Electronic Configurations

Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Atomic Structure and
Electronic Configurations
Chemistry for Earth Scientists
DM Sherman
University of Bristol
Discovery of the Electron!
• 
• 
• 
• 
A Crookes tube consists of a cathode
and anode in a sealed glass vessel in
which there is a very small gas
pressure.
When a voltage is applied, a glowing
discharge results.
In 1897 JJ Thomson discovered that
the discharge observed in a Crookes
Tube (“cathode rays”) consisted of
charged particles.
He proposed that these very small
particles (later called electrons),
were fundamental components of
matter.
Page 1
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Nuclear Model of the Atom!
Gold foil
Alpha particle
(+ve charge)
By bombarding gold foil with alpha
particles, Ernest Rutherford
(1871-1937) established that
atoms consist of mostly empty
space with a positively charged
nucleus at the center. The
electrons must surround the
nucleus in clouds(?) or orbits (?)…
Discovery of the Proton!
•  It was realized that atoms must
contain positively charged
particles to counteract the charge
of the electrons.
•  Rutherford (1919) was able to
generate positively charged
hydrogen nuclei from the collision
between alpha particles and
nitrogen gas.
•  It was inferred that positively
charged hydrogen nuclei must be
present in all atoms. These
fundamental particles were called
protons.
Page 2
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Discovery of the Neutron!
•  In 1920, Ernest Rutherford
hypothesized the existence of an
uncharged particle as a further
component of the atomic nucleus
in order to explain the variations
of isotopic masses.
Atomic Spectroscopy!
Atoms absorb and (when
thermally excited) emit light but at
discrete (quantized) wavelengths.
Different elements have their
own set of absorption/emission
lines. The absorption lines
observed in the spectrum of the
solar heliosphere allow us to
work out the elements present
in the sun.
Solar spectrum
Page 3
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Review of Waves!
y = A sin(kx + ωt + φ) where k = 2π/λ
The frequency (ν) of a wave is given by
ν = c/λ
Interference of Waves!
The wave nature of light can be
seen by interference patterns and
diffraction.
Constructive
Destructive
Page 4
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Photoelectric Effect: Light as particles (photons)!
The energy for the ejected electrons depends on the frequency of
the light but not on its intensity.
This led Einstein to propose that light occurs as packets or particles
called photons with energy E = hν.
The Bohr Model of the Atom!
• 
• 
• 
Electrons orbit the nucleus at fixed
radii with quantized energies.
Atomic absorption and emission
lines result from excitations between
orbitals.
However, this violates classical
physics; the model offers no
explanation for quantization and fails
to predict the spectra of other
elements.
Balmer Series (visible light)
5 -2
4 -2
3 -2
Page 5
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Wave Nature of Electrons!
If light could act as a particle, perhaps
particles could act like waves…Louis de
Broglie (1892-1987) proposed that very
small particles may display wave
properties with a wavelength of
λ = h/mv (where h is Planck’s constant).
In 1927, this was verified for
electrons by the experiment of
Davisson and Germer where they
showed that electrons can diffract.
Postulates of Quantum Mechanics!
• Any system can be described by a wavefunction Ψ
• The probability of finding the system at
coordinates (r1,r2..,t) is given by |Ψ(r1,r2..,t)|2
• For every observable λ, there corresponds and
operator Λ which operates on the wavefunction:
ΛΨ = λΨ
Page 6
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Examples of QM Operators!
•  The operator for momentum p (a vector) is
! ∂
∂
∂ $
p = i # i +
j+
k & = i ∇
∂y
∂z %
" ∂x
•  The operator for total energy (a scaler) is
E =−
∂
i ∂t
 = h / 2π = 1.05459 × 10−34 Joule − sec
€
€
The Schrodinger Equation!
Since kinetic energy (T) is p2/2m we get
T=−
2 2
∇
2m
Let V be the potential energy. Since total energy E is T + V
€
TΨ +VΨ = EΨ
We get the Schrodinger equation:
€
−
2 2
∇ Ψ +VΨ = EΨ
2m
€
Page 7
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Particle in a Box!
Consider a particle in a one-dimensional box with length L. Inside
the box, the potential energy V = 0; outside the box V = ∞
Inside the box, the S.E. takes the form
−
2 d 2
Ψ = EΨ
2m dx 2
€
The Particle in a Box (cont).!
The solution to the SE must be of the form
Ψ (x ) = Asin(kx ) + B cos(kx )
Since Ψ(0) = 0 , we get
Ψ(0) = A sin(0) + B cos(0) = 0 so that B = 0
€
Since Ψ(L) = 0
Ψ(L) = A sin(kL) = 0
€
kL = nπ
where n = 1,2,3,...
€
Page 8
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Particle in a Box (cont).!
Now, the energy of the particle is given by the SE
−
2 d 2
Ψ = EΨ
2m dx 2
Plugging in
€
Ψ(x) = A sin(
nπ
x)
L
gives the result that
€
h 2n 2
E =
8mL2
with n = 1, 2…
€
The Particle in a Box (cont).!
The quantum spacing gets smaller as the box gets bigger
or as the mass gets bigger relative to h. This is why we
don’t see quantized motion in our macroscopic world.
Page 9
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Free Particle!
The quantized energies of the particle in a box
result from the particle being bound. The
wavefunctions of the particle are standing waves.
Imagine if we made the box dimensions very large.
The spacing between the quantum levels would go
to zero.
For a free particle (an infinite box), there is no
quantization and the wavefunction is that of plane
wave with continuous energies.
The Hydrogen Atom!
Here, the potential energy (V) of the electron is
given by Coulombs Law:
electron
−e2
V=
r
-e
r
€
+e
nucleus (1 proton)
Page 10
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Hydrogen Atom!
It is convenient to transform the problem to spherical
coordinates:
z
x = r sinθ cos φ
y = r sinθ sinφ
z = r cos θ
(r,θ,φ)
r
θ
φ
y
x
€
The Hydrogen Atom!
The Schrodinger equation becomes:
−2 % 1 ∂ % 2 ∂ψ (
1
∂ %
∂ψ (
1
∂ 2ψ (
r
+
sin
θ
+
'
*
'
*
2m '& r 2 ∂r & ∂r ) r 2 sinθ ∂θ &
∂θ ) r 2 sin2 θ ∂φ 2 *)
−
Ze2
ψ = Eψ
r
We can use a separation of variables to write
€
ψ (r) = ψ (r ,θ ,φ ) = R(r )Θ(θ )Φ(φ )
€
Page 11
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Hydrogen Atom!
We then get separate equations for R, Θ and Φ
1 d 2Φ
= −ml2
2
Φ(φ ) dφ
€
1 + 1 d $
dΘ '
ml2 .
2
sin
θ
−
0=m
&
)
2
Θ(θ ) , sinθ dθ %
dθ ( sin θ /
1 ⎡ d ⎛ 2 dR ⎞ 2 2m ⎛
e2 ⎞ ⎤
⎢ ⎜ r
⎟ + r 2 ⎜E + ⎟R ⎥ = β
R(r ) ⎢⎣ dr ⎝ dr ⎠
r ⎠ ⎥⎦
 ⎝
€
€
The Hydrogen Atom: quantum numbers!
The three separate equations for R, Θ and Φ yield three
different quantum numbers n, l and ml
•  Principle quantum number (from r):
n =1, 2, 3…
•  Angular momentum quantum number (from θ):
l = 0, 1, 2 ,…, n-1.
(s, p, d and f for l = 0,1, 2 and 3)
•  Magnetic quantum number (from φ):
ml = -l,- l +1,…,+l.
However, we also have “Spin quantum number” from relativity:
ms = +1/2 and -1/2 (“up” and “down”)
Page 12
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
The Hydrogen Atom: orbitals!
The hydrogenic wavefunctions
(orbitals) are spherical waves
with shape determined by the l
quantum number and orientation
determined by the ml quantum
number.
The orbital energy is
determined from the n quantum
number:
1 ⎛ e2 ⎞
En = − 2 ⎜
⎟
n ⎝ 2a0 ⎠
€
What about the other atoms..!
If we have more than one electron, it gets very complicated.
Consider a He atom with two electrons at position r1 and r2.
The potential energy takes the form
V(r1,r2 ) =
€
−e2 −e2
e2
+
+
r1
r2
r1 − r2
Electron-electron repulsion term:
This is the problem!
Page 13
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Electronic Configurations: Pauli Exclusion
Principle
We can approximate the electronic
structures of multielectronic atoms
as electronic configurations over
one-electron (hydrogen-like)
orbitals.
The Pauli Exclusion principle
states that no two electrons in an
atom can have the same four
quantum numbers.
Hence, we can only put two
electrons (one spin-up and one
spin-down) in each nlm orbital.
The Quantum Numbers and the Periodic Table
of the Elements
Page 14
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Notation for X-ray Absorption/Emission!
Low level High level
Name of
emission
K (1s)
L3 (2p3/2)
Kα1
L2 (2p1/2)
Kα2
M3 (3p3/2)
Kβ1
L3 (2p3/2)
M5 (3d5/2)
Lα1
L2 (2p1/2)
M4 (3d3/2)
Lβ1
M5 (3d5/2)
N7 (5p3/2)
Mα1
The Electromagnetic Spectrum!
Vibrational
Modes
Electronic Transitions
Nuclear
Transitions
Page 15
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Spectroscopic Methods!
Method
X-Ray
Absorption
Energy
Range
E > 0.5
keV
Processes
Information Obtained
Atomic core
electron
excitations
Element abundances; coordination
environments via
XANES and EXAFS. Distinguish
surface complexation and
precipitation from solid solution
Identify and quantify aqueous
complexes via crystal field and
LMCT transitions.
Optical
0.5 < E < Electronic
Absorption and 4.5 eV
excitations of
Reflectance
transition metal
d-electrons.
Infrared and
< 0.5 eV Vibrations
Raman
and molecular
rotations.
Mossbauer
14.4 keV Nuclear
(for Fe) transitions
Identify complexes from vibrational
spectroscopy.
Site occupancies and oxidation
states of Fe; magnetic properties
Synchrotron Radiation!
This is the light emitted by electrons when moving in
curved trajectory. Synchrotron radiation provides a
powerful source of light from the IR to X-rays.
Diamond Light Source, UK
Page 16
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Beamlines at Diamond Light Source!
X-Ray Absorption Spectroscopy
O
O
4s,4p
3d
3p
3s
XANES = X-ray
absorption near-edge
spectroscopy
Ni
Photoelectron
O
O
EXAFS = Extended Xray absorption nearedge spectroscopy
X-ray photon
2p
2s
1s
Page 17
Chemistry for Earth Scientists
DM Sherman, University of Bristol
2011/2012
Summary
Important Concepts:
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Fundamental particles
Atomic spectra
Wave-particle duality
Quantum mechanical model of the hydrogen atom
Electronic configurations for multielectronic
atoms in terms of the hydrogenic orbitals but
obeying the Pauli Exclusion principle.
Page 18