Download Atomic Theory

Atomic Structure and the
Periodic Table
Development of the Periodic
Law
Development of atomic weights
Dalton's Atomic Theory
Elements consist of atoms
Each atom of the same element is identical
Atoms of different elements are different
Molecules of compounds are formed by the
combination of two or more different atoms, in fixed
whole-number atom ratios
† In a chemical reaction, atoms are not created or
destroyed, but only rearranged into different
combinations
†
†
†
†
The Consequences
Atomic (combining) weights
† Atomic (combining) weights
† Avogadro's combining volumes
† Cannizzaro's atomic weights
if a reaction is balanced as:
A + B --> C
then x grams of A and y grams of B must form z grams of
C, where x, y, and z are the atomic or molecular weights
of A, B, and C.
therefore if
1 g of hydrogen reacts with 8 g of oxygen to form 9 g of
water, and the atomic weight of hydrogen is 1, then the
atomic weight of oxygen is 8.
1
Avogadro's Combining Volumes
Ordering schemes
Hydrogen gas +
Oxygen gas →
Water
2 volumes
1 volume
2 volumes
Simple gaseous elements must be diatomic molecules
Combining weights are not always atomic weights
Dobereiner's Triads
Telluric Helix
† Similar elements occur in three's and
the atomic weight of the second is
approximately equal to the average of
the atomic weight of the first and
third.
A vertical cylinder with 16 equidistant lines
drawn parallel to its axis. Draw a helix at 45
to the axis and arrange the elements on the
spiral in order of increasing atomic weight. In
some cases, elements that lie directly under
each other show similarity.
The properties of the elements are the
properties of numbers
„ Cl 35; Br 80; I 127; average 81
„ S 32; Se 79; Te 128; average 80
„ Li 7; Na 23; K 39; average 23
Telluric Helix
The Law of Octaves
† Correlate the chemical properties of
the elements with their atomic
weights
2
The Law of Octaves
H
Li
G
B
C
Mendeleev's periodic law
N
O
F
Na
Mg
Al
Si
P
S
Cl
K
Ca
Cr
Ti
Mn
Fe
Co & Ni Cu
Zn
Y
In
As
Se
Br
Rb
Sr
Ce & La Zr
Di & Mo Ro & Ru
Pd
Ag
Cd
U
Sn
Sb
Te
I
Cs
Ba & V Ta
W
Nb
Au
Pt & Ir
Tl
Pb
Hg
Bi
Os
Th
† If the elements are ordered by atomic
weight, there is a similarity between every
eighth element.
† The properties of the elements can be
represented as periodic functions of
their atomic weights.
† emphasize group similarities,
reversing the order of atomic weights
where necessary.
† Leave vacancies for undiscovered
elements, predicting properties.
Development of a model of the
atom
Mosley
† X-ray spectra of the elements
† ν = K(Z - k)
† where K and k are constants, and Z is
the atomic number
Line Spectra
Thomson Model
n
wavelength
frequency
3
656.3 nm
4.57 x 1014 Hz
4
486.1
6.17 x 1014
5
434.0
6.91 x 1014
6
410.2
7.31 x 1014
† sphere with uniform positive charge
† negatively charged electrons imbedded in
the sphere, with a uniform distribution.
ν = R(1/22 - 1/n2)
where R is a constant (Rydberg constant), and n is an
integer greater than 2 which corresponds to a line.
3
Thomson Model
Thomson Model
† Line spectra are due to the oscillation of the
electrons
† Electron emission by active metals due to loss of
an imbedded electron.
Problems
Rutherford Model
†
†
†
scattering of alpha particles
Using this model, there
should be only very small
angle scattering of alpha
particles.
experimentally, small angle
and large angle scattering is
observed
† positive charge is
concentrated in a
small nucleus at
the center of the
atom
† electrons moving
around the nucleus
Problems
Bohr Model
† Classical physics
† electrons in orbits around the nucleus
† electrons are moving at a high velocity, such that
the Centrifugal force balances the electrostatic
attraction.
m v2
e2
=
r
r2
4
Bohr Model
Bohr Model
† Only certain orbitals
are possible, meaning
that only discrete
energy states are
allowed
† Energy is absorbed or
emitted only when
there is a change of
energy state
If a change of state occurs, then the energy
associated with it must be
Bohr Model
Bohr Model
the energy of an electron in the nth
level is found to be
the light emission is
En =
∆E = E2 - E1 = hν21
ν21 = (E2 - E1)/h
2π me Z
n2 h2
2
4
2
2 π 2 m e4 Z 2  1
1 
- 2
ν=

2
2
ch
 n2 n1 
this means that 2π2me4/ch3 must be the Rydberg constant
Problems
Bohr-Sommerfeld Model
† Fine structure in line spectra
† Allow orbitals to be elliptical
† The two degrees of freedom requires
two quantum numbers. The second
describing the elongation of the
ellipse.
5
Problems
New Approach
† Zeeman Effect—the splitting of
spectral lines in the presence of a
magnetic field.
deBroglie Relationship
Schrodinger Equation
† deBroglie considered equating the
energy of light with that of a moving
particle.
Using deBroglies idea, consider an
electron as a wave disturbance rather
than as a particle.
δ 2ψ + δ 2ψ + δ 2ψ + 8 π 2 m (E - V )ψ = 0
2
δ x 2 δ y2 δ z 2
h
Polar Coordinates
Separation of Variables
† assumption Ψ(r,θ,φ) = R(r)Y(θ,φ)
6
Radial Distribution Function
Radial Density Function
 2Z  (n - l - 1)! - ρ/2 ρ 2 l+1
ρ Ln +1 ( ρ )
R n ,l (r) = 
3 e

 n a o  2n [(n + 1)!]
3
ao =
h2
4 π 2 µ e2
determines relationship between n and l quantum numbers
n = 1, 2, 3,...
l = 0, 1, 2,... (n-1)
(n - l - 1)! >= 0 so l cannot be .GE. n
Spherical Harmonics
Spherical Harmonics (1)
θ ml =
Yml(θ,φ) = θml(θ) φ m (φ)
(2l + 1)(l − m )!
2(l + m )!
P (cosθ )
m
l
determines relationship between l and
m quantum numbers
l = 0, 1, 2, 3,...
m = 0, ±1, ±2,
±3,..., ±l
(l - |m|)! >= 0 so |m| cannot be .GE. l
Spherical Harmonics (2)
Φm =
Orbital Shapes-Total Density
Function
1 ± imφ
e
2π
m = 0, 1, 2, 3,...
7
Quantum Number Relationships
Orbital Filling
† Within a given atom, the lower the value of n, the
more stable (or lower in energy) the orbital
† There are n types of orbitals in the nth energy level
† There are 2l + 1 orbitals of each type
† There are n - l - 1 nodes in the radial distribution
functions of all orbitals (spherical nodes)
† There are l nodal surfaces in the angular distribution
function of all the orbitals
† There are n - 1 nodes in the total distribution
functions of all the orbitals
† Aufbau
principle--Starting
with the hydrogen
atom, and
successively adding
a proton to the
nucleus and an
electron to the
electron cloud.
Empirical Rules
Electron Spin
† Orbital energies increase as (n+l)
increases
† For two orbitals of the same (n+l),
the one with the smaller n lies lower
in energy
† There still were unexplained doublets
in the line spectra where single lines
were expected.
† explaination by allowing the electron
to spin with a spin angular
momentum of 1/2 in units of h/2π
Pauli Exclusion Principle
Correlation of Electron Motion
† No two electrons may have the same
four quantum numbers
† spin correlation(exchange
energy)--energy is lower when two
electrons have the same spin
† Charge correlation energy--lower
energy if electrons have different
angular functions
8
Electron Configuration
Aufbau Diagram
Ambiquities in the Aufbau Diagram
Writing Electron Configurations
Cr 3d54s1
Mo 4d55s1
W 5d36s2
† Using quantum numbers
† Using spdf notation
Cu
Ag
Au
3d104s1
4d105s1
5d106s1
Term Symbols
† The energy of the individual orbitals
are dependent on n and l
† Which is also influenced by the
coupling between the orbital
momentum and spin momentum
† And replusion between the electron
Total Orbital Angular Momentum
Quantum Number L
Assume that all of the angular momenta of the
different electrons, li, in an atom couple
together.
L = l1 + l2, l1 + l2 - 1, l1 + l2 - 2 ,..., |l1 - l2|
L is derived from Ml, which is the vectorial sum
of the Ml values for all electrons.
for n electrons Ml = ml(1) + ml(2) + ... ml(n)
and Ml = L, L-1, L-2, ..., -L
the number of possible values for Ml will be 2L+1
9
Total Spin Angular Momentum
Quantum Number S
Assume that the individual electron spins couple
together.
S is derived from Ms, which is the algebraic sum
of the s values for all electrons
for n electrons Ms = ms(1) + ms(2) + ... ms(n)
and Ms = S, S-1, S-2, ..., -S
the number of possible values for Ms will be 2S+1
Determining Term Symbols
† Closed shells and subshells are
ignored.
Term Symbols
Derive the term symbol for ground
state Li (1s22s1)
The term symbol is constructed as follows
2S+1X
where 2S+1 is the spin multiplicity and comes
from S, and X is a letter that comes from the
value of L
for L = 0, X is S
L = 1, X is P
L = 2, X is D
L = 3, X is F
Discount all closed shells, ie the 1s
Find the possible values for Ml
Ml = ml(1) = 0
Find the possible values for L
Ml = 0, therefore L = 0
Find the possible values for Ms
Ms = ms, ms = +1/2 or -1/2
Ms = +1/2 or -1/2
Derive the term symbol for ground
state Li (1s22s1)
Derive the term symbol for ground
state C (1s22s22p2)
Find the possible values for S
Ms = +1/2 or -1/2
S = 1/2
The term symbol
since L = 0 then X is S
S is 1/2
therefore the ground state of Li is
2S
† Draw the individual microstates
10
Derive the term symbol for ground
state C (1s22s22p2)
Derive the term symbol for ground
state C (1s22s22p2)
The possible values of
Ml is 2, 1, 0, -1, -2,
and values for Ms are
1, 0, -1
Use + and - to
indicate spin and 1, 0,
or -1 to indicate the
orbital
† The microstate with Ml
= 2 and Ms = 1 must
be eliminated due to
the Pauli exclusion
principle. To have Ml
of 2 means that the
electrons share the
same orbital, and a Ms
of 1 indicates that they
have the same spin.
Derive the term symbol for ground
state C (1s22s22p2)
Derive the term symbol for ground
state C (1s22s22p2)
If we start at the top, there is a
microstate with Ml = 1 and Ms = 1,
which is indicative of a state that has L
= 1 and S = 1, the 3P state. A 3P state
has a total of 9 microstates
Ml = 0, Ms = 1, Ml = -1, Ms = 1,
Ml = 1, Ms = 1
Ml = 0, Ms = 0, Ml = -1, Ms = 0,
Ml = 1, Ms = 0
Ml = 0, Ms = -1, Ml = -1, Ms = -1,
Ml = 1, Ms = -1
remove them from the array
If we look at the very top,
there is a microstate with
Ml = 2 and Ms = 0, which
is indicative of a state that
has L = 2 and S = 0, the
1D state.
A 1D state has a total of 5
microstates
Ml = 1, Ms = 0; Ml = 0, Ms = 0;
Ml = -1, Ms = 0
Ml = -2, Ms = 0; Ml = 2, Ms = 0
Derive the term symbol for ground
state C (1s22s22p2)
Derive the term symbol for ground
state C (1s22s22p2)
What remains is a
single microstate with
Ml = 0 and Ms = 0.
This microstate
indicates that there is
a term having L = 0
and S = 0, or 1S.
Thus we have three possible term
symbols for the ground state
configuration of carbon, 3P, 1D, and 1S.
Which one is the ground state?
11
Hund's First Rule
Order of Excited States
the term with the maximum spin
multiplicity will be lowest in energy,
therefore the term symbol for the
ground state of carbon is 3P.
The 1D and 1S terms will be excited
states
Hund's second rule: If more than one
term has the same multiplicity, the term
with the highest L value will be lower in
energy.
Thus, the 1D state will be the first
excited state.
The Total Angular Quantum
Number J
Hund's Third Rule
J is determined by taking the sum and
difference between L and S, that is L +
S and L - S, and then take the sequence
of numbers from L + S to |L - S| in
integral steps.
Therefore, for the 3P state there are
three more specific states 3P0, 3P1, and
3P
2
the term with the lowest J value lies
lowest in energy if the partially filled
subshell is less than half-filled.
Splitting of Orbital Energy
An Easier Method
1S
1 microstate
1D
5 microstates
3P
9 microstates
p2
15 microstates
no electronelectron
repulsions
electronelectron
repulsions
1S
_
0
1 microstate
1D
_
2
5 microstates
3P
0
will be the ground state
Partial Terms
D. H. McDaniel, J. Chem. Ed. 1977,
54 147.
3P
_
2
5 microstates
3P
_
1
3 microstates
3P
_
0
1 microstate
spin-orbit
coupling
12
Derive the term symbol for ground
state C (1s22s22p2)
If we just consider the electron spin, in
a p2 configuration, three spin
configurations are possible
p0a p2b
p1a p1b
p2a p0b
Where pa has a spin of +½ and pb has a
spin of -½
Orbital Occupancy
Orbital
Set
s
p
d
f
g
0
1
2
3
S
S
S
S
S
4
5
P
D
F
G
P
PF
PFH
PFHK
S
PF
D
S
SDFGI SDFGI PFH
PF(2)G SD(2)F
HIKM G(2)HI(2)
KLN
6
7
F
S
Derive the term symbol for ground
state C (1s22s22p2)
Derive the term symbol for ground
state C (1s22s22p2)
Each spin configuration may be
identified with one or more partial terms
p2a p0b
P × S => P
P × P => S P D
p1a p1b
To determine the multiplicity, we look at the
maximum Ms value for each partial term.
P maximum Ms = 1, so the term symbol is 3P
D maximum Ms = 0, so the term symbol is 1D
P maximum Ms = 0, so the term symbol is 1P
S maximum Ms = 0, so the term symbol is 1S
product of partial terms gives terms
having L values in integer runs from
|La-Lb| through |La+Lb|
Derive the term symbol for ground
state C (1s22s22p2)
Derive the term symbol for ground
state C (1s22s22p2)
Flaw in partial terms
Some microstates will be assigned to
more than one term symbol
Thus we have three possible term
symbols for the ground state
configuration of carbon, 3P, 1D, and 1S.
For each term symbol of a higher
multiplicity, one may eliminate one of
the same value of L with a lower
multiplicity.
13
Determining the number of
microstates
number of microstate s =
n!
e! n!
where n is the number of possible positions
for an electron, e is the number of electrons
and h is the number of leftover positions
(n - e).
f3 configuration
three spin configurations are possible
f2a f1b
f3a f0b
and a total of 364 microstates are
possible (14!/(3!*11!)
f3 configuration
f3 configuration
f3a f0b (SDFGI) x S
=>
4S, 4D, 4F, 4G, 4I
† Term Symbols: 4I, 4G, 4F, 4D, 4S, 2L,
2K, 2H(2), 2G(2), 2F(2), 2D(2), 2P
† Ground State: 4I
f2a f1b (PFH) x F =>
2G, 2F, 2D
2I, 2H, 2G, 2F, 2D, 2P,
2S
2L, 2K, 2I, 2H, 2G, 2F,
2D
Polyelectron factors
Slater's Rules
Penetration (shielding):
outer electrons do not "see" the same
nuclear charge as electrons below them.
ie a different effective nuclear charge.
†
†
†
†
The electrons are grouped in the order 1s, 2s & 2p, 3s &
3p, 3d, 4s and 4p, 4d, 4f, etc., with the s and p orbitals
grouped together.
Electrons in groups lying above that of a particular electron
do not shield it at all.
A shielding of 0.35 is contributed by each other electron in
the same group, except for a 1s electron which contributes
0.30 to the shielding of the other 1s electron
For d and f electron the shielding from underlying groups is
1.00 for each electron in the underlying group. For s and p
electrons the shielding from the immediately underlying
shell (n - 1) is 0.85 for each electron, and that from groups
further in is 1.00 for each electron.
14
Slater's Rules
Ionization Energies
Energy level calculation
Outer-most electron leaves first.
Electron with largest n lost first
Within the nth shell the electron with
the largest l.
− ( Z − S ) 213 .6eV
E=
n2
Trends in Ionization Energies
Ionization Energy (MJ)
2.50
Energy released when an electron is
added to a neutral atom.
X + e- ---> X- + energy
He
Ne
2.00
F
N
1.50
H
1.00
Be
0.50
Li
C
Ar
Cl
P
O
S
Mg
B
Na
Si
Al
0.00
0
5
10
Atomic Number
Atomic Radii
Electron Affinity
15
20
E. A. decreases down a group
Cl (3.61) > Br (3.36)
E. A. increases across a group
Covalent Radii
One half the bond length in a non-polar
homonuclear bond
Factors effecting: bond polarity, # of ein outermost orbital, nuclear charge
15
Ionic Radii
16