Download is the “quantum number”

Quantum-Mechanical View of Atoms
Since we cannot say exactly where an electron is, the Bohr picture
of the atom, with electrons in neat orbits, cannot be correct.
Quantum theory describes an
electron probability distribution;
this figure shows the distribution
for the ground state of hydrogen:
Quantum particle in a box
1-dimensional box
U(x)
potential
energy
L
0
U x   

x
0 x L
x  0 or x  L
We have standing wave:
k
n
L

x  C sin kx
2 2 L

k
n
C
2
L
n=1,2… - is the “quantum number”
h
nh
Energy and momentum are quantized: pn  
 2L
p 2 n2h2
En 
 2
2m 8 L m
Example: What is the energy difference between the first excited state and the
ground state of an electron in the “box” of size L=1nm?
2 2 h 2 12 h 2
3h 2
E  E2  E1  2  2  2
8L m 8L m 8L m


3  6.63
19
19
E 

10
J

1
.
8

10
J  1.1eV
2
9
31
8  10 m  9.1110 kg 8  9.11

3  6.63 10 34 J  s
 
2
2

3-dimensional box
We have 3 independent standing waves, and 3 independent quantum numbers.
E
p x2  p y2  p z2
2m

n

2
x

 n y2  nz2 h 2
8L2 m
The hydrogen atom
potential energy:
e2
U x   
40 r
1
•The electron is moving in 3-dimensional space.
•Because of that, we can expect 3 independent external quantum numbers.
•However, the potential energy is function of one coordinate, r.
•Because of that, we can expect that the energy depends only from one of
these 3 numbers.
•In addition, the electron has one internal quantum number.
The hydrogen atom
There are four different quantum numbers needed to specify the state of an
electron in an atom.
1) Principal quantum number n gives the total energy:
2) Orbital quantum number l gives the magnitude of the angular momentum.
(l can take on integer values from 0 to n – 1)
l  0, 1,... n  1
3) The magnetic quantum number, ml, gives the “direction” of the electron’s
angular momentum. ( ml can take on integer values from –l to +l )
ml  0,  1,...  l
4) The spin quantum number, ms, which for an electron can take on the
values +½ and -½.
The need for this quantum number was found by experiment; spin is an
intrinsically quantum mechanical quantity, although it mathematically
behaves as a form of angular momentum.
Angular momentum
This plot indicates the quantization of
angular momentum direction for l = 2.
The other two components of the
angular momentum are undefined.
L  l l  1  22  1  6
Lz  ml ,
ml  0,1,2
The angular momentum quantum numbers do
not affect the energy level of the hydrogen atom,
but they do change the spatial distribution of the
electron cloud.
Zeeman effect
In a magnetic field, the spectral lines are
split into several very closely spaced lines.
This splitting, known as the Zeeman effect,
demonstrates that the atoms energy levels
are split. This means that, in magnetic
field, the energy of state depend not only
on principal quantum number, n but also
on the “magnetic quantum number” ml.
Fine structure
A careful study of the spectral lines showed that each actually consist of several
very closely spaced lines even in the absence of an eternal magnetic field. This
splitting is called “fine structure”. It is related to the spin of electron.
Transitions between energy levels
“Allowed” transitions between energy levels occur between states whose
value of l differ by one:
Other, “forbidden,” transitions also occur but with much lower probability.
Photon has a spin angular momentum of 1ħ.
Complex Atoms
Complex atoms contain more than one electron, so the interaction
between electrons must be accounted for in the energy levels.
A neutral atom has Z electrons, as well as Z protons in its nucleus. Z
is called the atomic number.
Four quantum numbers: n, l, ml , ms can be used to describe an
electron in atom.
The energy depends mainly on n and l.
This table summarizes the four quantum numbers
The Pauli exclusion principle:
No two electrons in an atom can occupy the same quantum state.
More generally: No two identical particles whose spin quantum number
is a half-integer (1/2, 3/2,…), including electrons, protons and neutrons
can occupy the same quantum state.
The quantum state of an electron in atom is specified by the four quantum
numbers. According to the Pauli principle no two electrons can have the
same set.
The Periodic Table of the Elements
Electrons are grouped into shells and subshells:
•Electrons with the same n are in the same shell.
•Electrons with the same n and l are in the same subshell.
•The exclusion principle limits the maximum number of electrons in each
subshell to 2(2l + 1).
ml  0,1,...  l
ms   12
Example 1:
For n  1  l  0, ml  0, ms   12 
2 differnt states (maximum 2 electrons)
Example 2:
For n  2  l  0,1;
for l  0  ml  0;
ms   12  2 differnt states
for l  1  ml  0,1; ms   12  6 differnt states
total : 8 differnt states (maximum 8 electrons)
m
l
l
 2l  1
Electron configurations
Electron configurations are written by :
•the value for n
•the letter code for l
•and the number of electrons in
the subshell as a superscript
Notations:
Each value of l is given
its own letter symbol.
Example: The ground-state configuration of
sodium:
Sodium has 11 electrons (Z=11). Ten of them
form a closed neon-like core. The remaining
electron is the valence electron.
Example: A neutral atom of a certaine element has configuration given by :
1s 2 2s 2 2 p 6 3s 2 3 p 6 4s 2 3d 6 .
What is the atomic number of this element?
This table shows the configuration of the outer electrons only
Atoms with the same number of electrons in their outer shells have similar
chemical behavior. They appear in the same column of the periodic table.
The outer columns – those with full, almost full, or almost empty outer
shells – are the most distinctive.
The inner columns, with partly filled shells, have more similar chemical
properties.
Example: The electron configuration of the neutral fluorine atom in its ground state is
1s 2 2s 2 2 p 5
Make a list of the four quantum numbers of each electron in the fluorine atom.
n
1
1
2
2
2
2
2
2
2
l
ml
ms
orbital designation
Summary
• n, the principal quantum number, can have any integer value, and
gives the energy of the level
• l, the orbital quantum number, can have values from 0 to n – 1
• ml, the magnetic quantum number, can have values from –l to +l
• ms, the spin quantum number, can be +½ or -½
• Energy levels depend on n and l, except in hydrogen. The other
quantum numbers also result in small energy differences
• Pauli exclusion principle: no two electrons in the same atom can be in the
same quantum state
• Electrons are grouped into shells and subshells
• Periodic table reflects shell structure
Atoms with the same number of electrons in their outer shells have similar
chemical behavior. They appear in the same column of the periodic table.