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7. 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:
1
Standing waves on a string
n = 1,2,3...
121 L L
L2
1
21
2222 L L
L
2
22
323 L
3
23
L
nnL 2
n
Ln
2
n
xCy
2sin
x
y
2
Quantum particle in a box
xL
U(x)
Lxx
LxxU
or 0
0 0
Standing wave:
n
xCx
2sin
1- dimensional box
2
n
Ln
Quantum number: n = 1,2…
Momentum and energy are quantized:mL
hn
m
pE
L
nhhp n
nn 2
222
82
2
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?
eVJJ
kgm
sJE
mL
h
mL
h
mL
hEEE
1.1108.11011.98
63.63
1011.9108
1063.63
8
3
8
1
8
2
19192
3129
234
2
2
2
22
2
22
12
potential energy
Wavelength:
3
3-dimensional box
We have 3 independent standing waves, and 3 independent quantum numbers.
The hydrogen atom
r
exU
2
04
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, electron’s energy depends only on one of these 3 numbers.•In addition, an electron has one internal quantum number.
mL
hnnn
m
pppE zyxzyx
2
2222222
82
Potential energy:
4
The hydrogen atom
1) Principal quantum number n gives the total energy:
There are four different quantum numbers needed to specify the state of an electron in an atom.
2) Orbital quantum number l gives the magnitude of the angular momentum. (l can take on integer values from 0 to n – 1)
1 ,...1 ,0 nl
3) Magnetic quantum number, ml, gives the “direction” of the electron’s
angular momentum. (ml can take on integer values from –l to +l )
lml ,...1 ,0
4) 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, althoughit mathematically behaves as a form of angular momentum. 5
Angular momentum
This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined.
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.
61221 llL
,lz mL 2,1,0 lm
6
Zeeman effectIn 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 structureA 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.
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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
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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.
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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).
21
,...1,0
s
l
m
lm 12 lml
l
Example 1:
electrons) 2 (maximum statesdiffernt 2
,0 0, 1For 21 sl mmln
Example 2:
electrons) 8 (maximum statesdiffernt 8 :total
statesdiffernt 6 ;1,01for
statesdiffernt 2 ;00for
0,1; 2For
21
21
sl
sl
mml
mml
ln
10
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
Electron configurations
Example: A neutral atom of a certain element has configuration given by:
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.
What is the atomic number of this element?
6262622 3433221 dspspss
Notations:
Each value of l is given its own letter symbol.
11
This table shows the configuration of the outer electrons only
12
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:
522 221 pss
Make a list of the four quantum numbers of each electron in the fluorine atom.
n l ml ms s
1
1
2
2
2
2
2
2
2 13
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 –1 to +1
•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.
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