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Lecture 25
Quantum Mechanics of Atoms
and Atomic Systems
Chapter 28.10  28.14
Outline
• Quantum Mechanics and Periodic Table
• Atomic Structure and Conductivity
Quantum Numbers
Three quantum numbers determine the size and
shape of the probability cloud of an atomic electron.
n  the principal quantum number (0,1,2, … n)
l  the orbital quantum number (0,1,2, …n 1)
ml  the magnetic quantum number (l, … l)
The 4th, spin quantum number (ms) determines the
maximum number of electrons allowed on an orbit.
ms = +1/2 or 1/2
Energy Level Diagram for Hydrogen
Electron Shells and Subshells
All states with the same principal quantum number n
form a shell. K-shell corresponds to n=1.
States with a given n and l form a subshell.
Subshell s has n = 1, l = 0; p  n = 2, l = 1
The maximum number of electrons in a subshell is
2(2l+1).
Examples: shell K (n=1) has 1 subshell s (l=0) and the
maximum possible number of electrons 2 (ms=1/2).
Shell L (n=2) has 2 subshells (s and p, l=0,1). Two
electrons can be on s and 6 on p  total of 8 electrons.
Problems
Find the radius of the second orbit, ionization energy
from the 2nd orbit, and transition wavelength between
the 2nd and 3rd orbit in a singly ionized helium atom.
He+,
Atom:
Z=2
Orbits: nf = 2, ni = 3
Solution:
En =  13.6 Z2/n2 eV
rn = 0.0529 n2/Z nm
1/= RH Z2(1/nf2  1/ni2)
r2 = 0.0529 22/2 nm = 0.1058 nm
E2 = 13.6 22/22 eV = 13.6 eV
1/ = 1.097 107 22 (1/22  1/32) nm   = 164 nm
The Exclusion Principle
In 1925 Wolfgang Pauli proposed that:
No two electrons in an atom can have the same set
of the four quantum numbers.
This exclusion principle explains the electronic
structure of complex atoms.
The more electrons the atom has, the more levels is
filled.
Completely filled shells are called closed.
The Periodic Table
Halogens are followed in atomic number by noble
gases, then by alkali metals.
This is suggestive of a periodical law in properties
of chemical elements.
The Periodic Law
Elements (listed in order of atomic number) with
similar chemical and physical properties appear at
regular intervals.
The periodic law was first formulated by Dimitri
Mendeleev of Russia in 1869.
Predictions: unknown elements
Unexpected success: noble gases
Chemical Properties of Elements
Inert (Noble) Gases:
Outer shells are closed  the electrons are tightly
bound to the atom  chemically passive
Hydrogen and Alkali Metals:
Single outer electrons, which “see” only +e
charge and are loosely bound to the nucleus,
 lose the outermost electrons in chemical
reactions, have similar behavior, and are
chemically active
Chemical Properties of Elements
Halogen Atoms:
Outer shells lack 1 electron  tendency to pick up
such an electron through the strong attraction of the
poorly shielded nuclear charge.
Metals and Nonmetals:
Metals have 1 or several electrons outside the closed
shells  combine chemically by losing these
electrons to nonmetals
Nonmetals lack 1 or few electrons in the outermost
shell  combine by picking up electrons from metals
or by sharing electrons with other nonmetals
3D Periodic Table
Energy Bands
Closely spaced atoms become one quantum system.
The electron energy levels of different atoms form
continuous bands, separated by band gaps.
Electrons may have any energy within bands, but not
within band gaps.
Electrons are distributed from the lowest energy
levels up in agreement with the exclusion principle.
In metals, the highest energy band is partially full.
In insulators, it is completely full.
Summary
The exclusion principle allows to determine the
maximum possible number of electrons in each
shell and subshell.
Quantum mechanics explains chemical properties
of the existing variety of elements and large
atomic systems.