Download Section 2 Notes

In 1924, the French scientist Louis de Broglie pointed out that in many
ways the behavior of Bohr’s quantized electron orbits was similar to the
known behavior of waves.
The Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle states that it is impossible to
determine the position and velocity of an electron at the same time.
The Schrödinger Wave Equation Provides a mathematical model to
describe the motion of particles like electron as waves which supports
quantum theory.
The Bohr model was a one-dimensional model that used one quantum
number to describe the electrons in the atom. Only the size of the orbit
was important in the Bohr Model, which was described by the n
quantum number.
Schrödinger described an atomic model with electrons in three
dimensions. This model required three coordinates, or three quantum
numbers, to describe where electrons could be found. The three
coordinates from Schrödinger's wave equations are the principal
quantum number (n), angular quantum number (l), and magnetic
quantum number (m). These quantum numbers describe the size,
shape, and orientation in space of the orbitals of an atom.
We will come back to the Quantum Numbers shortly, but first a word
from Our Sponsor:
The Standing wave: de Broglie Wave
Wave Behavior Interference: When waves
overlap each other and change
to become one final wave
Diffraction: The bending of a
wave as it passes by the edge
of an object
Derived from the wave Particle Duality
The Basis for Quantum
Theory
wave properties of electrons
and other very small particles
are described mathematically
The Bohr model: The first
model to incorporate all of the
particles of an atom, it is a
stepping stone to the modern
model of the atom. Bohr used
hydrogen for his example; this
meant that he only modeled
the principle quantum number.
Schrödinger described an
atomic model with electrons in
three dimensions. He took the
Bohr model to the next level.
Electron Wave Behavior as
a standing wave “de Broglie
Wave”.
Electrons as de Broglie waves
Returning now to the problem of the atom, it was realized
that if, for a moment, we pictured the electron not as a
particle but as a wave, then it was possible to get stable
configurations. Imagine trying to establish a wave in a
circular path about a nucleus. One possibility might look
like the illustration to the left. Figure 12.2: This is an
unstable wave orbit; for this configuration, the wave starts at a given
point of the orbit, and ends up after one complete revolution at a different
point on the wave. The incoming wave will then be out of phase with the
original wave, and destructive interference will occur; which will destroy
the wave
Imagine a rollercoaster ride;
the operators have a switch
that can divert the
rollercoaster cars off the
rollercoaster. This is like the
unstable wave orbit, it cannot
return to the same spot in the
wave. This is what happens in
destructive interference, the
“electron” car is knocked out
of orbit.
However, certain stable configurations are possible, as is
illustrated below; the wave ends up in phase with the
original wave after one complete revolution, and
constructive interference results. Such a pattern would
result in a stable orbit. These types of wave are called a
standing wave, and are common in other contexts; for
example, they can be established on a string attached to a
wall if the string is moved up and down at exactly the
right speed (such a wave would appear not to be moving, which is why
it's called a standing wave).
HOWEVER
Imagine a rollercoaster ride
again; this time the operators
leave the rollercoaster cars on
the rollercoaster. This is like
the stable wave orbit, it will
return to the same spot in the
wave. The “electron” car will
follow the orbit – the electron
stays in the energy level (on
the ladder rung).
Now back to Quantum Numbers.
Schrödinger’s Quantum Numbers
1. Principal (shell) quantum number - n
• Describes the energy level within the atom.
• Energy levels are 1 to 7
• Maximum number of electrons in n is 2 n2
Schrödinger’s Quantum #’s
Principal: the higher the n value
the higher the energy; and the
lower in the periodic table the
element is. For example if n=3
the element’s highest energy is
more than an element with n=2
2. Momentum (subshell) quantum number - l
• Describes the sublevel in n
 Sublevels in the atoms of the known elements are s - p - d - f
 Each energy level has n sublevels.
 Sublevels of different energy levels may have overlapping
energies.
• The momentum quantum number
also describes the shape of the
orbital.
 Orbitals have shapes that are best
described as spherical (l = 0),
polar (l = 1), or cloverleaf (l = 2).
 Orbitals take on even more
complex shapes as the value of
the angular quantum number becomes larger.
Momentum quantum number: l
States the shape of “subshells”
The number of “subshells” is
equal to the energy level (The
Principal Quantum # - n)
The subshells are: s (l=0);
p (l=1); d (l=2); f (l=3)
Subshells describe the shape of
the orbitals
The d and f orbitals are
complex
3. Magnetic quantum number - m
Describes the orbital within a sublevel
 s has 1 orbital
 d has 5 orbitals
 p has 3 orbitals
 f has 7 orbitals
Orbitals contain 0, 1, or 2 electrons, never more.
m also describes the direction, or orientation in space for the
orbital.
This diagram shows the three possible orientations of a p orbital - px,
py, pz.
Atomic orbitals are studied in more detail in college.
4. Spin quantum number - s This fourth quantum number describes
the spin of the electron.
• Electrons in the same orbital must have opposite spins.
• Possible spins are clockwise or counterclockwise.
• The spin number is arbitrarily assigned the numbers + ½ and – ½
# of
Sublevel orbitals
s (l=0)
p (l=1)
d (l=2)
“m” values
1
3
5
7
0
-1, 0, 1
-2, -1, 0, 1, 2
-3, -2, -1, 0,
f (l=3)
1, 2, 3
9
-4, -3, -2, -1,
g (l=4)
0, 1, 2, 3, 4
Orbitals contain 0, 1, or 2
electrons
# of
Max # of
orbitals
electrons
Sublevel
s (l=0) 1
2
p (l=1) 3
6
d (l=2) 5
10
f (l=3)
7
14
g (l=4) 9
18
spin number + ½ called up spin
spin number – ½ called down
spin
In Conclusion:
The first three quantum numbers: the principal quantum (n), the angular quantum number (l) and
the magnetic quantum number (m) are integers.
The principal quantum number (n) cannot be zero: n must be 1, 2, 3, etc.
The angular quantum number (l) can be any integer between 0 and (n – 1).
For n = 3, l can be 0, 1, or 2. (only one of these values)
The magnetic quantum number (m) can be any integer between -l and +l.
For l = 2, m can be – 2, – 1, 0, +1, or +2.
The spin quantum number (s) is arbitrarily assigned the numbers + ½ or – ½