Download Ch. 5 Sections 7.5-7.8 Powerpoint

Quantum Mechanical
Model of the Atom
Mathematical laws can identify the
regions outside of the nucleus where
electrons are most likely to be found.
These laws are beyond the scope of this
class…
Schrodinger Wave Equation

d

V 
8  m dx
h
2
2
Erwin Schrodinger
2
2
 E
Equation for probability of
a single electron being
found along a single axis
(x-axis)
Heisenberg Uncertainty Principle
“One cannot simultaneously
determine both the position
and momentum of an electron.”
You can find out where the
electron is, but not where it is
going.
OR…
Werner
Heisenberg
You can find out where the
electron is going, but not
where it is!
QUANTUM NUMBERS
We will focus now on polyelectronic atoms – atoms with
more than one electron.
The shape, size, and energy of each orbital is a function
of 3 quantum numbers which describe the location of
an electron within an atom or ion
n (principal) ---> energy level
l (orbital) ---> shape of orbital
ml (magnetic) ---> designates a particular
suborbital
The fourth quantum number is not derived from the
wave function
s (spin) ---> spin of the electron
(clockwise or counterclockwise: ½ or – ½)
Principal Quantum Number
The principal quantum number, n, denotes the probable
distance of an electron from the nucleus.
n = energy levels (shells) = 1, 2, 3, 4, ….
distance of e- from the nucleus
Angular Momentum Quantum Number
An orbital is a region within an energy level where there is a
probability of finding an electron.
The angular momentum quantum number, l, denotes the
orbital (subshell or sublevel) in which an electron is located.
for a given value of n, l = 0, 1, 2, 3, … n-1
n = 1, l = 0
n = 2, l = 0 or 1
n = 3, l = 0, 1, or 2
l=0
l=1
l=2
l=3
s orbital
p orbital
d orbital
f orbital
Shape of the “volume” of space that the e- occupies
s Orbital shape
The s orbital
(l = 0) has a spherical
shape centered
around
the origin of the
three axes in space.
p orbital shape
Planar node
There are three dumbbell-shaped p orbitals (l = 1)
in each energy level above n = 1, each assigned to
its own axis (x, y and z) in space.
Note: there is a planar node through the nucleus,
which is a area of zero probability of finding an
electron.
d orbital shapes
Things get a bit more
complicated with the five d
orbitals (l = 2) that are found
in the d sublevels beginning
with n = 3. To remember the
shapes, think of “double
dumbells”
…and a “dumbell
with a donut”!
Shape of f (l = 3) orbitals
Magnetic Quantum Number
The magnetic quantum number m, denotes the orientation of
an electron’s orbital with respect to the three axes in space.
for a given value of l
ml = -l, …., 0, …. +l
if l = 1 (p orbital), ml = -1, 0, or 1
if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2
orientation of the orbital in space
Assigning the Numbers
 The three quantum numbers (n, l, and m) are integers.
 The principal quantum number (n) cannot be zero.
 n must be 1, 2, 3, etc.
 The angular momentum quantum number (l ) can be
any integer between 0 and n - 1.
 For n = 3, l can be either 0, 1, or 2.
 The magnetic quantum number (ml) can be any integer
between -l and +l.
 For l = 2, m can be either -2, -1, 0, +1, +2.
Quantum numbers for the first four levels of orbitals in the hydrogen atom
n
l
Orbital
designation
ml
# of
orbitals
1
0
1s
0
1
2
0
2s
0
1
1
2p
-1, 0, 1
3
0
3s
0
1
1
3p
-1, 0, 1
3
2
3d
-2, -1, 0, 1, 2
5
0
4s
0
1
1
4p
-1, 0, 1
3
2
4d
-2, -1, 0, 1, 2
5
3
4f
-3, -2, -1, 0, 1, 2, 3
7
3
4
Spin Quantum Number
The spin quantum number, ms, describes the behavior of
an electron in a magnetic field.
spin quantum number ms
ms = +½ or -½
ms = +½
ms = -½
Pauli exclusion principle - no two electrons in an
atom can have the same four quantum numbers.
Each seat is uniquely identified (E,
R12, S8)
Each seat can hold only one
individual at a time
Shell – electrons with the same value of n
Subshell or sublevel – electrons with the same values
of n and l
Orbital – electrons with the same values of n, l, and
ml
How many electrons can an orbital hold?
If n, l, and ml are fixed, then ms = ½ or - ½
An orbital can hold 2 electrons
How many electrons can be in a sublevel
subshell?
Remember: A maximum of two electrons can be
placed in an orbital.
s orbitals p orbitals d orbitals f orbitals
Number of
orbitals
Number of
electrons
1
3
5
7
2
6
10
14