Week 1 F - Par$cle
in
a
box
 a
 a
 • 

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Unformatted text preview: Par$cle
in
a
box
 a
 a
 •  The
par$cle
in
a
box
model
(also
known
as
the
infinite
poten$al
well
or
the
infinite
 square
well)
describes
a
par$cle,
which
is
free
to
move
in
a
small
space
surrounded
 by
impenetrable
barriers.

 •  One
dimension
only
and
the
poten$al
energy
V(x)
within
the
box
is
equal
to
0.
 •  This
means
that
the
par$cle
is
completely
trapped
inside
the
box,
no
forces
are
 ac$ng
on
the
it
inside
the
box.
 Difference
between
classical
and
 quantum
mechanics:
 •  In
classical
system,
the
par$cle
is
no
more
likely
to
be
found
at
 one
posi$on
than
another.

 •  However,
when
the
well
becomes
very
narrow
(on
the
scale
of
 a
few
nanometers),
quantum
effects
become
important.

 •  The
par$cle
may
only
occupy
certain
posi$ve
energy
levels
and
 it
can
never
have
zero
energy,
meaning
that
the
par$cle
can
 never
"sit
s$ll".

 •  Addi$onally,
it
is
more
likely
to
be
found
at
certain
posi$ons
 than
at
others,
depending
on
its
energy
level.
The
par$cle
may
 never
be
detected
at
certain
posi$ons,
known
as
spa$al
nodes.
 Par$cle
in
a
box
 Total
solu$on

 •  AMer
normaliza$on

 The
squared
wave
func$ons
are
the
probability
 densi$es,
and
they
show
the
difference
 between
classical
and
quantum
mechanical
 behavior.
 Classical
mechanics
predict
that
the
electron
 has
equal
probability
of
being
at
any
point
in
 the
box.
 The
wave
nature
of
the
electron
gives
it
 extremes
of
high
and
low
probability
at
 different
loca$on
in
the
box
 •  The
par$cle‐in‐a‐box
example
shows
how
a
wave
 func$on
operates
in
one
dimension.
 •  The
same
methods
used
for
the
one‐dimensional
box
 can
be
expanded
to
three
dimensions
for
atoms.
 •  Mathema$cally,
atomic
orbitals
are
discrete
solu$ons
 of
the
three‐dimensional
Schrodinger
equa$ons.
 Polar
Coordinates
 Ψ
consists
of
a
radial
component
R(r)
and
an
angular
component
A(θ,ϕ).
 The
hydrogen
atom
 •  The
solu$on
of
the
Schrodinger
equa$on
for
the
 hydrogen
atom
is
a
formidable
mathema$cal
problem.
 •  The
electron
in
the
hydrogen
atom
sees
a
spherically
 symmetric
poten$al.
The
poten$al
energy
is
simply
 that
of
a
point
charge.
 V(r)
=
 •  The
solu$on
is
managed
by
separa$ng
the
variables
so
 that
the
wave
func$on
is
represented
by
the
product.
 Quantum Numbers from Hydrogen Equations
 •  The
hydrogen
atom
solu$on
requires
finding
solu$ons
to
the
 separated
equa$ons
which
obey
the
constraints
on
the
 wavefunc$on.

 •  The
solu$on
to
the
radial
equa$on
can
exist
only
when
a
 constant
which
arises
in
the
solu$on
is
restricted
to
integer
 values.
This
gives
the
principal
quantum
number:
 Quantum
numbers
(summary)
 Quantum
numbers
and
their
proper$es
 •  the
quantum
number
n
(
1,
2,
3…)
 –  primarily
responsible
for
determining
the
overall
energy
of
an
atomic
orbital.
 •  The
quantum
number
l
(0,
1,
2,
…
n‐1)
 –  determines
the
angular
momentum
and
the
shape
of
an
orbital;

 –  has
a
small
effect
on
the
energy
 •  The
quantum
number
ml
(‐l
≤
ml
≤
l)
 –  determines
the
orienta$on
of
the
angular
momentum
vector
in
a
magne$c
 field;
or
the
posi$on
of
the
orbital
in
space.
 •  The
quantum
number
ms
(
+/‐
½)
 –  determines
the
orienta$on
of
the
electron’s
magne$c
moment
in
a
magne$c
 field,
either
in
the
direc$on
of
the
field
(+1/2)
or
opposed
to
it
(‐1/2).
 •  The
inner
quantum
number
j
(value
of
abs(l+s)
to
abs(l‐s))
 •  Total
angular
momentum,
spin‐orbit
coupling.
 When
no
magne$c
field
is
present,
all
ml
values
(all
 three
p
orbitals
and
all
five
d
orbitals)
have
the
same
 energy
(degenerate).

 Together,
the
quantum
numbers
n,
l,
ml
define
an
 atomic
orbital,
the
quantum
number
ms
describes
the
 electron
spin
within
the
orbital
 Hydrogen 1s Radial Probability
 The
Bohr
radius
a0
=
52.9
pm

 • 
It
is
the
value
of
r
at
the
 maximum
of
Ψ2
for
a
 hydrogen
1s
orbital
 • 
It
is
also
the
radius
of
a
1s
 orbital
according
to
the
Bohr
 model
 •  The
most
probable
radius
is
the
ground
state
radius
 obtained
from
the
Bohr
theory.

 •  The
Schrodinger
equa$on
confirms
the
first
Bohr
radius
as
 the
most
probable
radius
but
goes
further
to
describe
in
 detail
the
profile
of
probability
for
the
electron.
 Radial
probabili$es
 Boundary
surface
for
the
angular
part
 A(θ,ϕ)
 d‐Orbitals
 Schrodinger
Equa$on
and
Bohr’s
 model
of
the
hydrogen
atom
 •  Besides
providing
informa$on
about
the
 wavefunc$ons,
solu$ons
of
Schrodinger
 Equa$on
give
orbital
energies,
E.
 •  Shows
also
the
dependence
of
E
on
the
 principal
quantum
number
n
for
hydrogen‐like
 species
and
Z
the
atomic
number.
 E
=
‐kZ2/n2
with
k
=
1.312
x
103
kJ
mol‐1
 k
=
hcR
with
h=Planck’s
constant,
c:speed
of
light
and
R:
 Rydberg
constant
 For
each
value
of
n
there
is
only
one
energy
soluCon
;
all
atomic
 orbitals
with
the
same
n
are
degenerates
(valid
only
for
hydrogen‐like
 orbitals
H,
He+,
etc..)
 NOT
TRUE
FOR
MULTI‐ELECTRON
ATOM
 Shielding
effect
/
effec$ve
nuclear
charge
 •  The
effec$ve
nuclear
charge
is
the
net
posi$ve
 charge
experienced
by
an
electron
in
a
mul$‐ electron
atom.

 •  The
term
"effec$ve"
is
used
because
the
shielding
 effect
of
nega$vely
charged
electrons
prevents
 higher
orbital
electrons
from
experiencing
the
full
 nuclear
charge
by
the
repelling
effect
of
inner‐ layer
electrons.
 •  Slater’s
rule
to
es$mate
effec$ve
nuclear
charge
 experienced
by
electrons.
 Slater’s
Rules
 to
evaluate
the
shielding
effect
 Zeff
=
Z
–
S
 •  Electrons
are
grouped
as
follows:
(1s),
(2s,
2p),
(3s,
3p),
(3d),
(4s,
 4p),
(4d),
(4f),
(5s,
5p)
 •  Electrons
in
a
group
higher
than
electron
considered
contribute
 nothing
to
the
shielding
 •  For
an
electron
in
ns
or
np
orbital
each
of
the
other
electrons
in
(ns,
 np)
group
contributes
S
=
0.35;
each
of
the
electrons
in
n‐1
shell
 contributes
S
=
0.85;
each
of
the
electrons
in
shell
n‐2
or
lower
 contributes
S
=
1.00
 •  These
values
are
calculated
from
the
electron
probability
curves
of
 the
orbitals.
 •  For
an
electron
in
nd
or
nf
orbital,
each
of
the
electrons
in
the
same
 group
contributes
S
=
0.35;
each
of
the
electrons
in
a
lowergroup
 contribute
S
=
1.00
 Example
of
the
Slater’s
Rule
 •  For
K,
Z=19.
is
it
4s1
or
3d1??
 •  1s22s22p63s23p64s1
 –  Zeff
=
Z
–
S
=
19
–
(8*(0.85)
+
10*1)
=
2.20.
 •  
1s22s22p63s23p63d1
 –  Zeff
=
Z
–
S
=
19
–
(18*1)
=
1.
 Periodic
Table
of
Elements
 Electron
Configura$ons
of
Atoms
 •  Electrons
fill
energy
levels
star$ng
from
the
 sublevels
with
lowest
energy.
 




































1s
 




































2s



2p
 




































3s



3p


3d
 4s


4p


4d


4f
 5s


5p


5d


5f
 6s


6p


6d


6f
 7s


7p


7d


7f
 
8s


8p


8d



8f
 Aubfau
(building
up)
Principle
 •  Is
the
principle
of
building
up
electronic
ground
state
 configura$ons
used
in
conjunc$on
with
Hund’s
rules
and
 Pauli’s
exclusion
principle:
 –  electrons
are
placed
in
orbitals
to
give
the
lowest
total
energy
 to
the
atom
(lowest
values
of
n
and
l
filled
first).
Because
the
 orbitals
within
each
set
(p,
d…)
have
the
same
energy,
the
 orders
for
values
of
ml
and
ms
are
indeterminate.
 –  Pauli’s
exclusion
principle:
No
two
electrons
in
the
same
atom
 may
have
the
same
set
of
4
quantum
numbers
 –  Hund’s
rule:
In
a
set
of
degenerate
orbitals,
electron
may
not
be
 spin
paired
in
an
orbital
un$l
each
orbital
in
a
set
contains
one
 electron;
electrons
singly
occupying
orbitals
in
a
degenarate
set
 have
parallel
spins
 Examples
 •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  [He]2s22p2



 [Ne]3s1
 [Ar]4s23d2
 [Ar]4s13d5
 [Kr]5s24d2
 [Kr]5s04d10
 [Xe]4f146s25d7
 C





 Na
 Ti
 Cr
 Zr
 Pd
 Ir
 For
more
prac$ce
check
Table
1.3
 Hund’s
rule
explana$on
 •  Πc
:
Coulombic
energy
of
 repulsion
favors
electrons
in
 different
orbitals.
 •  Πe
:
Exchange
energy,
this
 energy
depends
on
the
 number
of
possible
 exchanges
between
two
 electrons
with
the
same
 energy
and
the
same
spin.

 Example
12C
1s22s22p2
 Exchange
Energy
 •  Exchange
energy
:
 –  number
of
possible
exchanges
between
two
electrons
 with
the
same
energy
and
the
same
spin.

 two
possible
ways
 only
one
possible
way
 The
higher
the
number
of
possible
exchanges,
the
lower
the
energy
 Total
Pairing
Energy,
Π

 •  Π
=
Πc
+
Πe
 –  Πc
is
posi$ve
and
is
nearly
constant
for
each
pair
of
 electrons.
 –  Πe
is
nega$ve
and
is
nearly
constant
for
each
pair
of
 electrons.
 •  If
there
is
a
difference
in
energy
between
the
 orbital
levels
involved,
this
difference
in
 combinaCon
with
the
total
pairing
energy
Π
 determines
the
final
configuraCon.
 Valence
and
core
electrons
 •  Valence
electrons
=
outer
electrons.
 •  Lower
energy
quantum
levels
electrons
=
core
 electrons.
 Trends
in
the
Periodic
Table
 •  Many
atomic
proper$es
correlate
with
 electronic
structure
and
so
also
with
their
 posi$on
in
the
periodic
table
 –  atomic
size
 –  ion
size
 –  ioniza$on
energy
 –  electron
affinity
 Atomic
size
 •  As
the
nuclear
charge
increases,
the
electrons
are
 pulled
in
toward
the
center
of
the
atom,
and
the
 size
of
any
par$cular
orbital
decreases.
 •  On
the
other
hand,
as
the
nuclear
charge
 increases,
more
electrons
are
added
to
the
atom,
 and
their
mutual
repulsion
keeps
the
outer
 orbitals
large.
 •  The
interac$on
of
the
two
effects
results
in
 gradual
decrease
in
atomic
size
across
a
period.
 Shielding
effect.
Slater
Rule.
 Varia$on
in
Size
of
Atoms
 Rela$ve
Size
of
Select
Ions
and
Their
 Parent
Atoms
 ...
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