Transcript orbital

Polar Coordinates
One way to give someone directions is to tell them to
go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a
half mile in that direction.”
Polar graphing is like the second method of giving
directions. Each point is determined by a distance and
an angle.
r

Initial ray
A polar coordinate pair
 r , 
determines the location of
a point.
Polar Coordinates
To define the Polar Coordinates of a plane we need first to fix a point
which will be called the Pole (or the origin) and a half-line starting
from the pole. This half-line is called the Polar Axis.
P(r, θ)
r
θ
Polar Axis
Polar Angles
The Polar Angle θ of a point P, P ≠ pole, is the
angle between the Polar Axis and the line
connecting the point P to the pole. Positive values
of the angle indicate angles measured in the
counterclockwise direction from the Polar Axis.
A positive angle.
Polar Coordinates
The Polar Coordinates (r,θ) of the point P,
P ≠ pole, consist of the distance r of the point P from the Pole and of
the Polar Angle θ of the point P. Every (0, θ) represents the pole.
More than one coordinate pair can refer to the same point.
 2,30 
o
2
210 o
30
o
150o
o
  2, 210

  2, 150
o

All of the polar coordinates of this point are:
o
o
2,30

n

360


o
o

2,

150

n

360


n  0,  1,  2 ...
The connection between
Polar and Cartesian coordinates
(x,y)
r
θ
x
y
From the right angle triangle in the picture
one immediately gets the following
correspondence between the Cartesian
Coordinates (x,y) and the Polar Coordinates
(r,θ) assuming the Pole of the Polar
Coordinates is the Origin of the Cartesian
Coordinates and the Polar Axis is the positive
x-axis.
x = r cos(θ)
r2 = x2 + y2
y = r sin(θ)
tan(θ) = y/x
Using these equations one can easily switch between the
Cartesian and the Polar Coordinates.
3) The hydrogen atom orbitals
Radial components of the hydrogen atom wavefunctions look as
follows:
R(r) =
R(r) =
R(r) =
Z
2 
 a0 
 Z 


na
 0
3/ 2
3/ 2
  Zr 

exp 
a
 0 

  Zr 
Zr 
 2   exp 

a
na
0 

 0 
 1  Z 



na
3

 0 
3/ 2
 Zr 
  Zr 
  exp 

a
na
 0
 0 
for n = 1, l = 0
1s orbital
for n = 2, l = 0
2s orbital
for n = 2, l = 1, ml = 0
2pz orbital
3/ 2
 Z   2 Zr 2 Z 2 r 2 
  Zr 





 n = 3, l = 0
2
1


exp
R(r) =  na   3a
2 
27
a
na
0
0 
 0 
 0 
3/ 2
  Zr 
8  Z   Zr Z 2 r 2 





 n = 3, l = 1, m = 0

exp
R(r) = 9 2  na   a
2 
l
6
a
na
0 
 0  0
 0 
3s orbital
3pz orbital
h2
Here a0 =
, the first orbit radius (0.529Å). Z is the nuclear charge (+1)
4 2 me 2
THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM
1) Theoretical description of the hydrogen atom:
Schrödinger equation:
H
HY = EY
 h2 
  2   
 
1  2  Ze 2

 2  sin   r

 sin 

8 m 
r  r   
  sin   2  r
H is the Hamiltonian operator (kinetic + potential energy of an electron)
E – the energy of an electron whose distribution in space is described by
Y - wave function
YY* is proportional to the probability of finding the electron in a given point of space
If Y is normalized, the following holds:
YY *dV  1

allspace
The part of space where the probability to find electron is non-zero is called orbital
When we plot orbitals, we usually show the smallest volume of space where this
probability is 90%.
:
2) Solution of the Schrödinger equation for the hydrogen atom.
Separation of variables r, , f
For the case of the hydrogen atom the Schrödinger equation can be solved
exactly when variables r, , f are separated like
Y = R(r) Q Ff
Three integer parameters appear in the solution,
n
(principal quantum number),
equation
l
(orbital quantum number),
equation
ml
(magnetic quantum number),
equation
n =
1, 2, … ∞; defines energy of an electron
while solving Rwhile solving Qwhile solving F-
4) Radial components of the hydrogen atom orbitals
The exponential decay of R(r) is slower for greater n
At some distances r where R(r) is equal to 0, we have radial nodes, n-l1 in total
1000*R(r)
•
•
6.00
5.00
4.00
1s
3.00
2s
Z
2 
 a0 
3/ 2
2.00
2p
  Zr 
 1.00
exp 
 a0 
 Z 


 na0 
3/ 2

  Zr 
Zr 
 2   exp 

a0 

 na0 
r
0.00
0
-1.00
node
1
2
3
4
5
6
7
8
5) Radial probability function
1s
14.00
12.00
10.00
8.00
6.00
1000*[r*R(r)]2
Radial probability function is defined as [r R(r)]2 – probability of finding an
electron in a spherical layer at a distance r from nucleus
2p
2s
4.00
r
2.00
0.00
0
2
4
6
8
6) Angular components
Angular components of the hydrogen
atom wave functions, QF
z
for l = 0, ml = 0;
s orbitals
QF =
 1 


 4 
1/ 2
z
for l = 1, ml = 0;
pz orbitals
QF =
 3 


 4 
1/ 2
cos 
4) Radial components of the hydrogen atom orbitals
The exponential decay of R(r) is slower for greater n
At some distances r where R(r) is equal to 0, we have radial nodes, n-l1 in total
1000*R(r)
•
•
6.00
5.00
1s
4.00
3.00
2s
Z
2 
 a0 
3/ 2
  Zr  2.00

exp 
 a0 
2p
 Z 


 na0 
3/ 2

  Zr 
Zr 
 2   exp 

a0 

 na0 
1.00
r
0.00
0
-1.00
node
1
2
3
4
5
6
7
8
5) Radial probability function
1s
14.00
12.00
10.00
8.00
6.00
1000*[r*R(r)]2
Radial probability function is defined as [r R(r)]2 – probability of finding an
electron in a spherical layer at a distance r from nucleus
2p
2s
4.00
r
2.00
0.00
0
2
4
6
8
Summary
• Exact energy and spatial distribution of an electron
in the hydrogen atom can be found by solving
Shrödinger equation;
• Three quantum numbers n, l, ml appear as integer
parameters while solving the equation;
• Each hydrogen orbital can be characterized by
energy (n), shape (l), spatial orientation (ml),
number of nodal surfaces (n-1)
Electrons are part of what
makes an atom an atom
atom
But where exactly are the
electrons inside an atom?
Orbitals
are areas within atoms
where there is a high probablility
of finding electrons.
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Knowing how electrons are
arranged in an atom is
important
because that governs
how atoms interact
with each other
Let’s say you have a
room with marbles in it
The marbles are not just
anywhere in the room.
They are inside boxes in
the room.
You know where the
boxes are, and you
know the marbles are
inside the boxes, but…
you don’t know exactly
where the marbles are
inside the boxes
The room is an atom
The marbles are electrons
The boxes are orbitals
The room is an atom
The marbles are electrons
The boxes are orbitals
The room is an atom
The marbles are electrons
The boxes are orbitals
Science has determined where the
orbitals are inside an atom, but it
is never known precisely where the
electrons are inside the orbitals
So what are the
sizes and shapes
of orbitals?
The area where an electron can be found,
the orbital,
is defined mathematically,
but we can see it as a specific shape
in 3-dimensional space…
z
y
x
z
y
The 3 axes represent
3-dimensional space
x
z
y
For this presentation, the
nucleus of the atom is at
the center of the three axes.
x
The “1s” orbital is a
sphere, centered
around the nucleus
The 2s orbital is also
a sphere.
The 2s electrons have a
higher energy than the 1s
electrons. Therefore, the 2s
electrons are generally more
distant from the nucleus,
making the 2s orbital larger
than the 1s orbital.
1s orbital
2s orbital
Don’t forget:
an orbital is the shape of the
space where there is a high
probability of finding electrons
Don’t forget:
an orbital is the shape of the
space where there is a high
probability of finding electrons
The s orbitals are spheres
There are three
2p orbitals
The three 2p orbitals
are oriented
perpendicular
to each other
z
This is
one 2p orbital
(2py)
y
x
z
another 2p orbital
(2px)
y
x
z
the third 2p orbital
(2pz)
y
x
Don’t forget:
an orbital is the shape of the
space where there is a high
probability of finding electrons
Don’t forget:
an orbital is the shape of the
space where there is a high
probability of finding electrons
This is the shape of p orbitals
z
y
x
z
2px
y
x
z
2px and 2pz
y
x
z
The three 2p orbitals,
2px, 2py, 2pz
y
x
once the
1s orbital
is filled,
the 2s orbital
begins to fill
once the 2s
orbital is
filled,
the 2p orbitals
begin to fill
each 2p orbital
intersects the
2s orbital and
the 1s orbital
each 2p orbital
gets one electron
before pairing begins
once each 2p orbital
is filled with a pair
of electrons, then
the 3s orbital
gets the next
two electrons
the 3s electrons
have a higher energy
than 1s, 2s, or 2p
electrons,
so 3s electrons are
generally found
further from the
nucleus than 1s,
2s, or 2p electrons
The order of filling orbitals
Electrons fill low energy orbitals (closer to the
nucleus) before they fill higher energy ones. Where
there is a choice between orbitals of equal energy,
they fill the orbitals singly as far as possible.
This filling of orbitals singly where possible is known
as Hund's rule. It only applies where the orbitals
have exactly the same energies (as with p orbitals,
for example), and helps to minimise the repulsions
between electrons and so makes the atom more
stable.
Observing the Effect of Electron Spin
Table 8.2 Summary of Quantum Numbers of Electrons in Atoms
Name
principal
angular
momentum
Symbol
n
l
Permitted Values
Property
positive integers(1,2,3,…)
orbital energy (size)
integers from 0 to n-1
orbital shape (The l values
0, 1, 2, and 3 correspond to
s, p, d, and f orbitals,
respectively.)
magnetic
ml
integers from -l to 0 to +l
orbital orientation
spin
ms
+1/2 or -1/2
direction of e- spin
Pauli Exclusion Principle
• no two e- in an atom can have the
same four quantum numbers.
• Result – an orbital can hold a
maximum of 2 electrons
Figure 8.7
Order for filling energy sublevels with
electrons
Illustrating Orbital Occupancies
The electron configuration
order of filling
#
of electrons in the sublevel
n l
as s,p,d,f
The orbital diagram (box or circle)
Electron Configurations
order of filling
7s 7p .. .. .. .. ..
6s 6p 6d 6f .. ..
5s 5p 5d 5f ..
4s 4p 4d 4f
3s 3p 3d
2s 2p
1s
no color-empty
A vertical
orbital diagram
for the Li
ground state
light - half-filled
dark - filled, spin-paired
Figure 8.9
Orbital occupancy for the first 10 elements, H through Ne.
Figure 8.13
The relation between orbital filling and the periodic table
Categories of electrons
Core (inner) electrons – all those shared by the previous
noble gas
Outer electrons – those in the highest occupied energy level
Similar chemical properties of elements in groups is a
result of similar outer electron configurations
In the main group, the group number equals the number
of outer electrons
Valence electrons – those involved in bonding
In the main group, the outer electrons are valence
In the transition metals can include some d electrons
Atomic radii of the maingroup and transition
elements.
Figure 8.16
Periodicity of atomic radius
Ionization energy
• energy required to remove an electron from 1 mole of
the element (to make it positive or more positive)
• always positive value i.e. endothermic
• First ionization energy IE1– to remove 1st e– Elements with small IE1 tend to form cations, large
IE1 anions
Ionization energy
• Trends:
– Generally as size decreases, IE1 increases.
– 1. IE1 generally increases from left to right, some exceptions
– 2. IE1 generally decreases going down in a group
– 3. transition and f-block elements have much smaller
variances in IE1
• But Why?
– Across - increasing Zeff and smaller size(e- closer to
nucleus)
– exceptions - Be to B because s shields p and lowers Zeff
– N to O because repulsions in first paired electron
– Down - Zeff constant and larger so easier to ionize
Figure 8.17
Periodicity of first ionization
energy (IE1)
Figure 8.18
First ionization
energies of the
main-group
elements
Figure 8.21
Trends in three atomic properties
Trends in metallic behavior
Main-group ions and the noble
gas configurations
Magnetic properties
• Spectral and magnetic properties of atoms
can be used to confirm electron
configurations.
• Paramagnetism
– attraction by an external magnetic field
– elements with unpaired electrons
– The more unpaired electrons there are, the
stronger the attraction
• Diamagnetism
– Not attracted by an external magnetic field
– elements with only paired electrons
Apparatus for measuring the magnetic behavior of a sample
SAMPLE PROBLEM 8.7
PROBLEM:
Use condensed electron configurations to write the reaction for the
formation of each transition metal ion, and predict whether the ion is
paramagnetic.
(a) Mn2+(Z = 25)
PLAN:
Writing Electron Configurations and Predicting
Magnetic Behavior of Transition Metal Ions
(b) Cr3+(Z = 24)
(c) Hg2+(Z = 80)
Write the electron configuration and remove electrons starting with
ns to match the charge on the ion. If the remaining configuration
has unpaired electrons, it is paramagnetic.
SOLUTION:
(a) Mn2+(Z = 25) Mn([Ar]4s23d5)
(b) Cr3+(Z = 24) Cr([Ar])
Mn2+ ([Ar] 3d5) + 2e-
Cr3+ ([Ar] ) + 3e-
(c) Hg2+(Z = 80) Hg([Xe]6s24f145d10)
paramagnetic
paramagnetic
Hg2+ ([Xe] 4f145d10) + 2enot paramagnetic (is diamagnetic)
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